Recent zbMATH articles in MSC 32https://zbmath.org/atom/cc/322023-09-22T14:21:46.120933ZWerkzeugRamanujan-type systems of nonlinear ODEs for \(\Gamma_0 (2)\) and \(\Gamma_0 (3)\)https://zbmath.org/1517.110322023-09-22T14:21:46.120933Z"Nikdelan, Younes"https://zbmath.org/authors/?q=ai:nikdelan.younesSummary: This paper aims to introduce two systems of nonlinear ordinary differential equations whose solution components generate the graded algebra of quasi-modular forms on Hecke congruence subgroups \(\Gamma_0 (2)\) and \(\Gamma_0 (3)\). Using these systems, we provide the generated graded algebras with an \(\mathfrak{sl}_2 (\mathbb{C})\)-module structure. As applications, we introduce Ramanujan-type tau functions for \(\Gamma_0 (2)\) and \(\Gamma_0 (3)\), and obtain some interesting and non-trivial recurrence and congruence relations.Appearance of the Kashiwara-Saito singularity in the representation theory of \(p\)-adic \(\mathrm{GL}(16)\)https://zbmath.org/1517.110522023-09-22T14:21:46.120933Z"Cunningham, Clifton"https://zbmath.org/authors/?q=ai:cunningham.clifton-l-r"Fiori, Andrew"https://zbmath.org/authors/?q=ai:fiori.andrew"Kitt, Nicole"https://zbmath.org/authors/?q=ai:kitt.nicoleIn the paper under the review, the authors calculate certain ABV-packets for the general linear groups, using a method which should be useful for both algorithmic implementation and symbolic calculation. It is known that A-packets for general linear groups are \(L\)-packets which are singletons. The authors prove that the ABV-packets for general linear groups do not have to be singletons. They obtain an unramified Langlands parameter for the group \(\mathrm{GL}_{16}(F)\), for a \(p\)-adic field \(F\), such that the corresponding ABV-packet contains exactly two representations.
A consequence of this result is the existence of nonsingleton ABV-packets of \(\mathrm{GL}_n(F)\) for all \(n \geq 16\). It is also expected that the obtained example for the group \(\mathrm{GL}_{16}(F)\) is the simplest example of an admissible representation of the general linear group with a corona.
Reviewer: Ivan Matić (Osijek)On the Milnor fibration of certain Newton degenerate functionshttps://zbmath.org/1517.140032023-09-22T14:21:46.120933Z"Eyral, Christophe"https://zbmath.org/authors/?q=ai:eyral.christophe"Oka, Mutsuo"https://zbmath.org/authors/?q=ai:oka.mutsuo.1Let \(f = f_1 \cdots f_k\) be the product of \(k\) polynomials. Assume that for any subset \(k_1, \dots, k_m \in \{1, \dots , k\}\) the analytic set \(\{f_{k_1} = \dots = f_{k_m} = 0\}\) is a nondegenerate complete intersection. The authors show that the diffeomorphism type of the Milnor fibration of a (possibly degenerate) polynomial function \(f\) is uniquely determined by the Newton boundaries of the polynomials \(f_1, \dots, f_k\). They also remark that this statement is a generalization of the results from [\textit{M. Oka}, J. Math. Soc. Japan 34, 541--549 (1982; Zbl 0476.32016); \textit{C. Eyral} and \textit{M. Oka}, J. Algebr. Geom. 31, No. 3, 561--591 (2022; Zbl 1492.14004)].
Reviewer's remark: It should be noted that a similar class of nonisolated singularities was studied by \textit{D. Bescheron-Lebrigand}, who constructed the corresponding Milnor fibrations using Whitney stratification technique and calculated all Milnor numbers for some types of such singularities (see [C. R. Acad. Sci., Paris, Sér. I 293, 421--424 (1981; Zbl 0496.32008)]).
Reviewer: Aleksandr G. Aleksandrov (Moskva)Derivations of local \(k\)-th Hessian algebras of singularitieshttps://zbmath.org/1517.140042023-09-22T14:21:46.120933Z"Hussain, Naveed"https://zbmath.org/authors/?q=ai:hussain.naveed"Yau, Stephen S.-T."https://zbmath.org/authors/?q=ai:yau.stephen-shing-toung"Zuo, Huaiqing"https://zbmath.org/authors/?q=ai:zuo.huaiqingLet \(f\colon (\mathbb C^n, 0) \rightarrow (\mathbb C, 0)\) be a holomorphic function germ determining the isolated hypersurface singularity \(V=f^{-1}(0)\) and \(J(f)\) the Jacobian ideal of \(f\). Let denote by \(h_k(f)\) the ideal of \(\mathcal O_{(\mathbb C^n, 0)}\) generated by all minors of order \(k\) of the Hessian matrix of \(f\). Then \(H_k(V) = \mathcal O_{(\mathbb C^n, 0)}/(f, J(f), h_k(f))\) is a finite dimensional \(\mathbb C\)-algebra for any \(0\leqslant k\leqslant n\) and the Lie algebra \(L_k(V) =\mathrm{Der}(H_k(V))\) of \(\mathbb C\)-derivations of \(H_k(V)\) is defined.
In the paper under review, the authors put forward a number of hypotheses regarding the dimension range of \(L_k(V)\), similarly to their previous works (see [\textit{N. Hussain} et al., Math. Z. 294, No. 1--2, 331--358 (2020; Zbl 1456.14005); Math. Z. 298, No. 3--4, 1813--1829 (2021; Zbl 1467.32014)], and so on). As an illustration, they calculate the corresponding dimensions for binomial, trinomial, and fewnomial singularities and obtain some estimates in terms of the weight type of \(f\).
Reviewer: Aleksandr G. Aleksandrov (Moskva)Strict Arakelov inequality for a family of varieties of general typehttps://zbmath.org/1517.140202023-09-22T14:21:46.120933Z"Lu, Xin"https://zbmath.org/authors/?q=ai:lu.xin.1"Yang, Jinbang"https://zbmath.org/authors/?q=ai:yang.jinbang"Zuo, Kang"https://zbmath.org/authors/?q=ai:zuo.kangSummary: Let \(f: X \to Y\) be a semistable non-isotrivial family of \(n\)-folds over a smooth projective curve with discriminant locus \(S \subseteq Y\) and with general fiber \(F\) of general type. We show the strict Arakelov inequality
\[
\frac{\deg f_\ast\omega_{X/Y}^\nu} {\operatorname{rank} f_\ast\omega_{X/Y}^\nu} < \frac{n\nu}{2}\cdot\deg\Omega^1_Y(\log S),
\]
for all \(\nu\in\mathbb{N}\) such that the \(\nu\)-th pluricanonical linear system \(|\omega^\nu_F|\) is birational. This answers a question asked by \textit{M. Möller} et al. [in: Global aspects of complex geometry. Berlin: Springer. 417--450 (2006; Zbl 1112.14027)].A Bochnak-Siciak theorem for Nash functions over real closed fieldshttps://zbmath.org/1517.140402023-09-22T14:21:46.120933Z"Kucharz, Wojciech"https://zbmath.org/authors/?q=ai:kucharz.wojciech"Kurdyka, Krzysztof"https://zbmath.org/authors/?q=ai:kurdyka.krzysztofSummary: Let \(R\) be a real closed field. We prove that if \(R\) is uncountable, then a function \(f: U \rightarrow R\) defined on an open semialgebraic set \(U\) in \(R^n\), with \(n \geq 2\), is a Nash function whenever for every affine 2-plane \(Q\) in \(R^n\) the restriction \(f |_{U \cap Q}\) is a Nash function (some condition on the shape of \(U\) is required if \(R\) is not Archimedean). This is an analog of the Bochnak-Siciak theorem established in the real analytic setting. We also provide an example showing that uncountability of \(R\) is essential.Polynomial inequalities, o-minimality and Denjoy-Carleman classeshttps://zbmath.org/1517.260122023-09-22T14:21:46.120933Z"Pierzchała, Rafał"https://zbmath.org/authors/?q=ai:pierzchala.rafalSummary: We study several intimately related problems in the theory of multivariate polynomial inequalities. Firstly, given a map \(h\) in certain quasianalytic Denjoy-Carleman classes, we show how to decide whether the image under \(h\) of a set satisfying Markov's (resp. Nikolskii's) inequality satisfies Markov's (resp. Nikolskii's) inequality. Our approach relies heavily on the theory of o-minimal structures, particularly on the work of Rolin, Speissegger and Wilkie. Secondly, we establish the relation between Markov's inequality and Nikolskii's inequality (both in general setting and in o-minimal setting). Thirdly, we prove that each compact, definable (in an o-minimal structure) set satisfying Markov's inequality is fat. In particular, this solves in the o-minimal category the well-known problem of nonpluripolarity of sets satisfying Markov's inequality. And lastly, we develop a unified method of polynomial approximation for various classes of smooth functions of several variables.Multipolar Hardy inequalities in \(L^p\)-spaceshttps://zbmath.org/1517.260162023-09-22T14:21:46.120933Z"Metoui, Imen"https://zbmath.org/authors/?q=ai:metoui.imenSummary: The aim of this paper is twofold. On the one hand, we establish with two different methods the new multipolar Hardy inequality
\[
\begin{aligned}
&\frac{(N-2)^2}{np^2}\int \limits_{{\mathbb{R}}^N}\sum_{i=1}^n\frac{|u|^p}{|x-a_i|^2}dx\\
&\qquad +\frac{(N-2)^2}{2p^2n^2}\int \limits_{{\mathbb{R}}^N}\sum_{\substack{i,j=1 \\ i\ne j}}^n\frac{|a_i-a_j|^2}{|x-a_i|^2|x-a_j|^2}|u|^pdx\\
&\quad \le \int \limits_{{\mathbb{R}}^N}|\nabla u|^2|u|^{p-2}dx
\end{aligned}
\] for every \(u \in H^{1,p}({\mathbb{R}}^N)\), \(p\ge 2, a_1, a_2,\ldots, a_n \in{\mathbb{R}}^N\), \(N\ge 3\), and \(n\ge 2\). On the other hand, we prove the weighted multipolar Hardy inequality
\[
\begin{aligned}
&\frac{(N+k_2-2)^2}{np^2}\int \limits_{{\mathbb{R}}^N} \sum_{i=1}^n\frac{|u|^p}{|x-a_i|^2}d\mu \\
&\qquad +\frac{(N+k_2-2)^2}{2n^2p^2}\int \limits_{{\mathbb{R}}^N}\sum_{\substack{i,j=1\\ i\ne j}}^n\frac{|a_i-a_j|^2}{|x-a_i|^2|x-a_j|^2}|u|^pd\mu \nonumber \\
&\quad \le \int \limits_{{\mathbb{R}}^N}|\nabla u|^2|u|^{p-2}d\mu +k_1\int \limits_{{\mathbb{R}}^N}|u|^pd\mu
\end{aligned}
\] for every \(u \in H^{1,p}({\mathbb{R}}^N,d\mu)\), \(d\mu =\mu (x)dx\), \(p\ge 2\), \(a_1, a_2,\ldots, a_n \in{\mathbb{R}}^N\), \(N\ge 3\), \(n\ge 1\), and some constants \(k_1, k_2 \in{\mathbb{R}}, k_2>2-N\). The weighted functions \(\mu\) are of a quite general type.Weil-Petersson geodesics on the modular surfacehttps://zbmath.org/1517.300072023-09-22T14:21:46.120933Z"Gadre, Vaibhav"https://zbmath.org/authors/?q=ai:gadre.vaibhav-sAuthor's abstract: We consider the Weil-Petersson (WP) metric on the modular surface. We lift WP geodesics to the universal cover of the modular surface, and analyse geometric properties of a lift as a path in the hyperbolic metric on the universal cover. For any pair of distinct points in the thick part of the universal cover, we prove that the WP and hyperbolic geodesic segments that connect the pair, fellow-travel in the thick part and all deviations between these segments arise during cusp excursions. Furthermore, we give a quantitative analysis of the deviation during an excursion. We leverage the fellow traveling to derive a correspondence between recurrent WP and hyperbolic rays from a base-point. We show that the correspondence can be promoted to a homeomorphism on the circle of directions. By analysing cuspidal winding of a typical WP geodesic ray, we show that the homeomorphism pushes forward a Lebesgue measure on the circle to a singular measure. In terms of continued fraction coefficients, the singularity boils down to a comparison that we prove, namely, the average coefficient is bounded along a typical WP ray but unbounded along a typical hyperbolic ray.
Reviewer: Yaşar Sözen (Ankara)McShane identities for higher Teichmüller theory and the Goncharov-Shen potentialhttps://zbmath.org/1517.300082023-09-22T14:21:46.120933Z"Huang, Yi"https://zbmath.org/authors/?q=ai:huang.yi.1"Sun, Zhe"https://zbmath.org/authors/?q=ai:sun.zheSummary: We derive generalizations of McShane's identity for higher ranked surface group representations by studying a family of mapping class group invariant functions introduced by Goncharov and Shen, which generalize the notion of horocycle lengths. In particular, we obtain McShane-type identities for finite-area cusped convex real projective surfaces by generalizing the Birman-Series geodesic scarcity theorem. More generally, we establish McShane-type identities for positive surface group representations with loxodromic boundary monodromy, as well as McShane-type inequalities for general rank positive representations with unipotent boundary monodromy. Our identities are systematically expressed in terms of projective invariants, and we study these invariants: we establish boundedness and Fuchsian rigidity results for triple and cross ratios. We apply our identities to derive the simple spectral discreteness of unipotent-bordered positive representations, collar lemmas, and generalizations of the Thurston metric.Estimates of the Bergman kernel on Teichmüller spacehttps://zbmath.org/1517.300092023-09-22T14:21:46.120933Z"Hu, Guangming"https://zbmath.org/authors/?q=ai:hu.guangming"Miyachi, Hideki"https://zbmath.org/authors/?q=ai:miyachi.hidekiSummary: In this paper, a comparison between the Bergman kernel form and the pushforward measure of the Masur-Veech measure on the Teichmüller space of genus \(g\geq 2\) is obtained.Fully non-linear elliptic equations on compact manifolds with a flat hyperkähler metrichttps://zbmath.org/1517.300102023-09-22T14:21:46.120933Z"Gentili, Giovanni"https://zbmath.org/authors/?q=ai:gentili.giovanni"Zhang, Jiaogen"https://zbmath.org/authors/?q=ai:zhang.jiaogenThe authors study a class of fully non-linear elliptic equations on certain compact hyperhermitian manifolds. By adapting the approach of Székelyhidi to the hypercomplex setting, they prove some a priori estimates for solutions to such equations under the assumption of existence of \(C\)-subsolutions. The main result of the paper is as follows. Let \((M, I, J, \mathrm{Keg})\) be a compact flat hyperkähler manifold, \(\Omega\) a \(q\)-real \((2,0)\)-form, and \(\underline{\varphi}\) a \(C\)-subsolution of
\[
F(A) = h,
\]
where \(h\in C^{\infty}(M, \mathbb{R})\) is given and \(F(A)=f(\lambda(A))\) is a smooth symmetric operator of the eigenvalues of \(A\). Then there exist \(\alpha \in (0,1)\) and a constant \(C>0\), depending only on \((M, I, J, K, g)\), \(\Omega\), \(h\) and \(\underline{\varphi}\), such that any \(\Gamma\)-admissible solution \(\varphi\) with \(\sup_M\varphi = 0\) satisfies the estimate
\[
|| \varphi ||_{C^{2,\alpha}} \leq C.
\]
The desired bound is obtained in two ways, by using an analogue of the Evans-Krylov theory as developed by Tosatti-Wang-Weinkove-Yang and by adapting the argument of Błocki similarly to what was done by Alesker for the treatment of the quaternionic Monge-Ampère equation.
Reviewer: Swanhild Bernstein (Freiberg)Global multiplicity, special closure and non-degeneracy of gradient mapshttps://zbmath.org/1517.320012023-09-22T14:21:46.120933Z"Bivià-Ausina, Carles"https://zbmath.org/authors/?q=ai:bivia-ausina.carles"Huarcaya, Jorge A. C."https://zbmath.org/authors/?q=ai:huarcaya.jorge-a-cSummary: Given a polynomial map \(F: \mathbb{C}^n\longrightarrow\mathbb{C}^p\) with finite zero set, \(p \geqslant n\), we introduce the notion of global multiplicity \(\mathrm{m}(F)\) associated to \(F\), which is analogous to the multiplicity of ideals in Noetherian local rings. This notion allows to characterize numerically the Newton non-degeneracy at infinity of \(F\). This fact motivates us to study a combinatorial inequality concerning the normalized volume of global Newton polyhedra and to characterize the corresponding equality using special closures. We also study the Newton non-degeneracy at infinity of gradient maps.Translation invariant linear spaces of polynomialshttps://zbmath.org/1517.320022023-09-22T14:21:46.120933Z"Kiss, Gergely"https://zbmath.org/authors/?q=ai:kiss.gergely"Laczkovich, Miklós"https://zbmath.org/authors/?q=ai:laczkovich.miklosSummary: A set of polynomials is called a \textit{submodule} of \(\mathbb{C} [x_1,\ldots, x_n]\) if it is a translation invariant linear subspace of \(\mathbb{C} [x_1,\ldots, x_n]\). We present a description of the submodules of \(\mathbb{C} [x,y]\) in terms of a special type of submodules. We say that submodule \(M\) of \(\mathbb{C} [x,y]\) is an \(L\)-\textit{module of order} \(s\) if, whenever \(F(x,y)=\sum_{n=0}^N f_n (x) \cdot y^n \in M\) is such that \(f_0 =\cdots = f_{s-1}=0\), then \(F=0\). We show that the proper submodules of \(\mathbb{C} [x,y]\) are the sums \(M_d +M\), where \(M_d =\{ F\in \mathbb{C} [x,y] \colon \deg_x F < d\}\), and \(M\) is an L-module. We give a construction of L-modules parametrized by sequences of complex numbers.
A submodule \(M\subset \mathbb{C} [x_1,\ldots, x_n]\) is \textit{decomposable} if it is the sum of finitely many proper submodules of \(M\). Otherwise \(M\) is \textit{indecomposable}. In \(\mathbb{C} [x,y]\) every indecomposable submodule is either an L-module or equals \(M_d\) for some \(d\). In the other direction we show that \(M_d\) is indecomposable for every \(d\), and so is every L-module of order \(1\).
Finally, we prove that there exists a submodule of \(\mathbb{C} [x,y]\) (in fact, an L-module of order \(1)\) which is not relatively closed in \(\mathbb{C} [x,y]\). This answers a problem posed by\textit L. Székelyhidi [``1. Problem'', in: Report of Meeting, The Forty-ninth International Symposium on Functional Equations, 2011, Graz-Mariatrost, Austria; Problems and remarks, Aequationes Math. 84 (2012), p. 307].A note on polydegree \((n, 1)\) rational inner functions, slice matrices, and singularitieshttps://zbmath.org/1517.320032023-09-22T14:21:46.120933Z"Sola, Alan"https://zbmath.org/authors/?q=ai:sola.alan-aSummary: We analyze certain compositions of rational inner functions in the unit polydisk \(\mathbb{D}^d\) with polydegree \((n, 1)\), \(n\in \mathbb{N}^{d-1}\), and isolated singularities in \(\mathbb{T}^d\). Provided an irreducibility condition is met, such a composition is shown to be a rational inner function with singularities in precisely the same location as those of the initial function, and with quantitatively controlled properties. As an application, we answer a \(d\)-dimensional version of a question posed in [\textit{K. Bickel} et al., Am. J. Math. 144, No. 4, 1115--1157 (2022; Zbl 1510.32007)] in the affirmative.The representation of holomorphic functions on the quasi-circular domain and the Bergman kernel function on the symmetrized ballhttps://zbmath.org/1517.320042023-09-22T14:21:46.120933Z"Zhong, Chengchen"https://zbmath.org/authors/?q=ai:zhong.chengchen"Pan, Lishuang"https://zbmath.org/authors/?q=ai:pan.lishuang"Wang, An"https://zbmath.org/authors/?q=ai:wang.anSummary: In this paper, we construct the relationship between the circular domain and quasi-circular domain by using standard mapping and standard inverse mapping in order to give the quasi-homogeneous representation of holomorphic functions on quasi-circular domain. By using the above result, we obtain the form of orthonormal basis on quasi-circular domain. Especially, we give the Bergman kernel function on symmetrized ball.Bergman projection on the symmetrized bidiskhttps://zbmath.org/1517.320052023-09-22T14:21:46.120933Z"Chen, Liwei"https://zbmath.org/authors/?q=ai:chen.liwei"Jin, Muzhi"https://zbmath.org/authors/?q=ai:jin.muzhi"Yuan, Yuan"https://zbmath.org/authors/?q=ai:yuan.yuan.1Summary: We apply the Bekollé-Bonami estimate for the (positive) Bergman projection on the weighted \(L^p\) spaces on the unit disk. As the consequences, we obtain the boundedness of the Bergman projection on the weighted Sobolev space on the symmetrized bidisk. We also improve the boundedness result of the Bergman projection on the unweighted \(L^p\) space on the symmetrized bidisk in [the first author et al.,
J. Funct. Anal. 279, No. 2, Article ID 108522 (2020; Zbl 1457.32003)].Irregularity of the Bergman projection on smooth unbounded worm domainshttps://zbmath.org/1517.320062023-09-22T14:21:46.120933Z"Krantz, Steven G."https://zbmath.org/authors/?q=ai:krantz.steven-george"Monguzzi, Alessandro"https://zbmath.org/authors/?q=ai:monguzzi.alessandro"Peloso, Marco M."https://zbmath.org/authors/?q=ai:peloso.marco-maria"Stoppato, Caterina"https://zbmath.org/authors/?q=ai:stoppato.caterinaSummary: In this work, we consider smooth unbounded worm domains \(\mathcal{Z}_\lambda\) in \(\mathbb{C}^2\) and show that the Bergman projection, densely defined on the Sobolev spaces \(H^{s,p}(\mathcal{Z}_\lambda)\), \(p\in(1, \infty)\), \(s \geq 0\), does not extend to a bounded operator \(P_\lambda: H^{s, p}(\mathcal{Z}_\lambda)\rightarrow H^{s, p}(\mathcal{Z}_\lambda)\) when \(s > 0\) or \(p \neq 2\). The same irregularity was known in the case of the non-smooth unbounded worm. This improved result shows that the irregularity of the projection is not a consequence of the irregularity of the boundary but instead of the infinite windings of the worm domain.Boundedness of Bergman projections acting on weighted mixed norm spaceshttps://zbmath.org/1517.320072023-09-22T14:21:46.120933Z"Savkovic, Ivana"https://zbmath.org/authors/?q=ai:savkovic.ivanaSummary: We prove that Bergman projections on weighted mixed norm spaces on smoothly bounded domains in \(\mathbb{R}^n\) are bounded for a certain range of parameters of such spaces and assuming certain conditions on weights. The proof relies on estimates of integral means of \(M_p(P_\gamma f, r)\) in terms of integral means of \(f\). This result complements earlier result on boundedness of \(P_\gamma\) on a closely related space \(L^{p, q}_\alpha(\Omega)\).Norm estimates for the \(\bar{\partial}\)-equation on a non-reduced spacehttps://zbmath.org/1517.320082023-09-22T14:21:46.120933Z"Andersson, Mats"https://zbmath.org/authors/?q=ai:andersson.mats"Lärkäng, Richard"https://zbmath.org/authors/?q=ai:larkang.richardSummary: We study norm-estimates for the \(\bar{\partial}\)-equation on non-reduced analytic spaces. Our main result is that on a non-reduced analytic space, which is Cohen-Macaulay and whose underlying reduced space is smooth, the \(\bar{\partial}\)-equation for (0, 1)-forms can be solved with \(L^p\)-estimates.Extension operators preserving biholomorphic mappings on Hartogs domainshttps://zbmath.org/1517.320092023-09-22T14:21:46.120933Z"Cui, Yanyan"https://zbmath.org/authors/?q=ai:cui.yanyan"Yang, Heju"https://zbmath.org/authors/?q=ai:yang.heju"Qiao, Yuying"https://zbmath.org/authors/?q=ai:qiao.yuyingSummary: In this paper, the authors extend the Roper-Suffridge operator on the generalized Hartogs domains. They mainly research the properties of the extended operator. By the characteristics of Hartogs domains and the geometric properties of subclasses of spirallike mappings, they obtain the extended Roper-Suffridge operator preserving almost starlikeness of complex order \(\lambda\), almost spirallikeness of type \(\beta\) and order \(\alpha\), parabolic spirallikeness of type \(\beta\) and order \(\rho\) on the Hartogs domains in different conditions. They conclude that the corresponding extension operator preserves the same geometric invariance on the unit ball \(B^n\) in \(\mathbb{C}^n\). The conclusions provide a new approach to study these geometric mappings in \(\mathbb{C}^n\).Clark measures and de Branges-Rovnyak spaces in several variableshttps://zbmath.org/1517.320102023-09-22T14:21:46.120933Z"Aleksandrov, Aleksei B."https://zbmath.org/authors/?q=ai:aleksandrov.aleksei-b"Doubtsov, Evgueni"https://zbmath.org/authors/?q=ai:doubtsov.evgueniSummary: Let \(B_n\) denote the unit ball of \(\mathbb{C}^n\), \(n \geq 1\), and let \(\mathcal{D}\) denote a finite product of \(B_{nj}\), \(j \geq 1\). Given a non-constant holomorphic function \(b: \mathcal{D} \to B_1\), we study the corresponding family \(\sigma_{\alpha} [b]\), \(\alpha \in \partial B_1\), of Clark measures on the distinguished boundary \(\partial \mathcal{D}\). We construct a natural unitary operator from the de Branges-Rovnyak space \(\mathcal{H}(b)\) onto the Hardy space \(H^2 (\sigma_{\alpha})\). As an application, for \(\mathcal{D}=B_n\) and an inner function \(I : B_n \to B_1\), we show that the property \(\sigma_1 [I] \ll \sigma_1 [b]\) is directly related to the membership of an appropriate explicit function in \(\mathcal{H}(b)\).Weighted \(L^2\) holomorphic functions on ball-fiber bundles over compact Kähler manifoldshttps://zbmath.org/1517.320112023-09-22T14:21:46.120933Z"Lee, Seungjae"https://zbmath.org/authors/?q=ai:lee.seungjae"Seo, Aeryeong"https://zbmath.org/authors/?q=ai:seo.aeryeongSummary: Let \(\widetilde{M}\) be a complex manifold, \(\Gamma\) be a torsion-free cocompact lattice of \(\mathrm{Aut}(\widetilde{M})\) and \(\rho :\Gamma \rightarrow SU(N,1)\) be a representation. Suppose that there exists a \(\rho\)-equivariant totally geodesic isometric holomorphic embedding \(\imath :\widetilde{M}\rightarrow\mathbb{B}^N\). Let \(M:=\widetilde{M}/\Gamma\) and \(\Sigma :=\mathbb{B}^N /\rho (\Gamma)\). In this paper, we investigate a relation between weighted \(L^2\) holomorphic functions on the fiber bundle \(\Omega :=M\times_{\rho} \mathbb{B}^N\) and the holomorphic sections of the pull-back bundle \(\imath^* (S^m T^*_{\Sigma})\) over \(M\). In particular, \(A^2_{\alpha} (\Omega)\) has infinite dimension for any \(\alpha >-1\) and if \(n<N\), then \(A^2_{-1}(\Omega)\) also has the same property. As an application, if \(\Gamma\) is a torsion-free cocompact lattice in \(SU (n, 1), n\geq 2\), and \(\rho :\Gamma \rightarrow SU(N,1)\) is a maximal representation, then for any \(\alpha >-1, A^2_{\alpha} (\mathbb{B}^n \times_{\rho} \mathbb{B}^N)\) has infinite dimension. If \(n<N\), then \(A_{-1}^2 (\mathbb{B}^n \times_{\rho} \mathbb{B}^N)\) also has the same property.Area functions characterizations of weighted Bergman spaces and area operatorshttps://zbmath.org/1517.320122023-09-22T14:21:46.120933Z"Wu, Huanqin"https://zbmath.org/authors/?q=ai:wu.huanqin"Wang, Maocai"https://zbmath.org/authors/?q=ai:wang.maocai"Dai, Guangming"https://zbmath.org/authors/?q=ai:dai.guangmingSummary: We completely characterize the boundedness and compactness of the area operators on weighted Bergman spaces \(A^p (\omega)\) over the unit ball induced by Békollé weights. Using the characterization of the boundedness, we obtain some general area formulas in terms of the radial derivative, the complex gradient, and the invariant gradient for \(A^p (\omega)\). As an application of area operator and the general area formula related to the radial derivative, we characterize the boundedness and compactness of Volterra integral operators.Lipschitz type characterization of Fock type spaceshttps://zbmath.org/1517.320132023-09-22T14:21:46.120933Z"Cho, Hong Rae"https://zbmath.org/authors/?q=ai:cho.hong-rae"Ha, Jeong Min"https://zbmath.org/authors/?q=ai:ha.jeongminSummary: For setting a general weight function on \(n\) dimensional complex space \(\mathbb{C}^n\), we expand the classical Fock space. We define Fock type space \(F^{p,q}_{\phi,t} (\mathbb{C}^n)\) of entire functions with a mixed norm, where \(0 < p, q < \infty\) and \(t \in \mathbb{R}\) and prove that the mixed norm of an entire function is equivalent to the mixed norm of its radial derivative on \(F^{p,q}_{\phi,t}(\mathbb{C}^n)\). As a result of this application, the space \(F^{p,p}_{\phi,t} (\mathbb{C}^n)\) is especially characterized by a
Lipschitz type condition.Forelli-Rudin type operators on the space \(L^{p,q,s}(B)\) and some applicationshttps://zbmath.org/1517.320142023-09-22T14:21:46.120933Z"Zhang, Xuejun"https://zbmath.org/authors/?q=ai:zhang.xuejun"Chen, Hongxin"https://zbmath.org/authors/?q=ai:chen.hongxin"Zhou, Min"https://zbmath.org/authors/?q=ai:zhou.minSummary: Let \(\phi\) be a holomorphic automorphism on the unit ball \(B\) of \(\mathbb{C}^n\), \(\lambda\) and \(\tau\) be two real numbers. In this paper, we investigate the conditions such that Forelli-Rudin type operators \(T_{\lambda, \tau} C_\varphi\) and \(| T |_{\lambda, \tau} C_\varphi\) are bounded on \(L^{p, q, s}(B)\) for \(p \geq 1\) or from \(\mathcal{H}^{p, q, s}(B)\) to \(L^{p, q, s}(B)\) for \(p > 0\). As some applications of \(| T |_{\lambda, \tau} C_\varphi\), we give several characterizations for functions in \(F^{p, q, s}(B)\), and prove that \(H^{p, q, s}(B)\) is an automorphism invariant space.Some remarks on approximation in several complex variableshttps://zbmath.org/1517.320152023-09-22T14:21:46.120933Z"Falcó, Javier"https://zbmath.org/authors/?q=ai:falco.javier"Gauthier, Paul M."https://zbmath.org/authors/?q=ai:gauthier.paul-m"Manolaki, Myrto"https://zbmath.org/authors/?q=ai:manolaki.myrto"Nestoridis, Vassili"https://zbmath.org/authors/?q=ai:nestoridis.vassiliSummary: In [Adv. Math. 381, Article ID 107649, 32 p. (2021; Zbl 1462.32002)], in order to correct a false Mergelyan-type statement given in
[\textit{T. W. Gamelin} and \textit{J. Garnett}, Trans. Am. Math. Soc. 143, 187--200 (1969; Zbl 0193.03501)] on uniform approximation on compact sets \(K\) in \(\mathbb{C}^d\), the authors introduced a natural function algebra \(A_D(K)\) which is smaller than the classical one \(A(K)\). In the present paper, we investigate when these two algebras coincide and compare them with the classes of all plausibly approximable functions by polynomials or rational functions or functions holomorphic on open sets containing the compact set \(K\). Finally, we introduce a notion of \(O\)-hull of \(K\) and strengthen known results.A polynomial approximation result for free Herglotz-Agler functionshttps://zbmath.org/1517.320162023-09-22T14:21:46.120933Z"Kojin, Kenta"https://zbmath.org/authors/?q=ai:kojin.kentaSummary: In this paper, we prove a noncommutative (nc) analog of Schwarz lemma for the nc Schur-Agler class and prove that the regular nc Schur-Agler class and the regular free Herglotz-Agler class are homeomorphic. Moreover, we give a characterization of regular free Herglotz-Agler functions. As an application, we will show that any regular free Herglotz-Agler functions can uniformly be approximated by regular Herglotz-Agler free polynomials.When the medial axis meets the singularitieshttps://zbmath.org/1517.320172023-09-22T14:21:46.120933Z"Denkowski, Maciej Piotr"https://zbmath.org/authors/?q=ai:denkowski.maciej-piotrSummary: In this survey we present recent results in the study of the \textit{medial axes} of sets definable in polynomially bounded o-minimal structures. We take the novel point of view of singularity theory. Indeed, it has been observed only recently that the medial axis -- i.e. the set of points with more than one closest point to a given closed set \(X\subset\mathbb{R}^n\) (with respect to the Euclidean
distance) -- reaches some singular points of \(X\) bringing along some metric information about them.
For the entire collection see [Zbl 1429.00039].Regular projections and regular covers in o-minimal structureshttps://zbmath.org/1517.320182023-09-22T14:21:46.120933Z"Oudrane, M'hammed"https://zbmath.org/authors/?q=ai:oudrane.mhammedSummary: We prove that for any definable subset \(X\subset \mathbb{R}^n\) in a polynomially bounded o-minimal structure, with \(\mathrm{dim}(X) < n\), there is a finite set of regular projections (in the sense of Mostowski). We also give a weak version of this theorem in any o-minimal structure, and we give a counterexample in o-minimal structures that are not polynomially bounded. As an application we show that in any o-minimal structure there exists a regular cover in the sense of Parusiński.Note on algebraic irregular Riemann-Hilbert correspondencehttps://zbmath.org/1517.320192023-09-22T14:21:46.120933Z"Ito, Yohei"https://zbmath.org/authors/?q=ai:ito.yoheiSummary: The subject of this paper is an algebraic version of the irregular Riemann-Hilbert correspondence which was mentioned in [the author, Tsukuba J. Math. 44, No. 1, 155--201 (2020; Zbl 1460.32009); corrigendum ibid. 46, No. 2, 271--275 (2022; Zbl 1511.32008)]. In particular, we prove an equivalence of categories between the triangulated category \(\mathbf{D}_{\mathrm{hol}}^{\mathrm{b}} (\mathscr{D}_X)\) of holonomic \(\mathscr{D}\)-modules on a smooth algebraic variety \(X\) over \(\mathbb{C}\) and the triangulated category \(\mathbf{E}^b_{\mathbb{C}-c} (\mathbf{I} \mathbb{C}_{X \infty})\) of algebraic \(\mathbb{C}\)-constructible enhanced ind-sheaves on a bordered space \(X_{\mathrm{\infty}}^{\mathrm{an}}\). Moreover, we show that there exists a t-structure on the triangulated category \(\mathbf{E}^b_{\mathbb{C}-c} (\mathbf{I} \mathbb{C}_{X \infty})\) whose heart is equivalent to the abelian category of holonomic \(\mathscr{D}\)-modules on \(X\). Furthermore, we shall consider simple objects of its heart and minimal extensions of objects of its heart.Concavity property of minimal \(L^2\) integrals with Lebesgue measurable gain V-fibrations over open Riemann surfaceshttps://zbmath.org/1517.320202023-09-22T14:21:46.120933Z"Bao, Shijie"https://zbmath.org/authors/?q=ai:bao.shijie"Guan, Qi'an"https://zbmath.org/authors/?q=ai:guan.qian"Yuan, Zheng"https://zbmath.org/authors/?q=ai:yuan.zhengSummary: In this article, we present characterizations of the concavity property of minimal \(L^2\) integrals degenerating to linearity in the case of fibrations over open Riemann surfaces. As applications, we obtain characterizations of the holding of equality in optimal jets \(L^2\) extension problem from fibers over analytic subsets to fibrations over open Riemann surfaces, which implies characterizations of the fibration versions of the equality parts of Suita conjecture and extended Suita conjecture.On sharper estimates of Ohsawa-Takegoshi \(L^2\)-extension theorem in higher dimensional casehttps://zbmath.org/1517.320212023-09-22T14:21:46.120933Z"Kikuchi, Shota"https://zbmath.org/authors/?q=ai:kikuchi.shotaSummary: \textit{G. Hosono} [J. Math. Soc. Japan 71, No. 3, 909--914 (2019; Zbl 1439.32026)] obtained sharper estimates of the Ohsawa-Takegoshi \(L^2\)-extension theorem by allowing the constant depending on the weight function for a domain in \(\mathbb{C}\). In this article, we show the higher dimensional case of sharper estimates of the Ohsawa-Takegoshi \(L^2\)-extension theorem. To prove the higher dimensional case of them, we establish an analogue of Berndtsson-Lempert type \(L^2\)-extension theorem by using the pluricomplex Green functions with poles along subvarieties. As a special case, we consider the sharper estimates in terms of the Azukawa pseudometric and show that the higher dimensional case of sharper estimate provides the \(L^2\)-minimum extension for radial case.Construction of the holomorphic and bounded function in a holomorphic domain of \(\mathbb{C}^2\)https://zbmath.org/1517.320222023-09-22T14:21:46.120933Z"Kato, Kazuko"https://zbmath.org/authors/?q=ai:kato.kazukoSummary: We construct the bounded solution of the second problem of Cousin in the analytic polyedra and in the domain of holomorphy of \(\mathbb{C}^2\).Holomorphic convexity of pseudoconvex surfaceshttps://zbmath.org/1517.320232023-09-22T14:21:46.120933Z"Vîjîitu, Viorel"https://zbmath.org/authors/?q=ai:vijiitu.viorelSummary: We prove that an irreducible complex surface \(X\) is holomorphically convex provided that there exists a plurisubharmonic exhaustion function \(\varphi :X\rightarrow [-\infty,\infty)\), not necessary continuous, and \(\mathcal{O}(X) \neq \mathbb{C}\).On stable sampling and interpolation in Bernstein spaceshttps://zbmath.org/1517.320242023-09-22T14:21:46.120933Z"López Nicolás, José Alfonso"https://zbmath.org/authors/?q=ai:lopez-nicolas.jose-alfonsoSummary: We define the concepts of stable sampling set, interpolation set, uniqueness set and complete interpolation set for a quasinormed space of functions and apply these concepts to Paley-Wiener spaces and Bernstein spaces. We obtain a sufficient condition on a uniformly discrete set to be an interpolation set based on a lemma of convergence of series in Paley-Wiener spaces. We also obtain a result of transference, Kadec type, of the property of being a stable sampling set, from a set with this property to other uniformly discrete set, which we apply to Bernstein spaces.Measure bound for translation surfaces with short saddle connectionshttps://zbmath.org/1517.320252023-09-22T14:21:46.120933Z"Dozier, Benjamin"https://zbmath.org/authors/?q=ai:dozier.benjaminSummary: We prove that any ergodic \(SL_2(\mathbb{R})\)-invariant probability measure on a stratum of translation surfaces satisfies strong regularity: the measure of the set of surfaces with two non-parallel saddle connections of length at most \(\varepsilon_1,\varepsilon_2\) is \(O(\varepsilon_1^2\cdot\varepsilon_2^2)\). We prove a more general theorem which works for any number of short saddle connections. The proof uses the multi-scale compactification of strata recently introduced by \textit{M. Bainbridge} et al. [``The moduli space of multi-scale differentials'', Preprint, \url{arXiv:1910.13492}] and the algebraicity result of \textit{S. Filip} [Ann. Math. (2) 183, No. 2, 681--713 (2016; Zbl 1342.14015)].On limit sets for geodesics of meromorphic connectionshttps://zbmath.org/1517.320262023-09-22T14:21:46.120933Z"Novikov, Dmitry"https://zbmath.org/authors/?q=ai:novikov.dmitry"Shapiro, Boris"https://zbmath.org/authors/?q=ai:shapiro.boris-zalmanovich"Tahar, Guillaume"https://zbmath.org/authors/?q=ai:tahar.guillaumeSummary: Meromorphic connections on Riemann surfaces originate and are closely related to the classical theory of linear ordinary differential equations with meromorphic coefficients. Limiting behavior of geodesics of such connections has been studied by, e.g., \textit{M. Abate} and \textit{F. Bianchi} [Math. Z. 282, No. 1--2, 247--272 (2016; Zbl 1332.32021)] and \textit{M. Abate} and \textit{F. Tovena} [J. Differ. Equations 251, No. 9, 2612--2684 (2011; Zbl 1241.32012)]
in relation with generalized Poincaré-Bendixson theorems. At present, it seems still to be unknown whether some of the theoretically possible asymptotic behaviors of such geodesics really exist. In order to fill the gap, we use the branched affine structure induced by a Fuchsian meromorphic connection to present several examples with geodesics having infinitely many self-intersections and quite peculiar \(\omega\)-limit sets.Rigid manifolds of general type with non-contractible universal coverhttps://zbmath.org/1517.320272023-09-22T14:21:46.120933Z"Frapporti, Davide"https://zbmath.org/authors/?q=ai:frapporti.davide"Gleissner, Christian"https://zbmath.org/authors/?q=ai:gleissner.christianSummary: For each \(n\ge 3\) we give examples of infinitesimally rigid projective manifolds of general type of dimension \(n\) with non-contractible universal cover. We provide examples with projective and examples with non-projective universal cover.Semi-polarized meromorphic Hitchin and Calabi-Yau integrable systemshttps://zbmath.org/1517.320282023-09-22T14:21:46.120933Z"Lee, Jia Choon"https://zbmath.org/authors/?q=ai:lee.jia-choon"Lee, Sukjoo"https://zbmath.org/authors/?q=ai:lee.sukjooSummary: It was shown by Diaconescu, Donagi, and Pantev that Hitchin systems of type ADE are isomorphic to certain Calabi-Yau integrable systems. In this paper, we prove an analogous result in the setting of meromorphic Hitchin systems of type A, which are known to be Poisson integrable systems. We consider a symplectization of the meromorphic Hitchin integrable system, which is a semi-polarized integrable system in the sense of Kontsevich and Soibelman. On the Hitchin side, we show that the moduli space of unordered diagonally framed Higgs bundles forms an integrable system in this sense and recovers the meromorphic Hitchin system as the fiberwise compact quotient. Then we construct a family of quasi-projective Calabi-Yau three-folds and show that its relative intermediate Jacobian fibration, as a semi-polarized integrable system, is isomorphic to the moduli space of unordered diagonally framed Higgs bundles.Riemann moduli spaces are quantum ergodichttps://zbmath.org/1517.320292023-09-22T14:21:46.120933Z"Baskin, Dean"https://zbmath.org/authors/?q=ai:baskin.dean"Gell-Redman, Jesse"https://zbmath.org/authors/?q=ai:gell-redman.jesse"Han, Xiaolong"https://zbmath.org/authors/?q=ai:han.xiaolongIn this paper, quantum ergodicity is shown for Riemann moduli spaces with Weil-Petersson metric under some technical structural and analytic assumptions. Similar results are also obtained for some other singular spaces with ergodic geodesic flow.
Reviewer: Thomas B. Ward (Durham)Counting arcs on hyperbolic surfaceshttps://zbmath.org/1517.320302023-09-22T14:21:46.120933Z"Bell, Nick"https://zbmath.org/authors/?q=ai:bell.nickSummary: We give the asymptotic growth of the number of arcs of bounded length between boundary components on hyperbolic surfaces with boundary. Specifically, if \(S\) has genus \(g, n\) boundary components and \(p\) punctures, then the number of orthogeodesic arcs in each pure mapping class group orbit of length at most \(L\) is asymptotic to \(L^{6g - 6+2 (n+p)}\) times a constant. We prove an analogous result for arcs between cusps, where we define the length of such an arc to be the length of the sub-arc obtained by removing certain cuspidal regions from the surface.Isomonodromic deformations: confluence, reduction and quantisationhttps://zbmath.org/1517.320312023-09-22T14:21:46.120933Z"Gaiur, Ilia"https://zbmath.org/authors/?q=ai:gaiur.ilia-yu"Mazzocco, Marta"https://zbmath.org/authors/?q=ai:mazzocco.marta"Rubtsov, Vladimir"https://zbmath.org/authors/?q=ai:rubtsov.vladimir-nSummary: In this paper we study the isomonodromic deformations of systems of differential equations with poles of any order on the Riemann sphere as Hamiltonian flows on the product of co-adjoint orbits of the truncated current algebra, also called generalised Takiff algebra. Our motivation is to produce confluent versions of the celebrated Knizhnik-Zamolodchikov equations and explain how their quasiclassical solution can be expressed via the isomonodromic \(\tau \)-function. In order to achieve this, we study the confluence cascade of \(r+ 1\) simple poles to give rise to a singularity of arbitrary Poincaré rank \(r\) as a Poisson morphism and explicitly compute the isomonodromic Hamiltonians.Proper holomorphic maps in Euclidean spaces avoiding unbounded convex setshttps://zbmath.org/1517.320322023-09-22T14:21:46.120933Z"Drinovec Drnovšek, Barbara"https://zbmath.org/authors/?q=ai:drnovsek.barbara-drinovec"Forstnerič, Franc"https://zbmath.org/authors/?q=ai:forstneric.francSummary: We show that if \(E\) is a closed convex set in \(\mathbb{C}^n\) (\(n>1\)) contained in a closed halfspace \(H\) such that \(E\cap bH\) is nonempty and bounded, then the concave domain \(\Omega = \mathbb{C}^n \setminus E\) contains images of proper holomorphic maps \(f:X\rightarrow\mathbb{C}^n\) from any Stein manifold \(X\) of dimension \(<n\), with approximation of a given map on closed compact subsets of \(X\). If in addition \(2\dim X+1\leq n\) then \(f\) can be chosen an embedding, and if \(2\dim X=n\), then it can be chosen an immersion. Under a stronger condition on \(E\), we also obtain the interpolation property for such maps on closed complex subvarieties.Big Picard theorems and algebraic hyperbolicity for varieties admitting a variation of Hodge structureshttps://zbmath.org/1517.320332023-09-22T14:21:46.120933Z"Deng, Ya"https://zbmath.org/authors/?q=ai:deng.ya|deng.ya.1Summary: In this paper, we study various hyperbolicity properties for a quasi-compact Kähler manifold \(U\) which admits a complex polarized variation of Hodge structures so that each fiber of the period map is zero-dimensional. In the first part, we prove that \(U\) is algebraically hyperbolic and that the generalized big Picard theorem holds for \(U\). In the second part, we prove that there is a finite étale cover \(\overline{U}\) of \(U\) from a quasi-projective manifold \(\overline{U}\) such that any projective compactification \(X\) of \(\overline{U}\) is Picard hyperbolic modulo the boundary \(X-\overline{U}\), and any irreducible subvariety of \(X\) not contained in \(X-\overline{U}\) is of general type. This result coarsely incorporates previous works by Nadel, Rousseau, Brunebarbe and Cadorel on the hyperbolicity of compactifications of quotients of bounded symmetric domains by torsion-free lattices.On meromorphic solutions of nonlinear partial differential-difference equations of first order in several complex variableshttps://zbmath.org/1517.320342023-09-22T14:21:46.120933Z"Cheng, Qibin"https://zbmath.org/authors/?q=ai:cheng.qibin"Li, Yezhou"https://zbmath.org/authors/?q=ai:li.yezhou"Liu, Zhixue"https://zbmath.org/authors/?q=ai:liu.zhixueSummary: This paper is concerned with the value distribution for meromorphic solutions \(f\) of a class of nonlinear partial differential-difference equation of first order with small coefficients. We show that such solutions \(f\) are uniquely determined by the poles of \(f\) and the zeros of \(f-c\), \(f-d\) (counting multiplicities) for two distinct small functions \(c\), \(d\).Proper holomorphic mappings of quasi-balanced domains in \(\mathbb{C}^3\)https://zbmath.org/1517.320352023-09-22T14:21:46.120933Z"Haridas, Pranav"https://zbmath.org/authors/?q=ai:haridas.pranav"Janardhanan, Jaikrishnan"https://zbmath.org/authors/?q=ai:janardhanan.jaikrishnanSummary: The main goal of this paper is to study Alexander-type theorems in the case of smoothly bounded quasi-balanced domains of finite type in \(\mathbb{C}^3\). Explicitly, the result that is proved in the paper is that the proper holomorphic self-maps of such domains are necessarily automorphisms.Lacunary series, resonances, and automorphisms of \({\mathbb{C}^2}\) with a round Siegel domainhttps://zbmath.org/1517.320362023-09-22T14:21:46.120933Z"Berteloot, François"https://zbmath.org/authors/?q=ai:berteloot.francois"Cheraghi, Davoud"https://zbmath.org/authors/?q=ai:cheraghi.davoudSummary: We construct transcendental automorphims of \({\mathbb{C}^2}\) that have an unbounded and regular Siegel domain.A lower bound for the Hausdorff dimension of the Green currenthttps://zbmath.org/1517.320372023-09-22T14:21:46.120933Z"de Thélin, Henry"https://zbmath.org/authors/?q=ai:de-thelin.henryLet \(f\) be a birational self-map of a compact Kähler surface \(X\), and \(T^\pm\) be the Green currents of \(f\) and \(f^{-1}\). If \(f\) satisfies the so-called Bedford-Diller condition, then the measure \(\mu:=T^+\wedge T^-\) is well-defined and is hyperbolic with one positive and one negative Lyapunov exponent. The main result of this paper (Theorem 1) shows that if the first dynamical degree of \(f\) is greater than one then the Hausdorff dimension of the support of the Green current \(T^+\) is greater than two.
Theorem 1 follows from a more general result (Theorem 2) which is true in any dimension, while Theorem 2 is a consequence of Theorem 3 which is proven using Pesin's theory and Oseledets' theorem.
Reviewer: Feng Rong (Shanghai)Periodic points of weakly post-critically finite all the way down mapshttps://zbmath.org/1517.320382023-09-22T14:21:46.120933Z"Van Tu Le"https://zbmath.org/authors/?q=ai:van-tu-le.1Summary: We study eigenvalues along periodic cycles of post-critically finite endomorphisms of \(\mathbb{CP}^n\) in higher dimension. It is a classical result when \(n = 1\) that those values are either 0 or of modulus strictly bigger than 1. It has been conjectured in [the author, Ergodic Theory Dyn. Syst. 42, No. 7, 2382--2414 (2022; Zbl 1495.32052)] that the same result holds for every \(n \ge 2\). In this article, we verify the conjecture for the class of weakly post-critically finite all the way down maps which was introduced in [\textit{M. Astorg}, Ergodic Theory Dyn. Syst. 40, No. 2, 289--308 (2020; Zbl 1437.37059)]. This class contains a well-known class of post-critically finite maps constructed in [\textit{S. Koch}, Adv. Math. 248, 573--617 (2013; Zbl 1310.32016)]. As a consequence, we verify the conjecture for Koch maps.A survey on rational curves on complex surfaceshttps://zbmath.org/1517.320392023-09-22T14:21:46.120933Z"Barbaro, Giuseppe"https://zbmath.org/authors/?q=ai:barbaro.giuseppe"Fagioli, Filippo"https://zbmath.org/authors/?q=ai:fagioli.filippo"Ríos Ortiz, Ángel David"https://zbmath.org/authors/?q=ai:rios-ortiz.angel-davidSummary: In this survey, we discuss the problem of the existence of rational curves on complex surfaces, both in the Kähler and non-Kähler setup. We systematically go through the Enriques-Kodaira classification of complex surfaces to highlight the different approaches applied to the study of rational curves in each class. We also provide several examples and point out some open problems.Uniqueness of birational structures on Inoue surfaceshttps://zbmath.org/1517.320402023-09-22T14:21:46.120933Z"Zhao, Shengyuan"https://zbmath.org/authors/?q=ai:zhao.shengyuanSummary: We prove that the natural \((\mathrm{Aff}_2(\mathbf{C}),\mathbf{C}^2)\)-structure on an Inoue surface is the unique \((\mathrm{Bir}(\mathbb{P}^2),\mathbb{P}^2(\mathbf{C}))\)-structure, generalizing a result of Bruno Klingler which asserts that the natural \((\mathrm{Aff}_2(\mathbf{C}),\mathbf{C}^2)\)-structure is the unique \((\mathrm{PGL}_3(\mathbf{C}),\mathbb{P}^2(\mathbf{C}))\)-structure.Regularity of the equilibrium measure for meromorphic correspondenceshttps://zbmath.org/1517.320412023-09-22T14:21:46.120933Z"Dinh, Tien-Cuong"https://zbmath.org/authors/?q=ai:tien-cuong-dinh."Wu, Hao"https://zbmath.org/authors/?q=ai:wu.hao.8Summary: Let \(f\) be a meromorphic correspondence on a compact Kähler manifold \(X\) of dimension \(k\). Assume that its topological degree is larger than the dynamical degree of order \(k-1\). We obtain a quantitative regularity of the equilibrium measure of \(f\) in terms of its super-potentials.Compact complex \(p\)-Kähler hyperbolic manifoldshttps://zbmath.org/1517.320422023-09-22T14:21:46.120933Z"Haggui, Fathi"https://zbmath.org/authors/?q=ai:haggui.fathi"Marouani, Samir"https://zbmath.org/authors/?q=ai:marouani.samirSummary: In this paper, we generalize tow new notions of hyperbolicity introduced recently in [\textit{S. Marouani} and \textit{D. Popovici}, ``Balanced Hyperbolic and Divisorially Hyperbolic Compact Complex Manifolds'', Preprint, \url{arXiv:2107.08972}] and we give some properties of this new notions.Adjoint \((1,1)\)-classes on threefoldshttps://zbmath.org/1517.320432023-09-22T14:21:46.120933Z"Höring, A."https://zbmath.org/authors/?q=ai:horing.andreasThe article under review studies the analytic analogue of the basepoint-freeness theorem: it is known that if \(X\) is a projective variety (with mild singularities) and \(D\) a divisor such that \(D-K_X\) is nef and big, then the linear system \(|mD|\) is basepoint-free if \(m\gg 1\). In particular, the latter gives rise to a morphism \(\Phi_{mD}:X\to Z\) with connected fibres and (the numerical class of) \(D\) can be written \(D=\Phi^*(A)\) where \(A\) is an ample \(\mathbb{Q}\)-divisor on \(Z\). If \(D\) is replaced with a \((1,1\))-class, it makes perfectly sense to consider the following conjecture [\textit{S. Filip} and \textit{V. Tosatti}, Ann. Inst. Fourier 68, No. 7, 2981--2999 (2018; Zbl 1428.14065)]:
\textbf{Conjecture.} If \(\alpha\) is a nef \((1,1)\)-class on a compact Kähler manifold \(X\) such that \(\alpha-K_X\) is big and nef, then there exists \(\Phi:X\to Z\) (with connected fibres and \(Z\) normal) such that \(\alpha=\Phi^*(\alpha_Z)\) with \(\alpha_Z\) a Kähler class on \(Z\).
It was proved in [loc. cit.] that it holds in dimension \(2\) and this article addresses the case \(\dim(X)=3\). The above conjecture holds if \(\alpha-K_X\) is assumed to be Kähler (and not only nef and big). Actually, the precise statement (Theorem 1.3) is about compact Kähler threefolds with terminal singularities.
The canonical bundle playing a central role in the satement, it is always interesting to look at the special case of \(K\)-trivial spaces. In that case, the above conjecture states that a nef and big class is automatically semi-ample (and thus the pull-back of a Kähler class by a bimeromorphic map). Proposition 1.4 settles it for terminal threefolds. In that case, the Decomposition Theorem is used to reduce the problem to the surface case.
The proof (of Theorem 1.3) is obtained by running a MMP for Kähler threefold [the author and \textit{T. Peternell}, Invent. Math. 203, No. 1, 217--264 (2016; Zbl 1337.32031)]: it ends with a meromorphic map \(X\dashrightarrow Y\), the space \(Y\) being endowed with a nef and big class \(\alpha_Y\). This class satisfies \(\dim(\mathrm{Null}(\alpha))\le 1\) and the null-locus of \(\alpha_Y\) can then be contracted to obtain the sought \(Z\).
Reviewer: Benoît Claudon (Rennes)Lelong numbers of currents of full mass intersectionhttps://zbmath.org/1517.320442023-09-22T14:21:46.120933Z"Vu, Duc-Viet"https://zbmath.org/authors/?q=ai:vu.duc-vietSummary: We study Lelong numbers of currents of full mass intersection on a compact Kähler manifold in a mixed setting. Our main theorems cover some recent results due to \textit{T. Darvas} et al. [Anal. PDE 11, No. 8, 2049--2087 (2018; Zbl 1396.32011); Compos. Math. 154, No. 2, 380--409 (2018; Zbl 1398.32042)]. The key ingredient in our approach is a new notion of products of pseudoeffective \((1,1)\)-classes which captures some ``pluripolar part'' of the ``total intersection'' of given pseudoeffective \((1,1)\)-classes.Fixed points and normal automorphisms of the unit ball of bounded operators on \(\mathbb{C}^n\)https://zbmath.org/1517.320452023-09-22T14:21:46.120933Z"Aggarwal, Rachna"https://zbmath.org/authors/?q=ai:aggarwal.rachna"Gongopadhyay, Krishnendu"https://zbmath.org/authors/?q=ai:gongopadhyay.krishnendu"Mishra, Mukund Madhav"https://zbmath.org/authors/?q=ai:mishra.mukund-madhavSummary: We examine the group of isometries of the open unit ball of a complex Banach space of certain bounded linear operators equipped with the Carathéodory metric. Therein we obtain a characterization of the normal isometries in terms of their special type of fixed points.Operator valued analogues of multidimensional Bohr's inequalityhttps://zbmath.org/1517.320462023-09-22T14:21:46.120933Z"Allu, Vasudevarao"https://zbmath.org/authors/?q=ai:allu.vasudevarao"Halder, Himadri"https://zbmath.org/authors/?q=ai:halder.himadriSummary: Let \(\mathcal{B}(\mathcal{H})\) be the algebra of all bounded linear operators on a complex Hilbert space \(\mathcal{H}\). In this paper, we first establish several sharp improved and refined versions of the Bohr's inequality for the functions in the class \(H^{\infty}(\mathbb{D},\mathcal{B}(\mathcal{H}))\) of bounded analytic functions from the unit disk \(\mathbb{D}:=\{ z\in \mathbb{C}:|z|<1\}\) into \(\mathcal{B}(\mathcal{H})\). For the complete circular domain \(Q \subset \mathbb{C}^n\), we prove the multidimensional analogues of the operator valued Bohr-type inequality which can be viewed as a special case of the result by \textit{G. Popescu} [Adv. Math. 347, 1002--1053 (2019; Zbl 07044308)] for free holomorphic functions on polyballs. Finally, we establish the multidimensional analogues of several improved Bohr's inequalities for operator valued functions in \(Q\).Fekete-Szegö problems for spirallike mappings and close-to-quasi-convex mappings on the unit ball of a complex Banach spacehttps://zbmath.org/1517.320472023-09-22T14:21:46.120933Z"Hamada, Hidetaka"https://zbmath.org/authors/?q=ai:hamada.hidetakaSummary: In the first part of this paper, we will give the Fekete-Szegö inequality for various subfamilies of spirallike mappings of type \(\beta\) on the unit ball of a complex Banach space. Our results give extensions of those given by \textit{Y. Lai} and \textit{Q. Xu} [Result. Math. 76, No. 4, Paper No. 191, (2021; Zbl 1478.32011)] and Elin and Jacobzon (Results Math 77(3), Paper No. 137, 2022). We next give the Fekete-Szegö inequality for close-to-quasi-convex mappings of type \(B\) on the unit ball of a complex Banach space. Our results give extensions of that given by \textit{Q. Xu} et al. [Complex Var. Elliptic Equ. 68, No. 1, 67-80 (2023; Zbl 1505.30022)].Revisit of multi-dimensional Bohr radiushttps://zbmath.org/1517.320482023-09-22T14:21:46.120933Z"Kumar, Shankey"https://zbmath.org/authors/?q=ai:kumar.shankey"Manna, Ramesh"https://zbmath.org/authors/?q=ai:manna.rameshSummary: In this article, we obtain a new lower bound of the Bohr radius of the unit ball in \(\ell_p^n\) space, for \(1 \leq p < \infty\) and \(n \in \mathbb{N}\), \(n \geq 2\). Moreover, we observe that this new lower bound is sharper than previously all known lower bounds.The refined Schwarz-Pick estimates for positive real part holomorphic functions in several complex variableshttps://zbmath.org/1517.320492023-09-22T14:21:46.120933Z"Liu, Xiaosong"https://zbmath.org/authors/?q=ai:liu.xiaosongSummary: In this article, the refined Schwarz-Pick estimates for positive real part holomorphic functions
\[
p(x)=p(0)+\sum\limits_{m=k}^{\infty} \dfrac{D^m p(0)(x^m)}{m!} :G\rightarrow \mathbb{C}
\]
are given, where \(k\) is a positive integer, and \(G\) is a balanced domain in complex Banach spaces. In particular, the results of first order Fréchet derivative for the above functions and higher order Fréchet derivatives for positive real part holomorphic functions
\[
p(x)=p(0)+\sum\limits_{s=1}^{\infty} \dfrac{D^{sk}p(0)(x^{sk})}{(sk)!} :G\rightarrow\mathbb{C}
\]
are sharp for \(G = B\), where \(B\) is the unit ball of complex Banach spaces or the unit ball of complex Hilbert spaces. Their results reduce to the classical result in one complex variable, and generalize some known results in several complex variables.Bounds of all terms of homogeneous expansions for a subclass of \(g\)-parametric biholomorphic mappings in \(\mathbb{C}^n\)https://zbmath.org/1517.320502023-09-22T14:21:46.120933Z"Xiong, Liangpeng"https://zbmath.org/authors/?q=ai:xiong.liangpeng"Sima, Xiaoying"https://zbmath.org/authors/?q=ai:sima.xiaoying"Ouyang, Dongling"https://zbmath.org/authors/?q=ai:ouyang.donglingSummary: Let \(\mathbb{B_X}\) be the unit ball in a complex Banach space \(\mathbb{X}\) and \(\mathbb{D}^n\) be the unit polydisc in the space of \(n\)-dimensional complex variables. We obtain the bounds of all terms of homogeneous expansions for a subclass of \(g\)-parametric biholomorphic mappings on \(\mathbb{B_X}\) (resp. \(\mathbb{D}^n\)), where \(g\) is a convex function. Our results generalize some known works and can be regarded as an extended example to hold the weak version of the Bieberbach conjecture in several complex variables.Loewner chains applied to \(g\)-starlike mappings of complex order of complex Banach spaceshttps://zbmath.org/1517.320512023-09-22T14:21:46.120933Z"Zhang, Xiaofei"https://zbmath.org/authors/?q=ai:zhang.xiaofei"Feng, Shuxia"https://zbmath.org/authors/?q=ai:feng.shuxia"Liu, Taishun"https://zbmath.org/authors/?q=ai:liu.taishun"Wang, Jianfei"https://zbmath.org/authors/?q=ai:wang.jianfeiSummary: This paper is devoted to studying geometric and analytic properties of \(g\)-starlike mappings of complex order \(\lambda \). By using Loewner chains, we obtain the growth theorems for \(g\)-starlike mappings of complex order \(\lambda\) on the unit ball in reflexive complex Banach spaces, which generalize some results of Graham, Hamada and Kohr. As applications, several different kinds of distortion theorems for \(g\)-starlike mappings of complex order \(\lambda\) are obtained. Finally, we prove that the Roper-Suffridge extension operators preserve the property of \(g\)-starlike mappings of complex order \(\lambda\) in complex Banach spaces, which generalizes many classical results.On the dimensions of vector spaces concerning holomorphic vector bundles over elliptic orbitshttps://zbmath.org/1517.320522023-09-22T14:21:46.120933Z"Boumuki, Nobutaka"https://zbmath.org/authors/?q=ai:boumuki.nobutakaSummary: In this paper, we study the complex vector space of holomorphic cross-sections of a homogeneous holomorphic vector bundle over an elliptic adjoint orbit, and give a sufficient condition for the vector space to be finite-dimensional.Kodaira dimension and zeros of holomorphic one-forms, revisitedhttps://zbmath.org/1517.320532023-09-22T14:21:46.120933Z"Villadsen, Mads Bach"https://zbmath.org/authors/?q=ai:villadsen.mads-bachSummary: We give a new proof that every holomorphic one-form on a smooth complex projective variety of general type must vanish at some point, first proven by \textit{M. Popa} and \textit{C. Schnell} [Ann. Math. (2) 179, No. 3, 1109--1120 (2014; Zbl 1297.14011)] using generic vanishing theorems for Hodge modules. Our proof relies on \textit{C. T. Simpson}'s results [Publ. Math., Inst. Hautes Étud. Sci. 75, 5--95 (1992; Zbl 0814.32003);
Ann. Sci. Éc. Norm. Supér. (4) 26, No. 3, 361--401 (1993; Zbl 0798.14005)] on the relation between rank one Higgs bundles and local systems of one-dimensional complex vector spaces, and the structure of the cohomology jump loci in their moduli spaces.Sharp pointwise and uniform estimates for \(\bar{\partial} \)https://zbmath.org/1517.320542023-09-22T14:21:46.120933Z"Dong, Robert Xin"https://zbmath.org/authors/?q=ai:dong.robert-xin"Li, Song-Ying"https://zbmath.org/authors/?q=ai:li.songying"Treuer, John N."https://zbmath.org/authors/?q=ai:treuer.john-nSummary: We use weighted \(L^2\)-methods to obtain sharp pointwise estimates for the canonical solution to the equation \( \bar \partial u=f\) on smoothly bounded strictly convex domains and the Cartan classical domains when \(f\) is bounded in the Bergman metric \(g\). We provide examples to show our pointwise estimates are sharp. In particular, we show that on the Cartan classical domains \(\Omega\) of rank \(2\) the maximum blow-up order is greater than \(-\log \delta_\Omega(z)\), which was obtained for the unit ball case by Berndtsson. For example, for \( \Omega\) of type \(\operatorname{IV}(n)\) with \(n \geq 3\), the maximum blow-up order is \(\delta(z)^{1 -n/2}\) because of the contribution of the Bergman kernel. Additionally, we obtain uniform estimates for the canonical solutions on the polydiscs, strictly pseudoconvex domains and the Cartan classical domains under stronger conditions on \(f\).Integrable generators of Lie algebras of vector fields on \(\mathrm{SL}_2 (\mathbb{C})\) and on \(xy = z^2\)https://zbmath.org/1517.320552023-09-22T14:21:46.120933Z"Andrist, Rafael B."https://zbmath.org/authors/?q=ai:andrist.rafael-bSummary: For the special linear group \(\mathrm{SL}_2 (\mathbb{C})\) and for the singular quadratic Danielewski surface \(x y = z^2\) we give explicitly a finite number of complete polynomial vector fields that generate the Lie algebra of all polynomial vector fields on them. Moreover, we give three unipotent one-parameter subgroups that generate a subgroup of algebraic automorphisms acting infinitely transitively on \(x y = z^2\).Directed harmonic currents near non-hyperbolic linearizable singularitieshttps://zbmath.org/1517.320562023-09-22T14:21:46.120933Z"Chen, Zhangchi"https://zbmath.org/authors/?q=ai:chen.zhangchiSummary: Let \((\mathbb{D}^2,\mathscr{F},\{0\})\) be a singular holomorphic foliation on the unit bidisc \(\mathbb{D}^2\) defined by the linear vector field
\[
z\frac{\partial}{\partial z}+\lambda w\frac{\partial}{\partial w},
\]
where \(\lambda\in\mathbb{C}^*\). Such a foliation has a non-degenerate singularity at the origin \(0:=(0,0)\in\mathbb{C}^2\). Let \(T\) be a harmonic current directed by \(\mathscr{F}\) which does not give mass to any of the two separatrices \((z=0)\) and \((w=0)\). Assume \(T\neq 0\). The Lelong number of \(T\) at \(0\) describes the mass distribution on the foliated space. \textit{Viêt-Anh Nguyên} [Ergodic Theory Dyn. Syst. 38, No. 8, 3170--3187 (2018; Zbl 1401.37056)] proved that when \(\lambda\notin\mathbb{R}\), that is, when \(0\) is a hyperbolic singularity, the Lelong number at \(0\) vanishes. Suppose the trivial extension \(\tilde{T}\) across \(0\) is \(dd^c\)-closed. For the non-hyperbolic case \(\lambda\in\mathbb{R}^*\), we prove that the Lelong number at \(0\):
\begin{itemize}
\item[(1)] is strictly positive if \(\lambda>0\);
\item[(2)] vanishes if \(\lambda\in\mathbb{Q}_{<0}\);
\item[(3)] vanishes if \(\lambda<0\) and \(T\) is invariant under the action of some cofinite subgroup of the monodromy group.
\end{itemize}Global pluripotential theory on hybrid spaceshttps://zbmath.org/1517.320572023-09-22T14:21:46.120933Z"Pille-Schneider, Léonard"https://zbmath.org/authors/?q=ai:pille-schneider.leonardSummary: Let \((X,L)\) be a polarized scheme over a Banach ring \(A\). We define and study a class \(\mathrm{PSH}(X,L)\) of plurisubharmonic metrics on the Berkovich analytification \(X^{\mathrm{an}}\). We focus mainly on the case where \(A\) is a hybrid ring of power series, so that \(X^{\mathrm{an}}\) is the hybrid space associated to a degeneration of complex manifolds \(X\). We then prove that any plurisubharmonic metric on \((X,L)\) with logarithmic growth at zero admits a canonical plurisubharmonic extension to the hybrid space \(X^{\mathrm{hyb}}\). We also discuss the continuity of the family of Monge-Ampère measures associated to a continuous plurisubharmonic hybrid metric.The space of finite-energy metrics over a degeneration of complex manifoldshttps://zbmath.org/1517.320582023-09-22T14:21:46.120933Z"Reboulet, Rémi"https://zbmath.org/authors/?q=ai:reboulet.remiSummary: Given a degeneration of projective complex manifolds \(X\rightarrow\mathbb{D}^*\) with meromorphic singularities, and a relatively ample line bundle \(L\) on \(X\), we study spaces of plurisubharmonic metrics on \(L\), with particular focus on (relative) finite-energy conditions. We endow the space \(\widehat{\mathcal{E}}^1(L)\) of relatively maximal, relative finite-energy metrics with a \(d_1\)-type distance given by the Lelong number at zero of the collection of fiberwise Darvas \(d_1\)-distances. We show that this metric structure is complete and geodesic. Seeing \(X\) and \(L\) as schemes \(X_{\mathrm{K}}, L_{\mathrm{K}}\) over the discretely-valued field \(\mathrm{K}=\mathbb{C}(\!(t)\!)\) of complex Laurent series, we show that the space \(\mathcal{E}^1(L_{\mathrm{K}}^{\mathrm{an}})\) of non-Archimedean finite-energy metrics over \(L_{\mathrm{K}}^{\mathrm{an}}\) embeds isometrically and geodesically into \(\widehat{\mathcal{E}}^1(L)\), and characterize its image. This generalizes previous work of Berman-Boucksom-Jonsson, treating the trivially-valued case.A criterion for biholomorphicity of self-mappings of generalized Fock-Bargmann-Hartogs domainshttps://zbmath.org/1517.320592023-09-22T14:21:46.120933Z"Kodama, Akio"https://zbmath.org/authors/?q=ai:kodama.akioSummary: By making use of our previous result on a localization principle for biholomorphic mappings between equidimensional Fock-Bargmann-Hartogs domains in \({\mathbb{C}^n} \times{\mathbb{C}^m}\) with \(m \ge 2\) [Hiroshima Math. J. 48, No. 2, 171--187 (2018; Zbl 1401.32002)] and the same technique as in [loc. cit], in this paper we establish a characterization of biholomorphicity of holomorphic self-mappings of generalized Fock-Bargmann-Hartogs domains. As a special case of this, we obtain the main result of a recent paper by \textit{T. Guo} et al. [Czech. Math. J. 71, No. 2, 373--386 (2021; Zbl 07361074)].Automorphisms of hyper-Reinhardt free spectrahedrahttps://zbmath.org/1517.320602023-09-22T14:21:46.120933Z"McCullough, Scott"https://zbmath.org/authors/?q=ai:mccullough.scott-aSummary: The free automorphisms of a class of Reinhardt free spectrahedra are trivial.
For the entire collection see [Zbl 1517.47001].On the Kähler hyperbolicity with respect to the Bergman metric on a class of Hartogs domainshttps://zbmath.org/1517.320612023-09-22T14:21:46.120933Z"Wang, An"https://zbmath.org/authors/?q=ai:wang.an"Zhong, Cheng Chen"https://zbmath.org/authors/?q=ai:zhong.chengchen"Liu, Ya Nan"https://zbmath.org/authors/?q=ai:liu.yananSummary: In this paper, we consider the Kähler hyperbolicity with respect to the Bergman metric on a class of non-compact Kähler manifolds using the \(d\)-boundedness of the Bergman metric. Also, we give an \(L^2\) cohomology vanishing theorem using Gromov's criterion.On the rigidity of proper holomorphic self-mappings of the Hua domainshttps://zbmath.org/1517.320622023-09-22T14:21:46.120933Z"Wang, Lei"https://zbmath.org/authors/?q=ai:wang.lei.6Summary: Hua domain, named after Chinese mathematician Loo-Keng Hua, is defined as a domain in \(\mathbb{C}^n\) fibered over an irreducible bounded symmetric domain \(\Omega \subset\mathbb{C}^d\) with the fiber over \(z\in \Omega\) being a \((n-d)\)-dimensional generalized complex ellipsoid \(\Sigma (z)\). \textit{Z. Tu} and the author [Math. Ann. 363, No. 1--2, 1--34 (2015; Zbl 1330.32007)] obtained the rigidity result that proper holomorphic mappings between two equidimensional Hua domains are biholomorphisms when the sets consisting of boundary points of Hua domains which are not strongly pseudoconvex have complex codimension at least 2. In this article, we find a counter-example to show that the rigidity result is not true for Hua domains without this condition and obtain the rigidity of proper holomorphic self-mappings of the Hua domains in this case.Products of Toeplitz and Hankel operators on the Bergman spaces of generalized Hartogs triangleshttps://zbmath.org/1517.320632023-09-22T14:21:46.120933Z"Zhang, Shuo"https://zbmath.org/authors/?q=ai:zhang.shuo|zhang.shuo.1Summary: The generalized Hartogs triangles \(H_{\{k_i\},\gamma}^n\) are nonsmooth Reinhardt domains defined by \[H_{\{k_{i}\},\gamma}^n : = \Big\{ z = (\tilde{z}_1, \ldots, \tilde{z}_m, z_n) \in\mathbb{C}^n : \mathop{\max}_{1\leq i\leq m} \|\tilde{z}_i\|<|z_n| ^\gamma<1\Big\}, \] where \((\tilde{z}_1,\ldots, \tilde{z}_m) = (z_1, \ldots, z_{n-1})\) with \(\tilde{z}_i \in \mathbb{C}^{k_i}\), \(i=1, \ldots, m\) and \(i=k_1+\cdots+k_m=n-1\). We find some ``good'' holomorphic automorphisms of \(H_{\{k_{i}\},\gamma}^n\) and use them to obtain necessary conditions for the Toeplitz and Hankel products to be bounded on the Bergman space of \(H_{\{k_{i}\},\gamma}^n\).The Schwarz lemma in Kähler and non-Kähler geometryhttps://zbmath.org/1517.320642023-09-22T14:21:46.120933Z"Broder, Kyle"https://zbmath.org/authors/?q=ai:broder.kyleSummary: We introduce a new curvature constraint that provides an analog of the real bisectional curvature considered by \textit{X. Yang} and \textit{F. Zheng} [Trans. Am. Math. Soc. 371, No. 4, 2703--2718 (2019; Zbl 1417.32027)] for the Aubin-Yau inequality. A unified perspective of the various forms of the Schwarz lemma is given, leading to novel Schwarz-type inequalities in both the Kähler and Hermitian categories.A hall of statistical mirrorshttps://zbmath.org/1517.320652023-09-22T14:21:46.120933Z"Khan, Gabriel"https://zbmath.org/authors/?q=ai:khan.gabriel-j-h"Zhang, Jun"https://zbmath.org/authors/?q=ai:zhang.jun.4Summary: The primary objects of study in information geometry are \textit{statistical manifolds}, which are parametrized families of probability measures, induced with the Fisher-Rao metric and a pair of torsion-free conjugate connections. In recent work [Differ. Geom. Appl. 73, Article ID 101678 (2020; Zbl 1478.53022)], the authors considered parametrized probability distributions as \textit{partially-flat statistical manifolds admitting torsion} and showed that there is a complex-to-symplectic duality on the tangent bundles of such manifolds, based on the dualistic geometry of the underlying manifold.
In this paper, we explore this correspondence further in the context of Hessian manifolds, in which case the conjugate connections are both curvature- and torsion-free, and the associated dual pair of spaces are Kähler manifolds. We focus on several key examples and their geometric features. In particular, we show that the moduli space of univariate normal distributions gives rise to a correspondence between a Siegel domain and the Siegel-Jacobi space, which are spaces that appear in the context of automorphic forms.On holomorphic sectional curvature of Kähler quotientshttps://zbmath.org/1517.320662023-09-22T14:21:46.120933Z"Manikandan, S."https://zbmath.org/authors/?q=ai:manikandan.sreenath-k|manikandan.sreekanth-kSummary: In this article, we compute the holomorphic sectional curvature of non-singular Kähler quotients. As a corollary, we show that the holomorphic sectional curvature of the moduli space of finite-dimensional complex representations of a finite quiver, which are stable with respect to a fixed rational weight and have a fixed type, is non-negative.On partial uniqueness of complete non-compact Ricci flat metricshttps://zbmath.org/1517.320672023-09-22T14:21:46.120933Z"Wang, Yuanqi"https://zbmath.org/authors/?q=ai:wang.yuanqiSummary: Using techniques for Caccioppoli inequality, on a fairly general class of complete non-compact Kähler manifolds with sub-quadratic volume growth, we show uniqueness of bounded \(C^{1,1}\) solution to Monge-Ampere equation. This does not a priori require any decay of the solution.The method of potential rescaling: overview and the localizationhttps://zbmath.org/1517.320682023-09-22T14:21:46.120933Z"Choi, Young-Jun"https://zbmath.org/authors/?q=ai:choi.young-jun"Lee, Kang-Hyurk"https://zbmath.org/authors/?q=ai:lee.kang-hyurkSummary: In this paper, we introduce the method of potential rescaling to generates a 1-parameter family of automorphisms on negatively curved complete Kähler-Einstein manifolds. We will deal with the convergence of potential rescaling under a localization assumption.Globally conformally Kähler Einstein metrics on certain holomorphic bundleshttps://zbmath.org/1517.320692023-09-22T14:21:46.120933Z"Feng, Zhiming"https://zbmath.org/authors/?q=ai:feng.zhimingSummary: The subject of this paper is the explicit momentum construction of complete Einstein metrics by ODE methods. Using the Calabi ansatz, further generalized by Hwang-Singer, we show that there are non-trivial complete conformally Kähler-Einstein metrics on certain Hermitian holomorphic vector bundles and their subbundles over complete Kähler-Einstein manifolds. In special cases, we give the explicit expressions of these metrics. These examples show that there are a compact Kähler manifold \(M\) and its subvariety \(N\) whose codimension is greater than 1 such that there is a complete conformally Kähler-Einstein metric on \(M-N\).Collapsing Calabi-Yau fibrations and uniform diameter boundshttps://zbmath.org/1517.320702023-09-22T14:21:46.120933Z"Li, Yang"https://zbmath.org/authors/?q=ai:li.yang|li.yang.4|li.yang.6|li.yang.11|li.yang.2|li.yang.12|li.yang.17|li.yang.5|li.yang.7|li.yang.8|li.yang.9|li.yang.13Summary: We study Calabi-Yau metrics collapsing along a holomorphic fibration over a Riemann surface. Assuming at worst canonical singular fibres, we prove a uniform diameter bound for all fibres in the suitable rescaling. This has consequences on the geometry around the singular fibres.Moduli theory, stability of fibrations and optimal symplectic connectionshttps://zbmath.org/1517.320712023-09-22T14:21:46.120933Z"Dervan, Ruadhaí"https://zbmath.org/authors/?q=ai:dervan.ruadhai"Sektnan, Lars Martin"https://zbmath.org/authors/?q=ai:sektnan.lars-martinSummary: K-polystability is, on the one hand, conjecturally equivalent to the existence of certain canonical Kähler metrics on polarised varieties, and, on the other hand, conjecturally gives the correct notion to form moduli. We introduce a notion of stability for families of K-polystable varieties, extending the classical notion of slope stability of a bundle, viewed as a family of K-polystable varieties via the associated projectivisation. We conjecture that this is the correct condition for forming moduli of fibrations.
Our main result relates this stability condition to Kähler geometry: we prove that the existence of an optimal symplectic connection implies semistability of the fibration. An optimal symplectic connection is a choice of fibrewise constant scalar curvature Kähler metric, satisfying a certain geometric partial differential equation. We conjecture that the existence of such a connection is equivalent to polystability of the fibration. We prove a finite-dimensional analogue of this conjecture, by describing a GIT problem for fibrations embedded in a fixed projective space, and showing that GIT polystability is equivalent to the existence of a zero of a certain moment map.Relative Ding and K-stability of toric Fano manifolds in low dimensionshttps://zbmath.org/1517.320722023-09-22T14:21:46.120933Z"Nitta, Yasufumi"https://zbmath.org/authors/?q=ai:nitta.yasufumi"Saito, Shunsuke"https://zbmath.org/authors/?q=ai:saito.shunsuke"Yotsutani, Naoto"https://zbmath.org/authors/?q=ai:yotsutani.naotoSummary: The purpose of this article is to clarify all of the uniformly relatively Ding stable toric Fano threefolds and fourfolds as well as unstable ones. The key player in our classification result is the Mabuchi constants, which can be calculated by combinatorial data of the associated moment polytopes due to the work of \textit{Y. Yao} [Int. Math. Res. Not. 2022, No. 24, 19790--19853 (2022; Zbl 1510.53084)]. In this article, we give the list of uniform relative Ding stability of all toric Fano manifolds in dimension up to four with the values of the Mabuchi constants. As an application of our main theorem, we clarify the difference between relative \(K\)-stability and relative Ding stability by considering some specific toric Fano manifolds. In the proof, we used Bott tower structure of relatively Ding unstable toric Fano manifolds.Limits of an increasing sequence of complex manifoldshttps://zbmath.org/1517.320732023-09-22T14:21:46.120933Z"Balakumar, G. P."https://zbmath.org/authors/?q=ai:balakumar.g-p"Borah, Diganta"https://zbmath.org/authors/?q=ai:borah.diganta"Mahajan, Prachi"https://zbmath.org/authors/?q=ai:mahajan.prachi"Verma, Kaushal"https://zbmath.org/authors/?q=ai:verma.kaushalSummary: Let \(M\) be a complex manifold which admits an exhaustion by open subsets \(M_j\) each of which is biholomorphic to a fixed domain \(\Omega \subset \mathbb{C}^n\). The main question addressed here is to describe \(M\) in terms of \(\Omega\). Building on work of Fornaess-Sibony, we study two cases, namely \(M\) is Kobayashi hyperbolic and the other being the corank one case in which the Kobayashi metric degenerates along one direction. When \(M\) is Kobayashi hyperbolic, its complete description is obtained when \(\Omega\) is one of the following domains -- (i) a smoothly bounded Levi corank one domain, (ii) a smoothly bounded convex domain, (iii) a strongly pseudoconvex polyhedral domain in \(\mathbb{C}^2\), or (iv) a simply connected domain in \(\mathbb{C}^2\) with generic piecewise smooth Levi-flat boundary. With additional hypotheses, the case when \(\Omega\) is the minimal ball or the symmetrized polydisc in \(\mathbb{C}^n\) can also be handled. When the Kobayashi metric on \(M\) has corank one and \(\Omega\) is either of (i), (ii) or (iii) listed above, it is shown that \(M\) is biholomorphic to a locally trivial fibre bundle with fibre \(\mathbb{C}\) over a holomorphic retract of \(\Omega\) or that of a limiting domain associated with it. Finally, when \(\Omega = \Delta \times \mathbb{B}^{n-1}\), the product of the unit disc \(\Delta \subset \mathbb{C}\) and the unit ball \(\mathbb{B}^{n-1} \subset \mathbb{C}^{n-1}\), a complete description of holomorphic retracts is obtained. As a consequence, if \(M\) is Kobayashi hyperbolic and \(\Omega = \Delta \times \mathbb{B}^{n-1}\), it is shown that \(M\) is biholomorphic to \(\Omega\). Further, if the Kobayashi metric on \(M\) has corank one, then \(M\) is globally a product; in fact, it is biholomorphic to \(Z \times \mathbb{C}\), where \(Z \subset \Omega = \Delta \times \mathbb{B}^{n-1}\) is a holomorphic retract.Geometric generalized Wronskians: applications in intermediate hyperbolicity and foliation theoryhttps://zbmath.org/1517.320742023-09-22T14:21:46.120933Z"Etesse, Antoine"https://zbmath.org/authors/?q=ai:etesse.antoineSummary: In this paper, we introduce a sub-family of the usual generalized Wronskians, which we call geometric generalized Wronskians. It is well known that one can test linear dependance of holomorphic functions (of several variables) via the identical vanishing of generalized Wronskians. We show that such a statement remains valid if one tests the identical vanishing only on geometric generalized Wronskians. It turns out that geometric generalized Wronskians allow to define intrinsic objects on projective varieties polarized with an ample line bundle: in this setting, the lack of existence of global functions is compensated by global sections of powers of the fixed ample line bundle. Geometric generalized Wronskians are precisely defined so that their local evaluations on such global sections globalize up to a positive twist by the ample line bundle. We then give three applications of the construction of geometric generalized Wronskians: one in intermediate hyperbolicity and two in foliation theory. In intermediate hyperbolicity, we show the algebraic degeneracy of holomorphic maps from \(\mathbb{C}^p\) to a Fermat hypersurface in \(\mathbf{P}^N\) of degree \(\delta>(N+1)(N-p)\): this interpolates between two well-known results, namely for \(p=1\) (first proved via Nevanlinna theory) and \(p=N-1\) (in which case the Fermat hypersurface is of general type). The first application in foliation theory provides a criterion for algebraic integrability of leaves of foliations: our criterion is not optimal in view of current knowledge, but has the advantage of having an elementary proof. Our second application deals with positivity properties of adjoint line bundles of the form \(K_{\mathcal{F}}+L\), where \(K_{\mathcal{F}}\) is the canonical bundle of a regular foliation \(\mathcal{F}\) on a smooth projective variety \(X\), and where \(L\) is an ample line bundle on \(X\).Complex-analytic intermediate hyperbolicity, and finiteness propertieshttps://zbmath.org/1517.320752023-09-22T14:21:46.120933Z"Etesse, Antoine"https://zbmath.org/authors/?q=ai:etesse.antoineSummary: Motivated by the finiteness of the set of automorphisms \(\mathrm{Aut}(X)\) of a projective manifold of general type \(X\), and by Kobayashi-Ochiai's conjecture that a projective manifold \(\dim(X)\)-analytically hyperbolic (also known as strongly measure hyperbolic) should be of general type, we investigate the finiteness properties of \(\mathrm{Aut}(X)\) for a complex manifold satisfying a (pseudo-) intermediate hyperbolicity property. We first show that a complex manifold \(X\) which is \((\dim(X)-1)\)-analytically hyperbolic has indeed finite automorphisms group. We then obtain a similar statement for a pseudo-\((\dim(X)-1)\)-analytically hyperbolic, strongly measure hyperbolic projective manifold \(X\), under an additional hypothesis on the size of the degeneracy set. Some of the properties used during the proofs lead us to introduce a notion of intermediate Picard hyperbolicity, which we last discuss.Bott-Chern cohomology of compact Vaisman manifoldshttps://zbmath.org/1517.320762023-09-22T14:21:46.120933Z"Istrati, Nicolina"https://zbmath.org/authors/?q=ai:istrati.nicolina"Otiman, Alexandra"https://zbmath.org/authors/?q=ai:otiman.alexandraThe authors prove several striking cohomological properties of compact Vaisman manifolds, a special class of locally conformally Kähler (lcK) manifolds for which the Lee vector field is parallel. Since the \(\partial\overline{\partial}\)-lemma does not hold on Vaisman manifolds, it is relevant to consider the Chern-Bott cohomology as its (finite-dimensional) failure:
\[
H^{\mathrm{BC}^{\bullet,\bullet}} = \ker(\partial)\cap \ker(\overline{\partial}) / \mathrm{Im}(\partial \overline{\partial})
\]
The Lee vector field and its rotation by \(J\) generate an integrable holomorphic distribution. The authors compute the Bott-Chern cohomology spaces in terms of the basic cohomology of the transverse distribution to this holomorphic foliation. The main technical tool used in the paper is Hodge theory for elliptic complexes over compact manifolds, identifying the various cohomology spaces with the kernels of certain elliptic differential operators.
As consequences, the authors show that, surprisingly, the Dolbeault and the Bott-Chern numbers of a Vaisman manifold determine each other. This result clearly has no counterpart in the Kähler setting. They show that the Bott-Chern numbers of a Vaisman manifold are constant under small holomorphic deformations through lcK manifolds. They also exhibit a finite-dimensional bi-graded algebra, defined in terms of the basic cohomology, which provides a simultaneous model for the Bott-Chern, the Dolbeault and the de Rham cohomology algebras. They finally prove that the positive invariant \(\Delta_3\) introduced by \textit{D. Angella} and \textit{A. Tomassini} [Invent. Math. 192, No. 1, 71--81 (2013; Zbl 1271.32011)], which vanishes when the \(\partial\overline{\partial}\)-lemma holds, can become arbitrarily large on Vaisman manifolds of fixed dimension \(n\geq 3\).
Reviewer: Sergiu Moroianu (Bucureşti)Morse-Novikov cohomology for blow-ups of complex manifoldshttps://zbmath.org/1517.320772023-09-22T14:21:46.120933Z"Meng, Lingxu"https://zbmath.org/authors/?q=ai:meng.lingxuThis paper is dedicated to the study of Morse-Novikov cohomology for blow-ups of complex manifolds through the weight \(\theta\)-sheaf \(\underline{\mathbb{R}}_{X,\theta}\). The author gives elementary properties of the weight \(\theta\)-sheaf \(\underline{\mathbb{R}}_{X,\theta}\) and also establishes several theorems of Künneth and Leray-Hirsch types. Then the computation of Morse-Novikov cohomologies of projective bundles as well as the independence between the \(\theta\)-Lefschetz number and \(\theta\) are given. Finally blow-up formulae on complex manifolds are investigated.
Reviewer: Guokuan Shao (Zhuhai)Bott-Chern hypercohomology and bimeromorphic invariantshttps://zbmath.org/1517.320782023-09-22T14:21:46.120933Z"Yang, Song"https://zbmath.org/authors/?q=ai:yang.song"Yang, Xiangdong"https://zbmath.org/authors/?q=ai:yang.xiangdong.1Summary: The aim of this article is to study the geometry of Bott-Chern hypercohomology from the bimeromorphic point of view. We construct some new bimeromorphic invariants involving the cohomology for the sheaf of germs of pluriharmonic functions, the truncated holomorphic de Rham cohomology, and the de Rham cohomology. To define these invariants, by using a sheaf-theoretic approach, we establish a blow-up formula together with a canonical morphism for the Bott-Chern hypercohomology. In particular, we compute the invariants of some compact complex threefolds, such as Iwasawa manifolds and quintic threefolds.Holomorphic factorization of mappings into \(\mathrm{Sp}_4 (\mathbb{C})\)https://zbmath.org/1517.320792023-09-22T14:21:46.120933Z"Ivarsson, Björn"https://zbmath.org/authors/?q=ai:ivarsson.bjorn"Kutzschebauch, Frank"https://zbmath.org/authors/?q=ai:kutzschebauch.frank"Løw, Erik"https://zbmath.org/authors/?q=ai:low.erikSummary: We prove that any null-homotopic holomorphic map from a Stein space \(X\) to the symplectic group \(\mathrm{Sp}_4 (\mathbb{C})\) can be written as a finite product of elementary symplectic matrices with holomorphic entries.On the algebra generated by \(\overline{\mu}\), \(\overline{\partial}\), \(\partial\), \(\mu\)https://zbmath.org/1517.320802023-09-22T14:21:46.120933Z"Auyeung, Shamuel"https://zbmath.org/authors/?q=ai:auyeung.shamuel"Guu, Jin-Cheng"https://zbmath.org/authors/?q=ai:guu.jin-cheng"Hu, Jiahao"https://zbmath.org/authors/?q=ai:hu.jiahaoSummary: In this note, we determine the structure of the associative algebra generated by the differential operators \(\overline{\mu}\), \(\overline{\partial}\), \(\partial\), and \(\mu\) that act on complex-valued differential forms of almost complex manifolds. This is done by showing that it is the universal enveloping algebra of the graded Lie algebra generated by these operators and determining the structure of the corresponding graded Lie algebra. We then determine the cohomology of this graded Lie algebra with respect to its canonical inner differential \([d, -]\), as well as its cohomology with respect to all its inner differentials.An almost complex structure with Norden metric on the phase spacehttps://zbmath.org/1517.320812023-09-22T14:21:46.120933Z"Bejan, Cornelia-Livia"https://zbmath.org/authors/?q=ai:bejan.cornelia-livia"Nakova, Galia"https://zbmath.org/authors/?q=ai:nakova.galia(no abstract)Almost complex manifolds from the point of view of Kodaira dimensionhttps://zbmath.org/1517.320822023-09-22T14:21:46.120933Z"Cattaneo, Andrea"https://zbmath.org/authors/?q=ai:cattaneo.andreaSummary: In complex geometry a classical and useful invariant of a complex manifold is its Kodaira dimension. Since its introduction by Iitaka in the early 70's, its behavior under deformations was object of study and it is known that Kodaira dimension is invariant under holomorphic deformations for smooth projective manifolds, while there are examples of holomorphic deformations of non-projective manifolds for which the Kodaira dimension is non-constant. Recently this concept has been generalized to almost complex manifolds, we want to present here some of its main features in the non-integrable case, mainly with respect to deformations. At the end we conclude with some speculations on the theory of meromorphic functions on almost complex manifolds.Kodaira dimensions of almost complex manifolds. Ihttps://zbmath.org/1517.320832023-09-22T14:21:46.120933Z"Chen, Haojie"https://zbmath.org/authors/?q=ai:chen.haojie"Zhang, Weiyi"https://zbmath.org/authors/?q=ai:zhang.weiyi.3|zhang.weiyi.1|zhang.weiyiSummary: This is the first of a series of papers in which we study the plurigenera, the Kodaira dimension, and more generally the Iitaka dimension on compact almost complex manifolds. Based on the Hodge theory on almost complex manifolds, we introduce the plurigenera, Kodaira dimension and Iitaka dimension on compact almost complex manifolds. We show that plurigenera and the Kodaira dimension are birational invariants in almost complex category, at least in dimension 4, where a birational morphism is defined to be a degree one pseudoholomorphic map. However, they are no longer deformation invariants, even in dimension 4 or under tameness assumption. On the way to establish the birational invariance, we prove the Hartogs extension theorem in the almost complex setting by the foliation-by-disks technique. Some interesting phenomena of these invariants are shown through examples. In particular, we construct non-integrable compact almost complex manifolds with large Kodaira dimensions. Hodge numbers and plurigenera are computed for the standard almost complex structure on the six sphere \(S^6\), which are different from the data of a hypothetical complex structure.Cup and Massey products on the cohomology of compact almost complex manifoldshttps://zbmath.org/1517.320842023-09-22T14:21:46.120933Z"Cirici, Joana"https://zbmath.org/authors/?q=ai:cirici.joanaSummary: The cohomology of any compact almost complex manifold carries bidegree decompositions induced by a Frölicher-type spectral sequence. In this note we give some restrictions on the possible decompositions on a given manifold and study how cup and Massey products behave with respect to such decompositions.Geometry of solutions to the c-projective metrizability equationhttps://zbmath.org/1517.320852023-09-22T14:21:46.120933Z"Flood, Keegan J."https://zbmath.org/authors/?q=ai:flood.keegan-j"Gover, A. Rod"https://zbmath.org/authors/?q=ai:gover.ashwin-rodSummary: On an almost complex manifold, a quasi-Kähler metric, with canonical connection in the c-projective class of a given minimal complex connection, is equivalent to a nondegenerate solution of the c-projectively invariant metrizability equation. For this overdetermined equation, replacing this maximal rank condition on solutions with a nondegeneracy condition on the prolonged system yields a strictly wider class of solutions with non-vanishing (generalized) scalar curvature. We study the geometries induced by this class of solutions. For each solution, the strict point-wise signature partitions the underlying manifold into strata, in a manner that generalizes the model, a certain Lie group orbit decomposition of \(\mathbb{CP}^m\). We describe the smooth nature and geometric structure of each strata component, generalizing the geometries of the embedded orbits in the model. This includes a quasi-Kähler metric on the open strata components that becomes singular at the strata boundary. The closed strata inherit almost CR-structures and can be viewed as a c-projective infinity for the given quasi-Kähler metric.Almost Kähler Kodaira-Spencer problemhttps://zbmath.org/1517.320862023-09-22T14:21:46.120933Z"Holt, Tom"https://zbmath.org/authors/?q=ai:holt.tom"Zhang, Weiyi"https://zbmath.org/authors/?q=ai:zhang.weiyi.3|zhang.weiyi|zhang.weiyi.1Summary: We show that the almost complex Hodge number \(h^{0,1}\) varies with different choices of almost Kähler metrics. This answers the almost Kähler version of a question of Kodaira and Spencer.Degenerate multilinear forms and Hermitian and para-Hermitian structureshttps://zbmath.org/1517.320872023-09-22T14:21:46.120933Z"Kornev, E. S."https://zbmath.org/authors/?q=ai:kornev.e-sSummary: We describe some method for obtaining families of complex and paracomplex structures on real manifolds by using degenerate skew-symmetric multilinear forms. To construct these structures, we employ a skew-symmetric form with nontrivial radical and obtain a family of almost complex structures on the six-dimensional sphere different from the Cayley structure and families of Hermitian and para-Hermitian structures on some six-dimensional manifolds.Harmonic \((1,1)\)-forms on compact almost Hermitian 4-manifoldshttps://zbmath.org/1517.320882023-09-22T14:21:46.120933Z"Piovani, Riccardo"https://zbmath.org/authors/?q=ai:piovani.riccardoSummary: We recall the most recent results concerning the spaces of Dolbeault and Bott-Chern harmonic \((1,1)\)-forms on a compact almost Hermitian 4-manifold and compute the dimension of the space of Dolbeault harmonic \((1,1)\)-forms in some explicit examples.Semi-continuity in real analytic familieshttps://zbmath.org/1517.320892023-09-22T14:21:46.120933Z"Yeung, Sai-Kee"https://zbmath.org/authors/?q=ai:yeung.sai-keeSummary: For a real analytic family of almost complex structures \((M, J_t)\), \(t\in T\), on an even dimensional compact real analytic manifold \(M\), we prove upper semi-continuity with respect to \(t\) of the dimension of the space of \(\Delta_{\overline{\partial}_t}\)-harmonic \((p, q)\)-forms in Zariski sense. More general statements in terms of the dimension of the kernel of a family of self-adjoint strongly elliptic operators on a vector bundle over a manifold in real analytic settings are proved.Tamed exhaustion functions and Schwarz type lemmas for almost Hermitian manifoldshttps://zbmath.org/1517.320902023-09-22T14:21:46.120933Z"Yu, Weike"https://zbmath.org/authors/?q=ai:yu.weikeSummary: In this paper, we study a special exhaustion function on almost Hermitian manifolds and establish the existence result by using the Hessian comparison theorem. From the viewpoint of the exhaustion function, we establish a related Schwarz type lemma for almost holomorphic maps between two almost Hermitian manifolds. As corollaries, we deduce several versions of Schwarz and Liouville type theorems for almost holomorphic maps.Almost complex Hodge theoryhttps://zbmath.org/1517.320912023-09-22T14:21:46.120933Z"Zhang, Weiyi"https://zbmath.org/authors/?q=ai:zhang.weiyi.3|zhang.weiyi|zhang.weiyi.1Summary: We review the recent development of Hodge theory for almost complex manifolds. This includes the determination of whether the Hodge numbers defined by \(\overline{\partial}\)-Laplacian are almost complex, almost Kähler, or birational invariants in dimension four.Non-degeneracy of cohomological traces for general Landau-Ginzburg modelshttps://zbmath.org/1517.320922023-09-22T14:21:46.120933Z"Doryn, Dmitry"https://zbmath.org/authors/?q=ai:doryn.dmitry"Lazaroiu, Calin Iuliu"https://zbmath.org/authors/?q=ai:lazaroiu.calin-iuliuSummary: We prove non-degeneracy of the cohomological bulk and boundary traces for general open-closed Landau-Ginzburg models associated to a pair \((X, W)\), where \(X\) is a non-compact complex manifold with trivial canonical line bundle and \(W\) is a complex-valued holomorphic function defined on \(X\), assuming only that the critical locus of \(W\) is compact (but may not consist of isolated points). These results can be viewed as certain ``deformed'' versions of Serre duality. The first amounts to a duality property for the hypercohomology of the sheaf Koszul complex of \(W\), while the second is equivalent with the statement that a certain power of the shift functor is a Serre functor on the even subcategory of the \(\mathbb{Z}_2\)-graded category of topological D-branes of such models.Zariski multiplicity conjecture in families of non-degenerate singularitieshttps://zbmath.org/1517.320932023-09-22T14:21:46.120933Z"Brzostowski, Szymon"https://zbmath.org/authors/?q=ai:brzostowski.szymon"Krasiński, Tadeusz"https://zbmath.org/authors/?q=ai:krasinski.tadeusz"Oleksik, Grzegorz"https://zbmath.org/authors/?q=ai:oleksik.grzegorzSummary: We give a new, elementary proof of the Zariski multiplicity conjecture in \(\mu\)-constant families of non-degenerate singularities.
For the entire collection see [Zbl 1509.14002].Graded matrix factorizations of size two and reductionhttps://zbmath.org/1517.320942023-09-22T14:21:46.120933Z"Ebeling, Wolfgang"https://zbmath.org/authors/?q=ai:ebeling.wolfgang"Takahashi, Atsushi"https://zbmath.org/authors/?q=ai:takahashi.atsushi.3Summary: We associate a complete intersection singularity to a graded matrix factorization of size two of a polynomial in three variables. We show that we get an inverse to the reduction of singularities considered by \textit{C. T. C. Wall} [Proc. Lond. Math. Soc., III. Ser. 48, 461--513 (1984; Zbl 0541.14003)]. We study this for the full strongly exceptional collections in the triangulated category of graded matrix factorizations constructed by \textit{H. Kajiura} et al. [Adv. Math. 220, No. 5, 1602--1654 (2009; Zbl 1172.18002)].Contact exponent and the Milnor number of plane curve singularitieshttps://zbmath.org/1517.320952023-09-22T14:21:46.120933Z"García Barroso, Evelia Rosa"https://zbmath.org/authors/?q=ai:garcia-barroso.evelia-rosa"Płoski, Arkadiusz"https://zbmath.org/authors/?q=ai:ploski.arkadiuszSummary: We investigate properties of the contact exponent (in the sense of
\textit{H. Hironaka} [Introduction to the theory of infinitely near singular points. Madrid: Consejo Superior de Investigaciones Cientificas (1974; Zbl 0366.32007)]) of plane algebroid curve singularities over algebraically closed fields of arbitrary characteristic. We prove that the contact exponent is an equisingularity invariant and give a new proof of the stability of the maximal contact. Then we prove a bound for the Milnor number and determine the equisingularity class of algebroid curves for which this bound is attained. We do not use the method of Newton's diagrams. Our tool is the logarithmic distance developed in [the authors, Rev. Mat. Complut. 28, No. 1, 227--252 (2015; Zbl 1308.32032)].
For the entire collection see [Zbl 1429.00039].Knots of irreducible curve singularitieshttps://zbmath.org/1517.320962023-09-22T14:21:46.120933Z"Krasiński, Tadeusz"https://zbmath.org/authors/?q=ai:krasinski.tadeuszSummary: In the article the relation between irreducible curve plane singularities and knots is described. In these terms the topological classification of such singularities is given.
For the entire collection see [Zbl 1429.00039].Algebraic approximation of Cohen-Macaulay algebrashttps://zbmath.org/1517.320972023-09-22T14:21:46.120933Z"Patel, Aftab"https://zbmath.org/authors/?q=ai:patel.aftab.1|patel.aftabSummary: This paper shows that Cohen-Macaulay algebras can be algebraically approximated in such a way that their Cohen-Macaulayness and minimal Betti numbers are preserved. This is achieved by showing that finitely generated modules over power series rings can be algebraically approximated in a manner that preserves their diagrams of initial exponents and their minimal Betti numbers. These results are also applied to obtain an approximation result for flat homomorphisms from rings of power series to Cohen-Macaulay algebras.An estimation of the jump of the Milnor number of plane curve singularitieshttps://zbmath.org/1517.320982023-09-22T14:21:46.120933Z"Zakrzewska, Aleksandra"https://zbmath.org/authors/?q=ai:zakrzewska.aleksandraSummary: The jump of the Milnor number of an isolated singularity \(f_0\) is the minimal non-zero difference between the Milnor numbers of \(f_0\) and one of its deformations \(f_s\). We estimate the jump using the Enriques diagram of \(f_0\).
For the entire collection see [Zbl 1509.14002].Singularity of normal complex analytic surfaces admitting non-isomorphic finite surjective endomorphismshttps://zbmath.org/1517.320992023-09-22T14:21:46.120933Z"Nakayama, Noboru"https://zbmath.org/authors/?q=ai:nakayama.noboruSummary: For a non-isomorphic finite endomorphism of a germ of a complex analytic normal surface at a point, the pair of the surface and a completely invariant reduced divisor is shown to be log-canonical. It is also shown in many situations that the endomorphism or its square lifts to an endomorphism of another surface by an essential blowing up.Some notes on the Lê numbers in the family of line singularitieshttps://zbmath.org/1517.321002023-09-22T14:21:46.120933Z"Oleksik, Grzegorz"https://zbmath.org/authors/?q=ai:oleksik.grzegorz"Różycki, Adam"https://zbmath.org/authors/?q=ai:rozycki.adamSummary: In this paper we introduce the jumps of the Lê numbers of non-isolated singularity \(f\) in the family of line deformations. Moreover, we prove
the existence of a deformation of a non-degenerate singularity \(f\) such that the first Lê number is constant and the zeroth Lê number jumps down to zero. We also give estimations of the Lê numbers when the critical locus is one-dimensional. These give a version of the celebrated theorem of A. G. Kouchnirenko in this case.
For the entire collection see [Zbl 1509.14002].Weierstrass 1-forms and nondicritical generalized curve foliationshttps://zbmath.org/1517.321012023-09-22T14:21:46.120933Z"García Barroso, Evelia R."https://zbmath.org/authors/?q=ai:garcia-barroso.evelia-rosa"Hernandes, Marcelo Escudeiro"https://zbmath.org/authors/?q=ai:hernandes.marcelo-escudeiro"Iglesias, Mauro Fernando Hernández"https://zbmath.org/authors/?q=ai:iglesias.mauro-fernando-hernandezSummary: In this paper, we introduce a distinguished expression for a given 1-form with respect to a polynomial \(f\in\mathbb{C}\{x\}[y]\), called \textit{Weierstrass form}. We will use this form and the properties of plane analytical curves to give new characterizations of nondicritical generalized curve foliations.Foliations on smooth algebraic surfaces in positive characteristichttps://zbmath.org/1517.321022023-09-22T14:21:46.120933Z"Mendson, Wodson"https://zbmath.org/authors/?q=ai:mendson.wodsonSummary: We investigate the notion of the \(p\)-divisor for foliations on a smooth algebraic surface defined over a field of positive characteristic \(p\) and we study some of its properties. We present a structure theorem for the \(p\)-divisor of foliations in the projective plane and the Hirzebruch surfaces where we show that, under certain conditions, such \(p\)-divisors are reduced.Existence of geodesic spirals for the Kobayashi-Fuks metric on planar domainshttps://zbmath.org/1517.321032023-09-22T14:21:46.120933Z"Kar, Debaprasanna"https://zbmath.org/authors/?q=ai:kar.debaprasannaSummary: In this note, we discuss the following problem: Given a smoothly bounded strongly pseudoconvex domain \(D\) in \(\mathbb{C}^n\), can we guarantee the existence of geodesics for the Kobayashi-Fuks metric which ``spiral around'' in the interior of \(D\)? We find an affirmative answer to the above question for \(n = 1\) when \(D\) is not simply connected.Relating catlin and D'Angelo \(q\)-typeshttps://zbmath.org/1517.321042023-09-22T14:21:46.120933Z"Brinzanescu, Vasile"https://zbmath.org/authors/?q=ai:brinzanescu.vasile"Nicoara, Andreea C."https://zbmath.org/authors/?q=ai:nicoara.andreea-cSummary: We clarify the relationship between the two most standard ways to measure the order of contact of \(q\)-dimensional complex varieties with a real hypersurface, the Catlin and D'Angelo \(q\)-types, by showing that the former equals the generic value of the normalized order of contact measured along curves whose infimum is by definition the D'Angelo \(q\)-type.On smoothing of plurisubharmonic functions on unbounded domainshttps://zbmath.org/1517.321052023-09-22T14:21:46.120933Z"Harz, T."https://zbmath.org/authors/?q=ai:harz.tobiasLet \(\Omega\subset \mathbb C^n\) be a domain. If \(\Omega\) is bounded, then a bounded strictly plurisubharmonic function exists on \(\Omega\). If \(\Omega\) is unbounded, this is not true in general. The subset of \(\Omega\) where all from above plurisubharmonic functions on \(\Omega\) bounded fail to be strictly plurisubharmonic is called \textit{the core of} \(\Omega\). Cores have been studied for different subclasses of plurisubharmonic functions on \(\Omega\). The author provides examples showing that the core with respect to locally bounded and with respect to continuous plurisubharmonic functions are not the same in general and that the core with respect to continuous and with respect to \(\mathcal C^1\)-smooth plurisubharmonic functions are not the same in general, even if \(\Omega\) is pseudoconvex. The constructed examples also show that it is not possible in general to approximate bounded plurisubharmonic functions by continuous bounded plurisubharmonic functions. Furthermore, the author proves that the non-approximability of bounded from the above plurisubharmonic functions in different smoothness classes is a consequence of having different cores. The proof depends on the known structure of the core [\textit{Z. Slodkowski}, Contemp. Math. 735, 239--259 (2019; Zbl 1446.32024)].
Reviewer: Barbara Drinovec Drnovšek (Ljubljana)Core sets in Kähler manifoldshttps://zbmath.org/1517.321062023-09-22T14:21:46.120933Z"Göğüş, Nihat Gökhan"https://zbmath.org/authors/?q=ai:gogus.nihat-gokhan"Günyüz, Ozan"https://zbmath.org/authors/?q=ai:gunyuz.ozan"Yazıcı, Özcan"https://zbmath.org/authors/?q=ai:yazici.ozcanSummary: The primary objective of this paper is to study core sets in the setting of \(m\)-subharmonic functions on the class of (non-compact) Kähler manifolds. Core sets are investigated in different aspects by considering various classes of plurisubharmonic functions. One of the crucial concepts in studying the structure of this kind of sets is the pseudoconcavity. In a more general way, we will have the structure of core defined with respect to the \(m\)-subharmonic functions, which we call \(m\)-core in our setting, in terms of \(m\)-pseudoconcave sets. In the context of \(m\)-subharmonic functions, we define \(m\)-harmonic functions and show that, in \(\mathbb{C}^n(n \geq 2)\) and more generally in any Kähler manifold of dimension at least \(2\), \(m\)-harmonic functions are pluriharmonic functions for \(m \geq 2\).Kohn-Rossi cohomology class, Sasakian space form and CR Frankel conjecturehttps://zbmath.org/1517.321072023-09-22T14:21:46.120933Z"Chang, Der-Chen"https://zbmath.org/authors/?q=ai:chang.der-chen-e"Chang, Shu-Cheng"https://zbmath.org/authors/?q=ai:chang.shu-cheng"Kuo, Ting-Jung"https://zbmath.org/authors/?q=ai:kuo.ting-jung"Lin, Chien"https://zbmath.org/authors/?q=ai:lin.chienSummary: In this paper, we give a criterion of pseudo-Einstein contact forms and then affirm the CR analogue of Frankel conjecture in a closed, spherical, strictly pseudoconvex CR manifold of nonnegative pseudohermitian curvature on the space of smooth representatives of the first Kohn-Rossi cohomology group.The Kohn-Laplacian and Cauchy-Szegö projection on model domainshttps://zbmath.org/1517.321082023-09-22T14:21:46.120933Z"Chang, Der-Chen"https://zbmath.org/authors/?q=ai:chang.der-chen-e"Li, Ji"https://zbmath.org/authors/?q=ai:li.ji.1"Tie, Jingzhi"https://zbmath.org/authors/?q=ai:tie.jingzhi"Wu, Qingyan"https://zbmath.org/authors/?q=ai:wu.qingyanSummary: We study the Kohn-Laplacian and its fundamental solution on some model domains in \(\mathbb{C}^{n+1} \), and further discuss the explicit kernel of the Cauchy-Szegö projections on these model domains using the real analysis method. We further show that these Cauchy-Szegö kernels are Calderón-Zygmund kernels under the suitable quasi-metric.Five-dimensional para-CR manifolds and contact projective geometry in dimension threehttps://zbmath.org/1517.321092023-09-22T14:21:46.120933Z"Merker, Joël"https://zbmath.org/authors/?q=ai:merker.joel"Nurowski, Paweł"https://zbmath.org/authors/?q=ai:nurowski.pawelSummary: We study invariant properties of 5-dimensional para-CR structures whose Levi form is degenerate in precisely one direction and which are 2-nondegenerate. We realize that \textit{two}, out of three, primary (basic) para-CR invariants of such structures are the classical differential invariants known to \textit{G. Monge} [``Sur les équations différentielles des courbes du second degré'', Bull. Soc. Math.France, 87--88 (1810)] and
to \textit{K. Wünschmann} [Über Berührungsbedingungen bei Integralkurven von Differentialgleichungen.
Greifswald. Leipzig: B. G. Teubner (1905; JFM 36.0379.03)]:
\[
\begin{aligned}
M(G) := 40G^3_{ppp} - 45G_{pp} G_{ppp} G_{pppp} + 9G^2_{pp} G_{ppppp}, \\
W(H) := 9D^2 H_r - 27DH_p - 18H_r DH_r + 18H_p H_r + 4H^3_r + 54H_z.
\end{aligned}
\]
The vanishing \(M(G) \equiv 0\) provides a local necessary and sufficient condition for the graph of a function in the \((p, G)\)-plane to be contained in a conic, while the vanishing \(W(H) \equiv 0\) gives an \textit{if-and-only-if} condition for a \(3^\mathrm{rd}\) order ODE to define a natural Lorentzian geometry on the space of its solutions.
Mainly, we give a geometric interpretation of the \textit{third} basic invariant of our class of para-CR structures, the simplest one, of lowest order, and of mixed nature \(N(G,H) := 2G_{ppp} + G_{pp} H_{rr}\). We establish that the vanishing \(N(G,H) \equiv 0\) gives an \textit{if-and-only-if} condition for the \textit{two} 3-dimensional quotients of the para-CR manifold by its two canonical integrable rank-2 distributions, to be equipped with contact projective geometries.
A curious transformation between the Wünschmann invariant and the Monge invariant, first noted by us in a recent publication [the authors, ``On degenerate para-CR structures: Cartan reduction and homogeneous models'', Transformation Groups (to appear) (\url{doi:10.1007/s00031-022-09746-4})] is also discussed, and its mysteries are further revealed.Generalizations of the \(Q\)-prime curvature via renormalized characteristic formshttps://zbmath.org/1517.321102023-09-22T14:21:46.120933Z"Takeuchi, Yuya"https://zbmath.org/authors/?q=ai:takeuchi.yuyaA highly specialized investigation around CR-manifolds, GJMS operators (the author considers the reader informed that these are Cauchy-Riemann ones and differential operators as developed by Graham, Jenne, Mason \& Sparling in 1992, respectively), Monge-Ampère equations, Gauss-Bonnet-Chern formulae, pseudo-Einstein contacts, Graham-Lee connections, Sasakian manifolds, and Tanaka-Webster curvatures. My rough counting of 90 integration symbols, together with 24 reference entries supports the impression that this may be attempts in an important direction.
Reviewer: Gunther Schmidt (München)An upper bound for the first positive eigenvalue of the Kohn Laplacian on Reinhardt real hypersurfaceshttps://zbmath.org/1517.321112023-09-22T14:21:46.120933Z"Dall'ara, Gian Maria"https://zbmath.org/authors/?q=ai:dallara.gian-maria"Son, Duong Ngoc"https://zbmath.org/authors/?q=ai:duong-ngoc-son.In this paper, the authors consider compact connected real hypersurfaces \(M\subset\mathbb C^2\) that are invariant under the standard action of the \(2\)-dimensional torus, with the assumption that this
action is free and that \(M\) is strictly pseudoconvex. They consider then the Kohn Laplacian associated to the natural pseudohermitian structure and give a sharp upper bound for the first positive eigenvalue of it. See Theorem 1.1.
Reviewer: Francine Meylan (Lausanne)Improved \(L^p\)-Folland-Stein-Sobolev inequality under constraintshttps://zbmath.org/1517.321122023-09-22T14:21:46.120933Z"Ho, Pak Tung"https://zbmath.org/authors/?q=ai:ho.pak-tungSummary: Hang-Wang improved the Sobolev inequality on the unit sphere under the vanishing of higher-order moments of the volume element. In this note, we improve the Folland-Stein inequality on the CR sphere under the vanishing of higher order moments of the volume element.CR embeddings of CR manifoldshttps://zbmath.org/1517.321132023-09-22T14:21:46.120933Z"Cowling, M. G."https://zbmath.org/authors/?q=ai:cowling.michael-g"Ganji, M."https://zbmath.org/authors/?q=ai:ganji.masoud"Ottazzi, A."https://zbmath.org/authors/?q=ai:ottazzi.alessandro"Schmalz, G."https://zbmath.org/authors/?q=ai:schmalz.gerdIt is well known that analytic CR-manifolds can always be locally embedded in complex space. In the paper [Invent. Math. 82, 359--396 (1985; Zbl 0598.32019)] \textit{M. S. Baouendi} et al. consider the case where there is an abelian Lie algebra of real vector fields that is \textit{transverse} and \textit{normalizing} and construct an embedding into a complex space. In this paper, the authors address the case of a finite-dimensional Lie algebra extension of the CR bundle by nonvanishing complex vector fields satisfying some ``regularity'' properties and show that $M$ embeds into a CR manifold $\tilde{M}$ of type $(n+l, k-l)$ if the dimension of the extension has dimension \(n+l.\) See Theorem 1 for a precise statement.
Reviewer: Francine Meylan (Lausanne)On finitely Levi non degenerate homogeneous CR manifoldshttps://zbmath.org/1517.321142023-09-22T14:21:46.120933Z"Marini, Stefano"https://zbmath.org/authors/?q=ai:marini.stefanoSummary: A \(CR\) manifold \(M\) is a differentiable manifold together with a complex subbundle of the complexification of its tangent bundle, which is formally integrable and has zero intersection with its conjugate bundle. A fundamental invariant of a \(CR\) manifold \(M\) is its vector-valued Levi form. A Levi non degenerate \(CR\) manifold of order \(k\ge 1\) has non degenerate Levi form in a higher order sense. For a (locally) homogeneous manifold Levi non degeneracy of order \(k\) can be described in terms of its \(CR\) algebra, i.e. a pair of Lie algebras encoding the structure of (locally) homogeneous \(CR\) manifolds. I will give an introduction to these topics presenting some recent results.Rigid biholomorphic equivalences of rigid \(\mathfrak{C}_{2 , 1}\) hypersurfaces \(M^5 \subset \mathbb{C}^3\)https://zbmath.org/1517.321152023-09-22T14:21:46.120933Z"Foo, Wei-Guo"https://zbmath.org/authors/?q=ai:foo.wei-guo"Merker, Joël"https://zbmath.org/authors/?q=ai:merker.joel"Ta, The-Anh"https://zbmath.org/authors/?q=ai:ta.the-anhSummary: We study the local equivalence problem for real-analytic \(( \mathcal{C}^\omega )\) hypersurfaces \(M^5 \subset \mathbb{C}^3\) that, in some holomorphic coordinates \((z_1, z_2, w) \in \mathbb{C}^3\) with \(w = u + \sqrt{ - 1} v \), are \textit{rigid} in the sense that their graphing functions \[
u = F( z_1,z_2, \overline{z}_1, \overline{z}_2)
\]
are independent of \(v\). Specifically, we study the group \(\mathsf{Hol}_{\mathsf{rigid}}(M)\) of \textit{rigid} local biholomorphic transformations of the form
\[
(z_1,z_2,w) \longmapsto( f_1( z_1, z_2), f_2( z_1, z_2), aw + g( z_1, z_2)),
\]
where \(a \in \mathbb{R} \setminus \{0 \}\) and \(D( f_1, f_2) / D( z_1, z_2) \neq 0\), which preserve the rigidity of hypersurfaces.
After performing a Cartan-type reduction to an appropriate \(\{e\} \)-structure, we find exactly \textit{two} primary invariants \(I_0\) and \(V_0\), which we express explicitly in terms of the 5-jet of the graphing function \(F\) of \(M\). The identical vanishing \(0 \equiv I_0( J^5 F) \equiv V_0( J^5 F)\) then provides a necessary and sufficient condition for \(M\) to be locally \textit{rigidly biholomorphic} to the known model hypersurface
\[
M_{\mathsf{LC}} : u = \frac{ z_1 \overline{z}_1 + 1 / 2 z_1^2 \overline{z}_2 + 1 / 2 \overline{z}_1^2 z_2}{ 1 - z_2 \overline{z}_2}.
\]
We establish that always \(\dim \mathsf{Hol}_{\mathsf{rigid}}(M) \leq 7 = \dim \mathsf{Hol}_{\mathsf{rigid}}( M_{\mathsf{LC}})\).
If one of these two primary invariants \(I_0 \not\equiv 0\) or \(V_0 \not\equiv 0\) does not vanish identically, then on either of the two Zariski-open sets \(\{p \in M : I_0(p) \neq 0 \}\) or \(\{p \in M : V_0(p) \neq 0 \} \), we show that this rigid equivalence problem between rigid hypersurfaces reduces to an equivalence problem for a certain five-dimensional \(\{e \} \)-structure on \(M\), that is, we get an invariant absolute parallelism on \(M^5\). Hence \(\dim \mathsf{Hol}_{\mathsf{rigid}}(M)\) drops from 7 to 5, illustrating the \textit{gap phenomenon}.Convergent normal form for five dimensional totally nondegenerate CR manifolds in \(\mathbb{C}^4\)https://zbmath.org/1517.321162023-09-22T14:21:46.120933Z"Sabzevari, Masoud"https://zbmath.org/authors/?q=ai:sabzevari.masoudThe goal of the paper is to construct convergent normal forms for real-analytic 5-dimensional totally nondegenerate CR submanifolds in \(\mathbb{C}^4\) of CR codimension three. This normal form is obtained applying the equivariant moving frame method, in which a further normalization is applied depending on the vanishing of certain coefficients of the normal form. This approach leads to a partition of the class of CR manifolds treated into several biholomorphically inequivalent subclasses, each of them with the specified normal form no further normalizable in the sense above. This is used to show that the Beloshapka's cubic model is the unique, up to biholomorphisms, class with the maximum possible dimension of seven for the algebra of infinitesimal CR automorphisms. These results can be used to study the biholomorphic equivalence problem for the treated class of CR manifolds.
Reviewer: Stefano Marini (Parma)Unique continuation for \(\bar{\partial}\) with square-integrable potentialshttps://zbmath.org/1517.321172023-09-22T14:21:46.120933Z"Pan, Yifei"https://zbmath.org/authors/?q=ai:pan.yifei"Zhang, Yuan"https://zbmath.org/authors/?q=ai:zhang.yuanSummary: In this paper, we investigate the unique continuation property for the inequality \(|\bar{\partial} u| \le V|u|\), where \(u\) is a vector-valued function from a domain in \(C^n\) to \(C^N\), and the potential \(V \in L^2\). We show that the strong unique continuation property holds when \(n=1\), and the weak unique continuation property holds when \(n>1\). In both cases, the \(L^2\) integrability condition on the potential is optimal.Estimates for \(\overline{\partial}\) on domains in \(\mathbb{C}^n\) and \(\mathbb{CP}^n\)https://zbmath.org/1517.321182023-09-22T14:21:46.120933Z"Shaw, Mei-Chi"https://zbmath.org/authors/?q=ai:shaw.mei-chiSummary: In this paper, we survey some results of the Cauchy-Riemann equation on domains in \(\mathbb{C}^n\) and \(\mathbb{CP}^n\) related to Sibony's work. We also raise many questions on estimates for \(\overline{\partial}\) for smooth and non-smooth domains.A general estimate for the \(\bar \partial\)-Neumann problemhttps://zbmath.org/1517.321192023-09-22T14:21:46.120933Z"Tran Vu Khanh"https://zbmath.org/authors/?q=ai:khanh.tran-vuSummary: This paper especially focuses on a general estimate, called \((f-\mathcal M)^k\), for the \(\bar \partial\)-Neumann problem
\[
{(f-\mathcal M)^k} \qquad \| f({\varLambda})\mathcal M u\|^2\le c(\|\bar \partial u\|^2+\|\bar \partial^*u\|^2+\|u\|^2)+C_{\mathcal M}\|u\|^2_{-1}
\]
for any \(u\in C^{\infty}_c(U\cap \bar{\Omega})^k\cap \text{Dom}(\bar{\partial}^*)\), where \(f (\Lambda)\) is the tangential pseudodifferential operator with symbol \(f ((1 + |\xi |^2)^{1/2})\), \(\mathcal M\) is a multiplier, and \(U\) is a neighborhood of a given boundary point \(z_0\). Here the domain \(\Omega\) is \(q\)-pseudoconvex or \(q\)-pseudoconcave at \(z_0\). We want to point out that under a suitable choice of \(f\) and \(\mathcal M\), \((f{-}\mathcal M)^k\) is the subelliptic, superlogarithmic, compactness and so on. Generalizing the Property \((P)\) by \textit{D. W. Catlin } [Proc. Symp., Madison/Wis. 1982, Proc. Symp. Pure Math. 41, 39--49 (1984; Zbl 0578.32031)], we define Property \((f-\mathcal M-P)^k\). The result we obtain in here is: Property \((f-\mathcal M-P)^k\) yields the \((f-\mathcal M)^k\) estimate. The paper also aims at exhibiting some relevant classes of domains which enjoy Property \((f-\mathcal M-P)^k\).The fundamental solution to \(\Box_b\) on quadric manifolds with nonzero eigenvalueshttps://zbmath.org/1517.321202023-09-22T14:21:46.120933Z"Boggess, Albert"https://zbmath.org/authors/?q=ai:boggess.albert"Raich, Andrew"https://zbmath.org/authors/?q=ai:raich.andrew-sSummary: This paper is part of a continuing examination into the geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of \(\mathbb{C}^n\times \mathbb{C}^m\). The goal of this article is explore the complex Green operator in the case that the eigenvalues of the directional Levi forms are nonvanishing. We (1) investigate the geometric conditions on \(M\) which the eigenvalue condition forces, (2) establish optimal pointwise upper bounds on complex Green operator and its derivatives, (3) explore the \(L^p\) and \(L^p\)-Sobolev mapping properties of the associated kernels, and (4) provide examples.Semi-isometric CR immersions of CR manifolds into Kähler manifolds and applicationshttps://zbmath.org/1517.321212023-09-22T14:21:46.120933Z"Duong Ngoc Son"https://zbmath.org/authors/?q=ai:duong-ngoc-son.This paper studies semi-isometric CR immersions of CR manifolds into Kähler manifolds. The author first shows that the squared mean curvature function \(|H|^2\) and the transverse curvature \(r(\rho)\) of a defining function \(\rho\) coincide if \(\rho\) is chosen appropriately. Using the result above, he gives a reformulation (Theorem 1.1) for the theorem of \textit{S. Y. Li} and the author [Acta Math. Sin., Engl. Ser. 34, No. 8, 1248--1258 (2018; Zbl 1404.32071)] and gets an upper bound of the first positive eigenvalue \(\lambda_1\) of the Kohn Laplacian \(\square_b\). After that, the author establishes a global CR equivalence from semi-isometric CR immersion manifolds to spheres (Theorem 1.2) through the ``extremal'' situation for estimates of \(\lambda_1\) in Theorem 1.1. Next, he derives a generalization result of the ``first gap'' theorem by Webster, Cima-Suffridge, Faran, Huang. This result is based on \textit{X. Huang}'s lemma [J. Differ. Geom. 51, No. 1, 13--33 (1999; Zbl 1042.32008)] and Theorem 1.2. On the other hand, he also gives an inequality for the Levi-Fefferman determinant \(J(\rho)\) for a strictly pseudoconvex real hypersurfaces \(M\) in \(\mathbb{C}^{n+1}\) and shows the relation between umbilical points and critical points of the inequality, where \(\rho\) is the defining function of \(M\). At the end of this paper, the author provides an example to verify the necessity of the dimension condition in some theorems above.
Reviewer: Guokuan Shao (Zhuhai)The Dirichlet problem for the Monge-Ampère equation on Hermitian manifolds with boundaryhttps://zbmath.org/1517.321222023-09-22T14:21:46.120933Z"Kołodziej, Sławomir"https://zbmath.org/authors/?q=ai:kolodziej.slawomir"Nguyen, Ngoc Cuong"https://zbmath.org/authors/?q=ai:nguyen.ngoc-cuongAuthors' abstract: We study weak quasi-plurisubharmonic solutions to the Dirichlet problem for the complex Monge-Ampère equation on a general Hermitian manifold with non-empty boundary. We prove optimal subsolution theorems: for bounded and Hölder continuous quasi-plurisubharmonic functions. The continuity of the solution is proved for measures that are well dominated by capacity, for example measures with \(L^{p}\), \(p>1\) densities, or moderate measures in the sense of Dinh-Nguyen-Sibony.
Reviewer: Emil J. Straube (College Station)Complex Monge-Ampère equations for plurifinely plurisubharmonic functionshttps://zbmath.org/1517.321232023-09-22T14:21:46.120933Z"Nguyen Xuan Hong"https://zbmath.org/authors/?q=ai:nguyen-xuan-hong."Hoang Van Can"https://zbmath.org/authors/?q=ai:hoang-van-can."Nguyen Thi Lien"https://zbmath.org/authors/?q=ai:nguyen-thi-lien."Pham Thi Lieu"https://zbmath.org/authors/?q=ai:pham-thi-lieu.Summary: This paper studies the complex Monge-Ampère equations for \(\mathcal{F}\)-plurisubharmonic functions in bounded \(\mathcal{F}\)-hyperconvex domains. We give sufficient conditions for this equation to solve for measures with a singular part.Algebras of pseudo-differential operators acting on holomorphic Sobolev spaceshttps://zbmath.org/1517.321242023-09-22T14:21:46.120933Z"Winterrose, David Scott"https://zbmath.org/authors/?q=ai:winterrose.david-scottSummary: We search for pseudo-differential operators acting on holomorphic Sobolev spaces. The operators should mirror the standard Sobolev mapping property in the holomorphic analogues. The setting is a closed real-analytic Riemannian manifold, or Lie group with a bi-invariant metric, and the holomorphic Sobolev spaces are defined on a fixed Grauert tube about the core manifold. We find that every pseudo-differential operator in the commutant of the Laplacian is of this kind. Moreover, so are all the operators in the commutant of certain analytic pseudo-differential operators, but for more general tubes, provided that an old statement of Boutet de Monvel holds true generally. In the Lie group setting, we find even larger algebras, and characterize all their elliptic elements. These latter algebras are determined by global matrix-valued symbols.Typical properties of periodic Teichmüller geodesics: Lyapunov exponentshttps://zbmath.org/1517.370412023-09-22T14:21:46.120933Z"Hamenstädt, Ursula"https://zbmath.org/authors/?q=ai:hamenstadt.ursulaSummary: Consider a component \(\mathcal{Q}\) of a stratum in the moduli space of area-one abelian differentials on a surface of genus \(g\). Call a property \(\mathcal{P}\) for periodic orbits of the Teichmüller flow on \(\mathcal{Q}\) \textit{typical} if the growth rate of orbits with property \(\mathcal{P}\) is maximal. We show that the following property is typical. Given a continuous integrable cocycle over the Teichmüller flow with values in a vector bundle \(V\to\mathcal{Q}\), the logarithms of the eigenvalues of the matrix defined by the cocycle and the orbit are arbitrarily close to the Lyapunov exponents of the cocycle for the Masur-Veech measure.Gaussian Gabor frames, Seshadri constants and generalized Buser-Sarnak invariantshttps://zbmath.org/1517.420322023-09-22T14:21:46.120933Z"Luef, Franz"https://zbmath.org/authors/?q=ai:luef.franz"Wang, Xu"https://zbmath.org/authors/?q=ai:wang.xu.8|wang.xu.5|wang.xu.3|wang.xu.1|wang.xu|wang.xu.7Summary: We investigate the frame set of regular multivariate Gaussian Gabor frames using methods from Kähler geometry such as Hörmander's \({\overline{\partial}}\)-\(L^2\) estimate with singular weight, Demailly's Calabi-Yau method for Kähler currents [\textit{J.-P. Demailly}, Lect. Notes Math. 1507, 87--104 (1992; Zbl 0784.32024)] and a Kähler-variant generalization of the symplectic embedding theorem of McDuff-Polterovich [\textit{D. McDuff} and \textit{L. Polterovich}, Invent. Math. 115, No. 3, 405--429 (1994; Zbl 0833.53028)] for ellipsoids. Our approach is based on the well-known link between sets of interpolation for the Bargmann-Fock space and the frame set of multivariate Gaussian Gabor frames. We state sufficient conditions in terms of a certain extremal type Seshadri constant of the complex torus associated to a lattice to be a set of interpolation for the Bargmann-Fock space, and give also a condition in terms of the generalized Buser-Sarnak invariant of the lattice. In particular, we obtain an effective Gaussian Gabor frame criterion in terms of the covolume for almost all lattices, which is the first general covolume criterion in multivariate Gaussian Gabor frame theory. The recent Berndtsson-Lempert method [\textit{B. Berndtsson} and \textit{L. Lempert}, J. Math. Soc. Japan 68, No. 4, 1461--1472 (2016; Zbl 1360.32006)] and the Ohsawa-Takegoshi extension theorem [\textit{T. Ohsawa} and \textit{K. Takegoshi}, Math. Z. 195, 197--204 (1987; Zbl 0625.32011)] also allow us to give explicit estimates for the frame bounds in terms of certain Robin constant. In the one-dimensional case we obtain a sharp estimate of the Robin constant using Faltings' theta metric formula [\textit{G. Faltings}, Ann. Math. (2) 119, 387--424 (1984; Zbl 0559.14005)] for the Arakelov Green functions.Chaos for convolution operators on the space of entire functions of infinitely many complex variableshttps://zbmath.org/1517.470122023-09-22T14:21:46.120933Z"Caraballo, Blas M."https://zbmath.org/authors/?q=ai:caraballo.blas-melendez"Favaro, Vinicius V."https://zbmath.org/authors/?q=ai:favaro.vinicius-vieiraSummary: In sharp contrast to a classical result of \textit{G. Godefroy} and \textit{J. H. Shapiro} [J. Funct. Anal. 98, No. 2, 229--269 (1991; Zbl 0732.47016)], Mujica and the second author showed that no translation operator on the space \(\mathcal{H}(\mathbb{C}^\mathbb{N})\) of entire functions of infinitely many complex variables is hypercyclic [\textit{V. V. Fávaro} and \textit{J. Mujica}, J. Oper. Theory 76, No. 1, 141--158 (2016; Zbl 1399.47038)]. In an attempt to better understand the dynamics of such operators, in this work we show, firstly, that no convolution operator on \(\mathcal{H}(\mathbb{C}^\mathbb{N})\) is cyclic or \(n\)-supercyclic for any positive integer \(n\). In the opposite direction, we show that every non-trivial convolution operator on \(\mathcal{H}(\mathbb{C}^\mathbb{N})\) is mixing. Particularizing Arai's concept of Li-Yorke chaos to non-metrizable topological vector spaces, we show that non-trivial convolution operators on \(\mathcal{H}(\mathbb{C}^\mathbb{N})\) are also Li-Yorke chaotic.Operator theory on noncommutative polydomains. IIhttps://zbmath.org/1517.470162023-09-22T14:21:46.120933Z"Popescu, Gelu"https://zbmath.org/authors/?q=ai:popescu.geluThis paper continues the study of noncommutative polydomains and their universal operator models generated by admissible \(k\)-tuples of formal power series in several noncommuting indeterminates. As was announced in Part~I [\textit{G. Popescu}, Complex Anal. Oper. Theory 16, No. 4, Paper No. 50, 101 p. (2022; Zbl 1516.47024)], the results obtained there will be used here to develop a dilation theory for non-pure elements in admissible noncommutative polydomains and to obtain a complete description for the invariant subspaces of the corresponding universal operator models.
The paper starts with recalling some necessary notions and notations made in the first part, and introducing other ones. So, for the \(k\)-tuple \(\mathbf{g}=(\mathbf{g}_1,\dots,\mathbf{g}_k)\) of free holomorphic functions in a neighborhood of the origin in the operator ball \(B(\mathcal{H})^{n_i}\), the Hilbert space \(F^2(\mathbf{g})\) of formal power series is attached. \(\mathcal{M}_{\mathbf{g}}(\mathcal{H})\) is the noncommutative set of all \(k\)-tuples \(X=(X_1,\dots,X_k)\) in \(B(\mathcal{H})^{n_1}\times\cdots\times B(\mathcal{H})^{n_k}\), with \(X_i = (X_{i,1}, \dots, X_{i,n_i})\). The defect operator \(\Delta_{\mathbf{g}^{-1}}(X,X^*)\) is defined and a pure noncommutative polydomain \(\mathcal{D}^{\mathrm{pure}}_{\mathbf{g}^{-1}}(\mathcal{H})\) is a subset of \(B(\mathcal{H})^{n_1+\dots+n_k}\) as the set of pure solutions of operator inequation given by the defect operator \(\Delta_{\mathbf{g}^{-1}}(X,X^*)\ge0\). The universal model \(\mathbf{W}\) for \(X\) is defined and the pure part and the Cuntz part \(\mathcal{D}^{c}_{\mathbf{g}^{-1}}(\mathcal{H})\) in the noncommutative polydomain \(\mathcal{D}_{\mathbf{g}^{-1}}(\mathcal{H})\) are introduced. The role of universal operator model associated with \(\mathbf{g}\) will be played by a \(k\)-tuple \(\mathbf{W} = (\mathbf{W}_1,\dots,\mathbf{W}_k)\) with \(\mathbf{W}_i := (\mathbf{W}_{i,1},\dots,\mathbf{W}_{i,n_i})\), where \(\mathbf{W}_{i,j}\) are weighted left creation operators acting on the tensor products \(F^2(H_{n_1})\otimes \dots\otimes F^2(H_{n_k})\), where \(F^2(H_{n_i})\) is the full Fock space with \(n_i\) generators. A condition for \(\mathcal{D}^{\mathrm{pure}}_{\mathbf{g}^{-1}}(\mathcal{H})=\mathcal{M}_{\mathbf{g}}(\mathcal{H})\) was found, and in this case \(\mathcal{D}^{\mathrm{pure}}_{\mathbf{g}^{-1}}(\mathcal{H})\) is called an admissible polydomain and \(\mathbf{g}=(\mathbf{g}_1,\dots,\mathbf{g}_k)\) is an admissible \(k\)-tuple for operator model theory. Also, some examples of classes of \(k\)-tuples of formal power series admissible for operator model theory are recalled.
The second section of the paper is devoted to \(C^*\)-algebras associated with noncommutative polydomains and Wold decompositions. It is shown that there is a unique minimal nontrivial two-sided ideal of the \(C^*\)-algebra \(C^*(\mathbf{W})\) generated by the operators \(\mathbf{W}_{i,j}\) and the identity, namely, the ideal \(\mathcal{K}\) of all compact operators in \(B(\bigotimes_{s=1}^k F^2(H_{n_s}))\). A geometric version of the Wold decomposition for unital \(*\)-representations of the \(C^*\)-algebra \(C^*(W)\) is obtained. This extends the corresponding result previous obtained by the author in this more general setting.
In the third section, the notions of completely non-pure \(k\)-tuple \(X\) from \(\mathcal{D}_{\mathbf{g}^{-1}}(\mathcal{H})\), pure and completely non-pure representation \(\pi\), and Cuntz-type representation are defined. The Cuntz-type algebra \(\mathcal{O}(\mathbf{g})\) is introduced as the universal \(C^*\)-algebra generated by \(\pi(\mathbf{W}_{i,s})\) and the identity, where \(\pi\) is a completely non-pure \(*\)-representation of \(C^*(\mathbf{W})\). Using the Wold decompositions from the previous section, an exact sequence of \(C^*\)-algebras generalizing the well-known results obtained by \textit{L. A. Coburn} for the unilateral shift [Bull. Am. Math. Soc. 73, 722--726 (1967; Zbl 0153.16603)] and by \textit{J. Cuntz} for the left creation operators [Commun. Math. Phys. 57, 173--185 (1977; Zbl 0399.46045)] on the full Fock space with \(n\) generators is obtained.
In the fourth section, it is shown that, for any admissible noncommutative polydomain \(\mathcal{D}_{\mathbf{g}^{-1}}\), the corresponding universal model \(\mathbf{W} = (\mathbf{W}_1,\dots,\mathbf{W}_k)\) with \(\mathbf{W}_i := (\mathbf{W}_{i,1},\dots,\mathbf{W}_{i,n_i})\) admits a Beurling-type characterization of the joint invariant subspaces of \(\mathbf{W}_{i,j}\). In the fifth section, a~functional model for the pure elements in admissible noncommutative polydomains is provided. To each \(X\) in \(\mathcal{D}^{\mathrm{pure}}_{\mathbf{g}^{-1}}(\mathcal{H})\), a~characteristic function \(\Theta_{\mathbf{g},X} := (\Theta^{(1)}_{\mathbf{g},X},\dots, \Theta^{(k)}_{\mathbf{g},X})\) is associated, which consists of partially isometric multi-analytic operators, and it is shown that \(X\) is unitarily equivalent to \(G = (G_1,\dots, G_n)\) with \(G_i := (G_{i,1},\dots,G_{i,n_i})\) of an appropriate form. It is also proved that, if \(T\in \mathcal{D}^{\mathrm{pure}}_{\mathbf{g}^{-1}}(\mathcal{H})\) and \(T'\in \mathcal{D}^{\mathrm{pure}}_{\mathbf{g}^{-1}}(\mathcal{H}')\), then \(T\) and \(T'\) are unitarily equivalent if and only if their characteristic functions \(\Theta_{\mathbf{g},T}\) and \(\Theta_{\mathbf{g},T'}\) coincide in a certain sense.
The dilation theory on admissible noncommutative polydomains is analyzed in the last section of the paper. For an admissible tuple \(\mathbf{g}\), it is proved that, if \(X\in \overline{\mathcal{D}^{\mathrm{pure}}_{\mathbf{g}^{-1}}(\mathcal{H})}\), then there is a \(*\)-representation \(\pi : C^*(\mathbf{W}) \rightarrow\mathcal{K}_\pi\) on a separable Hilbert space \(\mathcal{K}_\pi\) such that \(\pi\) annihilates the compact operators in \(C^*(\mathbf{W})\) and \(\Delta_{\mathbf{g}^{-1}}(\pi(\mathbf{W}), \pi(\mathbf{W})^*) = 0\), and \(\mathcal{H}\) can be identified with a \(*\)-cyclic co-invariant subspace of \(\mathcal{K} := (\bigotimes^k_{s=1}F^2(H_{n_s} ) \otimes\mathcal{D})\oplus\mathcal{K}_\pi\) under the operators \((\mathbf{W}_{i,j}\otimes I_{\mathcal{D}})\oplus\pi(\mathbf{W}_{i,j})\) such that \(X^*_{i,j}=[(\mathbf{W}_{i,j}\otimes I_{\mathcal{D}})\oplus\pi(\mathbf{W}_{i,j})]_{\mathcal{H}}\) for \(i\in \{1,\dots,k\}\), \(j \in \{1,\dots,n_i\}\), where \(\mathcal{D} := \overline{\Delta_{\mathbf{g}^{-1}} (X, X^*)\mathcal{H}}\). The uniqueness of the dilation is also analyzed.
Reviewer: Ilie Valuşescu (Bucureşti)Joint Carleson measure for the difference of composition operators on the polydiskshttps://zbmath.org/1517.470412023-09-22T14:21:46.120933Z"Koo, Hyungwoon"https://zbmath.org/authors/?q=ai:koo.hyungwoon"Park, Inyoung"https://zbmath.org/authors/?q=ai:park.inyoung"Wang, Maofa"https://zbmath.org/authors/?q=ai:wang.maofaSummary: In [\textit{H. W. Koo} and \textit{M. F. Wang}, J. Math. Anal. Appl. 419, No. 2, 1119--1142 (2014; Zbl 1294.47038)], the first and the third author introduced a concept of joint Carleson measure and used it to characterize when the difference of two composition operators on weighted Bergman space over the unit ball is bounded or compact. In this paper, we extend the concept of joint Carleson measure to the polydisk setting and obtain analogue characterizations of the boundedness (compactness, resp.)\ of the difference of two composition operators on the weighted Bergman spaces over the unit polydisk, which may provide a unified approach for various ad hoc studies on the boundedness or the compactness of the difference of composition operators on polydisk. Moreover, we construct a concrete example to show that both the boundedness and the compactness depend on the index \(p\) when the dimension \(n \geq 2\), which is in sharp contrast with the one-variable case where the boundedness and the compactness of the difference of two composition operators are independent of \(p>0\). Due to the complexity of the Carleson measure on the unit polydisk, some new techniques are required in the polydisk setting.Components of the space of weighted composition operators between different Fock spaces in several variableshttps://zbmath.org/1517.470422023-09-22T14:21:46.120933Z"Le Hai Khoi"https://zbmath.org/authors/?q=ai:le-hai-khoi."Le Thi Hong Thom"https://zbmath.org/authors/?q=ai:le-thi-hong-thom."Pham Trong Tien"https://zbmath.org/authors/?q=ai:pham-trong-tien.Summary: In this paper, we completely solve the topological structure problem for the space of nonzero weighted composition operators acting between Fock spaces \(\mathcal{F}^p(\mathbb{C}^n)\) and \(\mathcal{F}^q(\mathbb{C}^n)\) with \(p, q\in(0, \infty]\). All (path) components of this space are explicitly described.Geometric Arveson-Douglas conjecture for the Hardy space and a related compactness criterionhttps://zbmath.org/1517.470532023-09-22T14:21:46.120933Z"Wang, Yi"https://zbmath.org/authors/?q=ai:wang.yi.33"Xia, Jingbo"https://zbmath.org/authors/?q=ai:xia.jingboSuppose that \(\mathcal{N}\) is a either a submodule or a quotient module of a Hilbert module \(\mathcal{H}\). Let \(P_{\mathcal{N}}:\mathcal{H}\to\mathcal{N}\) be the orthogonal projection and define \(\mathcal{Z}_{\mathcal{N},j}= P_{\mathcal{N}}M_{z_j}|\mathcal{N}\) for \(j=1,\dots,n\). Recall that \(\mathcal{N}\) is said to be \(p\)-essentially normal if all commutators \([\mathcal{Z}_{\mathcal{N},i}^*,\mathcal{Z}_{\mathcal{N},j}]\) with \(1\le i,j\le n\) are in the Schatten class \(\mathcal{C}_p\).
Let \(\mathbb{B}=\{z\in\mathbb{C}^n:|z|<1\}\) and \(\mathbb{S}=\{z\in\mathbb{C}^n:|z|=1\}\), where \(\mathbb{S}\) is equipped with the standard spherical measure \(d\sigma\). The Hardy space \(H^2(\mathbb{S})\) is the closure of the ring of analytic polynomials \(\mathbb{C}[z_1,\dots,z_n]\) in \(L^2(\mathbb{S},d\sigma)\). Let \(\Omega\) be a complex manifold. A~subset \(A\subset\Omega\) is called a complex analytic subset of \(\Omega\) if, for each \(a\in A\), there are a neighborhood \(U\) of \(a\) and functions \(f_1,\dots,f_N\) analytic in this neighborhood such that \(A\cap U=\{z\in U:f_1(z)=\ldots=f_N(z)=0\}\).
Let \(\widetilde{M}\) be an analytic subset of an open neighborhood of \(\overline{\mathbb{B}}\) with \(1\le\dim_{\mathbb{C}}\widetilde{M}\le n-1\) and \(M=\overline{B}\cap\widetilde{M}\). Consider a submodule \(\mathcal{R}=\{f\in H^2(\mathbb{S}):f=0 \text{ on } M\}\) and the corresponding quotient module \(\mathcal{Q}=H^2(\mathbb{S})\ominus\mathcal{R}\).
The authors prove the geometric Arveson-Douglas conjecture saying that the quotient module \(\mathcal{Q}\) is \(p\)-essentially normal for every \(p>d=\dim_{\mathbb{C}}\widetilde{M}\). Further, for a measure \(\mu\) on \(M\), one can define the Toeplitz operator \(T_\mu\) by \((T_\mu h)(z)=\int_M h(w)(1-\langle z,w\rangle)^{-n}\,d\mu(w)\). Let \(Q\) denote the orthogonal projection from \(L^2(\mathbb{S},d\sigma)\) onto \(\mathcal{Q}\). The second main result says that, for constants \(0<c\le C<\infty\), there exists a measure \(\mu\) on \(M\) such that \(cQ\le T_\mu\le CQ\) on \(L^2(\mathbb{S},d\sigma)\). For each \(f\in L^\infty(\mathbb{S},d\sigma)\), define \(Q_f=QM_f|\widetilde{Q}\). Let \(\mathcal{TQ}\) be the \(C^*\)-algebra generated by \(\{Q_f:f\in L^\infty(\mathbb{S},d\sigma)\}\). The third main result of the paper says that, if \(A\in\mathcal{TQ}\) and \(\lim_{z\in M,|z|\to 1}\langle Ak_z,k-z\rangle=0\), where \(k_z\) is the reproducing kernel for \(H^2(\mathbb{S})\), then \(A\) is a compact operator.
Reviewer: Oleksiy Karlovych (Lisboa)Embedding derivatives and integration operators on Hardy type tent spaceshttps://zbmath.org/1517.470732023-09-22T14:21:46.120933Z"Wang, Mao Fa"https://zbmath.org/authors/?q=ai:wang.maofa"Zhou, Lv"https://zbmath.org/authors/?q=ai:zhou.lvSummary: In this paper, we completely characterize the positive Borel measures \(\mu\) on the unit ball \(\mathbb{B}_n\) such that the differential type operator \(\mathcal{R}^m\) of order \(m \in \mathbb{N}\) is bounded from Hardy type tent space \(\mathcal{HT}_{q,\alpha}^p (\mathbb{B}_n)\) into \(L^s (\mu)\) for full range of \(p, q, s, \alpha\). Subsequently, the corresponding compact description of differential type operator \(\mathcal{R}^m\) is also characterized. As an application, we obtain the boundedness and compactness of integration operator \(J_g\) from \(\mathcal{HT}_{q,\alpha}^p (\mathbb{B}_n)\) to \(\mathcal{HT}_{s,\beta}^t (\mathbb{B}_n)\), and the methods used here are adaptable to the Hardy spaces.Twistors, quartics, and del Pezzo fibrationshttps://zbmath.org/1517.530012023-09-22T14:21:46.120933Z"Honda, Nobuhiro"https://zbmath.org/authors/?q=ai:honda.nobuhiroIn this well-written research monograph the author studies Moishezon twistor spaces on connected sums of complex projective planes.
Let us recall the notation and basic definitions before formulating the main result. Let \(M\) be a \(4\)-manifold equipped with a self-dual conformal structure and let \(Z\) be the associated twistor space. Write \(\pi : Z \rightarrow M\) for the projection. The fibers of \(\pi\) are complex submanifolds of \(Z\) and are isomorphic to \(\mathbb{P}^{1}_{\mathbb{C}}\). These lines are called the twistor lines of the twistor space \(Z\). Recall that the anti-canonical bundle \(-K_{Z}\) of \(Z\) is of degree \(4\) over a twistor line. In particular, \(\kappa(Z) = -\infty\) for the Kodaira dimension when \(Z\) is compact. In a neighborhood of a twistor line, there exists a \(4\)-th square root of \(-K_{Z}\) and it exists globally when the base \(4\)-manifold \(M\) admits a spin structure. However, even if \(M\) is not spin, from the construction of a twistor space, there always exists a global and natural square root of \(-K_{Z}\) as a holomorphic line bundle. This is called the vertical line bundle or the fundamental line bundle, and is of degree two over a twistor line. We denote it by \(F(=K_{Z}^{-1/2})\) and we call the complete linear system \(|F|\) the fundamental system. An element of the fundamental system is called a fundamental divisor. The main result is a classification and a description of Moishezon twistor spaces on \(n\mathbb{P}^{2}_{\mathbb{C}}\) with \(n\geq 4\), where \(n\mathbb{P}^{2}_{\mathbb{C}}\) denotes the connected sum of \(n\) copies of \(\mathbb{P}^{2}_{\mathbb{C}}\). It can be formulated as follows.
Theorem. Let \(n\geq 4\) and \(Z\) be a Moishezon twistor space on \(n\mathbb{P}^{2}_{\mathbb{C}}\). Suppose that \(\dim |F| = 1\) and \(Z\) is not a generalized LeBrun space. Then there exists \(m\geq 2\) such that the pluri-system \(|mF|\) includes an \((m+2)\)-dimensional sub-system whose meromorphic map \(\Phi : Z \rightarrow \mathbb{P}^{m+2}_{\mathbb{C}}\) satisfies the following properties:
(i) The image \(\Phi(Z)\) is a scroll of planes over a rational normal curve in \(\mathbb{P}^{m}_{\mathbb{C}}\);
(ii) The meromorphic map \(\Phi: Z \rightarrow \Phi(Z)\) is two-to-one and the branch divisor is a cut of the scroll by a quartic hypersurface in \(\mathbb{P}^{m+2}_{\mathbb{C}}\);
(iii) The quartic hypersurface is defined by an equation of the form
\[
h_{1}h_{2}h_{3}h_{4} = Q^{2},
\]
where \(h_{i}\) are linear and \(Q\) is quadratic.
Furthermore, the integer \(m\) necessarily satisfies \(m \geq n-2\).
Reviewer: Piotr Pokora (Kraków)On holomorphic isometries into blow-ups of \(\mathbb{C}^n\)https://zbmath.org/1517.530502023-09-22T14:21:46.120933Z"Loi, Andrea"https://zbmath.org/authors/?q=ai:loi.andrea"Mossa, Roberto"https://zbmath.org/authors/?q=ai:mossa.robertoSummary: We study the Kähler-Einstein manifolds which admits a holomorphic isometry into either the generalized Burns-Simanca manifold \((\tilde{\mathbb{C}}^n, g_S)\) or the Eguchi-Hanson manifold \((\tilde{\mathbb{C}}^2, g_{EH})\). Moreover, we prove that \((\tilde{\mathbb{C}}^n, g_S)\) and \((\tilde{\mathbb{C}}^2, g_{EH})\) are not relatives to any homogeneous bounded domain.Modular classes of Jacobi bundleshttps://zbmath.org/1517.530742023-09-22T14:21:46.120933Z"Diallo, Mamadou Lamarana"https://zbmath.org/authors/?q=ai:diallo.mamadou-lamarana"Wade, Aïssa"https://zbmath.org/authors/?q=ai:wade.aissaThis paper introduces and studies modular classes of Jacobi structures from the viewpoint of line bundles. The authors prove a one-to-one correspondence between Jacobi algebroids and Gerstenhaber-Jacobi algebras. They also prove that every nowhere vanishing Atiyah form of top degree gives rise to a generator of the Lie bracket of the Gerstenhaber-Jacobi algebra associated with the Atiyah algebroid \(DL\) of a line bundle \(L\to M\). Finally they study modular classes of Jacobi manifolds.
Reviewer: Miroslav Doupovec (Brno)Scalar curvature and deformations of complex structureshttps://zbmath.org/1517.530762023-09-22T14:21:46.120933Z"Scarpa, Carlo"https://zbmath.org/authors/?q=ai:scarpa.carlo|scarpa.carlo.1Let \((M,\omega)\) be a compact symplectic \(2n\)-dimensional manifold and denote by \(\mathcal{J}\) the set of all integrable complex structures on \(M\) that are compatible with \(\omega\). This set is actually a space of sections of a bundle over \(M\) and its fibres are all isomorphic to the Hermitian symmetric space \(\mathrm{Sp}(2n)/\mathrm{U}(n)\); hence it is an infinite-dimensional locally symmetric Kähler manifold. It is known that the group \(\mathcal{G}\) of exact symplectomorphisms of \((M,\omega)\) acts by pullbacks on \(\mathcal{J}\) and the action is Hamiltonian. A moment map is then defined by assigning to each complex structure \(J\in\mathcal{J}\) the scalar curvature of the Kähler metric \(g_J\) defined by \(\omega\) and \(J\).
In the present paper, the author considers the total space of the holomorphic cotangent bundle \(T^*(\mathcal{J})\). This space has a canonical holomorphic symplectic form \(\Theta\), given by the differential of the tautological 1-form on the cotangent bundle. It is invariant under the \(\mathcal{G}\)-action on \(T^*(\mathcal{J})\), and the action is also Hamiltonian with respect to \(\Theta\). The moment map is given by the adjoint of the Lichnerowicz operator of the Kähler metric \(g_J\) and for any choice of Kähler form coming from the Calabi Ansatz metrics on \(T^*(\mathcal{J})\), the author studies a system of two moment maps equations: both of them come from an infinite-dimensional Kähler reduction which is a hyper-Kähler reduction for a particular choice of the spectral function. The system is studied using a flat connection on the space of first-order deformations of the complex structure; a formal complexification of the moment map equations is then obtained. Using this connection, the author describes:
(a) A variational characterization of the equations;
(b) A Futaki invariant for the system;
(c) A generalisation of the notion of \(K\)-stability that is conjectured to characterize the existence of solutions.
After this, the author reasonably conjectures that the space of solutions of the system of moment map equations can be used to study the (hypothetical) moduli space of polarized manifolds together with a class of first-order deformations of the complex structure.
Reviewer: Ioannis D. Platis (Heraklion)Pseudo-holomorphic dynamics in the restricted three-body problemhttps://zbmath.org/1517.700152023-09-22T14:21:46.120933Z"Moreno, Agustin"https://zbmath.org/authors/?q=ai:moreno.agustin-sAuthor's abstract: In this paper, the author identifies the five-dimensional analogue of the finite energy Hofer-Wysocki-Zehnder foliations for the study of three dimensional Reeb flows, and shows that these exist for the spatial circular restricted three-body problem (SCR3BP) whenever the planar dynamic is convex. He introduces the notion of a fiberwise-recurrent point, which may be thought of as a symplectic version of the leafwise intersections introduced by \textit{J. Moser} [Acta Math. 141, 17--34 (1978; Zbl 0382.53035)], and shows that they exist in abundance for a perturbative regime in the SCR3BP. Then he uses this foliation to induce a Reeb flow on the standard 3-sphere, via the use of pseudoholomorphic curves, to be understood as the best approximation of the given dynamics that preserves the foliation. He discusses examples, further geometric structures, and speculates on possible applications.
Reviewer: Maria Gousidou-Koutita (Thessaloniki)Feral curves and minimal setshttps://zbmath.org/1517.700292023-09-22T14:21:46.120933Z"Fish, Joel"https://zbmath.org/authors/?q=ai:fish.joel-w"Hofer, Helmut"https://zbmath.org/authors/?q=ai:hofer.helmut-h-wSummary: We prove that for each Hamiltonian function \(H\in\mathcal{C}^\infty(\mathbb{R}^4,\mathbb{R})\) defined on the standard symplectic \((\mathbb{R}^4,\omega_0)\), for which \(M:=H^{-1}(0)\) is a non-empty compact regular energy level, the Hamiltonian flow on \(M\) is not minimal. That is, we prove that there exists a closed invariant subset of the Hamiltonian flow in \(M\) that is neither \(\emptyset\) nor all of \(M\). This answers the four-dimensional case of a more than twenty year old question of \textit{M. R. Herman} [NATO ASI Ser., Ser. C, Math. Phys. Sci. 533, 126 (1999; Zbl 0955.37502)], part of which can be regarded as a special case of the Gottschalk conjecture.
Our principal technique is the introduction and development of a new class of pseudoholomorphic curves in the ``symplectization'' \(\mathbb{R}\times M\) of framed Hamiltonian manifolds \((M,\lambda,\omega)\). We call these \textit{feral} curves because they are allowed to have infinite (so-called) Hofer energy, and hence may limit to invariant sets more general than the finite union of periodic orbits. Standard pseudoholomorphic curve analysis is inapplicable without energy bounds, and thus much of this paper is devoted to establishing properties of feral curves, such as area and curvature estimates, energy thresholds, compactness, asymptotic properties, etc.Riesz potential for \((k,1)\)-generalized Fourier transformhttps://zbmath.org/1517.810522023-09-22T14:21:46.120933Z"Ivanov, Valeriĭ Ivanovich"https://zbmath.org/authors/?q=ai:ivanov.valerii-ivanovichSummary: In spaces with weight \(|x|^{-1}v_k(x)\), where \(v_k(x)\) is the Dunkl weight, there is the \((k,1)\)-generalized Fourier transform. Harmonic analysis in these spaces is important, in particular, in problems of quantum mechanics. We define the Riesz potential for the \((k,1)\)-generalized Fourier transform and prove for it, a \((L^q,L^p)\)-inequality with radial power weights, which is an analogue of the well-known Stein-Weiss inequality for the classical Riesz potential. For the Riesz potential we calculate the sharp value of the \(L^p\)-norm with radial power weights. The sharp value of the \(L^p\)-norm with radial power weights for the classical Riesz potential was obtained independently by W. Beckner and S. Samko.Semiclassical asymptotics of oscillating tunneling for a quadratic Hamiltonian on the algebra \(\operatorname{su}(1,1)\)https://zbmath.org/1517.810562023-09-22T14:21:46.120933Z"Vybornyi, E. V."https://zbmath.org/authors/?q=ai:vybornyi.e-v"Rumyantseva, S. V."https://zbmath.org/authors/?q=ai:rumyantseva.s-vSummary: In this paper, we consider the problem of constructing semiclassical asymptotics for the tunnel splitting of the spectrum of an operator defined on an irreducible representation of the Lie algebra \(\operatorname{su}(1,1)\). It is assumed that the operator is a quadratic function of the generators of the algebra. We present coherent states and a unitary coherent transform that allow us to reduce the problem to the analysis of a second-order differential operator in the space of holomorphic functions. Semiclassical asymptotic spectral series and the corresponding wave functions are constructed as decompositions in coherent states. For some values of the system parameters, the minimal energy corresponds to a pair of nondegenerate equilibria, and the discrete spectrum of the operator has an exponentially small tunnel splitting of the levels. We apply the complex WKB method to prove asymptotic formulas for the tunnel splitting of the energies. We also show that, in contrast to the one-dimensional Schrödinger operator, the tunnel splitting in this problem not only decays exponentially but also contains an oscillating factor, which can be interpreted as tunneling interference between distinct instantons. We also show that, for some parameter values, the tunneling is completely suppressed and some of the spectral levels are doubly degenerate, which is not typical of one-dimensional systems.Towards super Teichmüller spin TQFThttps://zbmath.org/1517.810732023-09-22T14:21:46.120933Z"Aghaei, Nezhla"https://zbmath.org/authors/?q=ai:aghaei.nezhla"Pawelkiewicz, M. K."https://zbmath.org/authors/?q=ai:pawelkiewicz.m-k"Yamazaki, Masahito"https://zbmath.org/authors/?q=ai:yamazaki.masahitoSummary: The quantization of the Teichmüller theory has led to the formulation of the so-called Teichmüller TQFT for 3-manifolds. In this paper we initiate the study of ``supersymmetrization'' of the Teichmüller TQFT, which we call the super Teichmüller spin TQFT. We obtain concrete expressions for the partition functions of the super Teichmüller spin TQFT for a class of spin 3-manifold geometries, by taking advantage of the recent results on the quantization of the super Teichmüller theory. We then compute the perturbative expansions of the partition functions, to obtain perturbative invariants of spin 3-manifolds. We also comment on the relations of the super Teichmüller spin TQFT to 3-dimensional Chern-Simons theories with complex gauge groups, and to a class of 3d \(\mathcal{N}=2\) theories arising from the compactifications of the M5-branes.Relational analysis of Dirac equation in momentum representationhttps://zbmath.org/1517.830212023-09-22T14:21:46.120933Z"Solov'yov, Anton V."https://zbmath.org/authors/?q=ai:solovyov.anton-vSummary: In terms of the relational approach to space-time geometry and physical interactions, we show that the Dirac equation for a free fermion in the momentum representation can be obtained starting from a \textit{binary system of complex relations} (BSCR) between elements of two abstract sets. With the derivation performed, we show that the 4-dimensional pseudo-Euclidean momentum space is not needed \textit{a priori} but naturally emerges from considerations of rather general nature (2-spinor algebra). A bispinor wave function is constructed for a fermion with positive energy and an arbitrary distribution of momenta. Special attention is paid to physical assumptions that should be made to enable the construction.Weakly isolated horizons: \(3+1\) decomposition and canonical formulations in self-dual variableshttps://zbmath.org/1517.830342023-09-22T14:21:46.120933Z"Corichi, Alejandro"https://zbmath.org/authors/?q=ai:corichi.alejandro"Reyes, Juan D."https://zbmath.org/authors/?q=ai:reyes.juan-d"Vukašinac, Tatjana"https://zbmath.org/authors/?q=ai:vukasinac.tatjanaSummary: The notion of Isolated Horizons has played an important role in gravitational physics, being useful from the characterization of the endpoint of black hole mergers to (quantum) black hole entropy. In particular, the definition of weakly isolated horizons (WIHs) as quasilocal generalizations of event horizons is purely geometrical, and is independent of the variables used in describing the gravitational field. Here we consider a canonical decomposition of general relativity in terms of connection and vierbein variables starting from a first order action. Within this approach, the information about the existence of a (weakly) isolated horizon is obtained through a set of boundary conditions on an internal boundary of the spacetime region under consideration. We employ, for the self-dual action, a generalization of the Dirac algorithm for regions with boundary. While the formalism for treating gauge theories with boundaries is unambiguous, the choice of dynamical variables on the boundary is not. We explore this freedom and consider different canonical formulations for non-rotating black holes as defined by WIHs. We show that both the notion of horizon degrees of freedom and energy associated to the horizon is not unique, even when the descriptions might be self-consistent. This represents a generalization of previous work on isolated horizons both in the exploration of this freedom and in the type of horizons considered. We comment on previous results found in the literature.On the uniqueness of supersymmetric AdS(5) black holes with toric symmetryhttps://zbmath.org/1517.830452023-09-22T14:21:46.120933Z"Lucietti, James"https://zbmath.org/authors/?q=ai:lucietti.james"Ntokos, Praxitelis"https://zbmath.org/authors/?q=ai:ntokos.praxitelis"Ovchinnikov, Sergei G."https://zbmath.org/authors/?q=ai:ovchinnikov.sergei-gSummary: We consider the classification of supersymmetric \(\mathrm{AdS}_5\) black hole solutions to minimal gauged supergravity that admit a torus symmetry. This problem reduces to finding a class of toric Kähler metrics on the base space, which in symplectic coordinates are determined by a symplectic potential. We derive the general form of the symplectic potential near any component of the horizon or axis of symmetry, which determines its singular part for any black hole solution in this class, including possible new solutions such as black lenses and multi-black holes. We find that the most general known black hole solution in this context, found by \textit{Z.-W. Chong} et al. [Phys. Rev. Lett. 95, No. 16, Article ID 161301, 4 p. (2005; \url{doi:10.1103/PhysRevLett.95.161301})] (CCLP), is described by a remarkably simple symplectic potential. We prove that any supersymmetric and toric solution that is timelike outside a smooth horizon, with a Kähler base metric of Calabi type, must be the CCLP black hole solution or its near-horizon geometry.Topology of Born-Infeld AdS black holes in 4D novel Einstein-Gauss-Bonnet gravityhttps://zbmath.org/1517.830532023-09-22T14:21:46.120933Z"Yerra, Pavan Kumar"https://zbmath.org/authors/?q=ai:yerra.pavan-kumar"Bhamidipati, Chandrasekhar"https://zbmath.org/authors/?q=ai:bhamidipati.chandrasekharSummary: The topological classification of critical points of black holes in 4D Einstein-Gauss-Bonnet gravity coupled to Born-Infeld theory is investigated. Considered independently, Born-infeld corrections to the Einstein action alter the topological charge of critical points of the charged AdS black hole system, whereas the Gauss-Bonnet corrections do not. For the combined system though, the topological charge of the Einstein-Gauss-Bonnet theory is unaltered in the presence of Born-Infeld coupling.On-shell action for type IIB supergravity and superstrings on \(AdS_5 \times S^5\)https://zbmath.org/1517.830672023-09-22T14:21:46.120933Z"Chakrabarti, Subhroneel"https://zbmath.org/authors/?q=ai:chakrabarti.subhroneel"Gupta, Divyanshu"https://zbmath.org/authors/?q=ai:gupta.divyanshu"Manna, Arkajyoti"https://zbmath.org/authors/?q=ai:manna.arkajyotiSummary: AdS/CFT predicts that the value of the on-shell action for type IIB Supergravity (SUGRA) on \(AdS_5 \times S^5\) background must be a non-zero number completely determined from the boundary theory. We examine this statement within Sen's formalism for type IIB SUGRA and find that consistency with AdS/CFT requires us to add a specific boundary term to the action. We contrast our resolution with two other resolutions recently proposed in the literature in the context of different approaches to type IIB SUGRA. We explain how our resolution presents a strong benchmark for the possible boundary term of the complete spacetime action for type IIB superstring and how it may possibly lead to a piece of evidence for the strongest form of AdS/CFT conjecture in \(AdS_5 \times S^5\). We also comment on the fate of the on-shell action for general self-dual \(p\)-form fields in Sen's formalism in any curved backgrounds.Yang-Mills solutions on Minkowski space via non-compact coset spaceshttps://zbmath.org/1517.830812023-09-22T14:21:46.120933Z"Kumar, Kaushlendra"https://zbmath.org/authors/?q=ai:kumar.kaushlendra"Lechtenfeld, Olaf"https://zbmath.org/authors/?q=ai:lechtenfeld.olaf"Costa, Gabriel Picanço"https://zbmath.org/authors/?q=ai:costa.gabriel-picanco"Röhrig, Jona"https://zbmath.org/authors/?q=ai:rohrig.jonaSummary: We find a two-parameter family of solutions of the Yang-Mills equations for gauge group \(\mathrm{SO}(1, 3)\) on Minkowski space by foliating different parts of it with non-compact coset spaces with \(\mathrm{SO}(1, 3)\) isometry. The interior of the lightcone is foliated with hyperbolic space \(H^3\cong\mathrm{SO}(1, 3)/\mathrm{SO}(3)\), while the exterior of the lightcone employs de Sitter space \(\mathrm{dS}_3\cong\mathrm{SO}(1, 3)/\mathrm{SO}(1, 2)\). The lightcone itself is parametrized by \(\mathrm{SO}(1, 3)/\mathrm{ISO}(2)\) in a nilpotent fashion. Equivariant reduction of the \(\mathrm{SO}(1,3)\) Yang-Mills system on the first two coset spaces yields a mechanical system with inverted double-well potential and the foliation parameter serving as an evolution parameter. Its known analytic solutions are periodic or runaway except for the kink. On the lightcone, only the vacuum solution remains. The constructed Yang-Mills field strength is singular across the lightcone and of infinite action due to the noncompact cosets. Its energy-momentum tensor takes a very simple form, with energy density of opposite signs inside and outside the lightcone.Singularity scattering laws for bouncing cosmologies: a brief overviewhttps://zbmath.org/1517.830842023-09-22T14:21:46.120933Z"LeFloch, Philippe G."https://zbmath.org/authors/?q=ai:lefloch.philippe-gSummary: For contracting/expanding bouncing cosmologies, the formulation of junction conditions at a bouncing was recently revisited by the author in collaboration with B. Le Floch and G. Veneziano. The regime of interest here is the so-called quiescent regime, in which a monotone behavior of the metric is observed and asymptotic expansions can be derived. Here, we overview our new methodology based on the notion of \textit{singularity scattering maps} and \textit{cyclic spacetimes}, and we present our main conclusions. In particular, we provide a classification of all allowed bouncing junction conditions, including \textit{three universal laws}.
For the entire collection see [Zbl 1497.53003].Matter and space. New theory of fields and particleshttps://zbmath.org/1517.830892023-09-22T14:21:46.120933Z"Zhuravlev, V. M."https://zbmath.org/authors/?q=ai:zhuravlev.viktor-mikhailovichSummary: The paper presents a theory giving a unified geometric description of space and matter on the basis of a new concept related to general relativity (GR). The theory is built on the basis of a critical analysis of GR. The principle of materiality of space is introduced. The description of matter is based on the idea of space as a three-dimensional material hypersurface embedded in a four-dimensional Euclidean space. Matter particles are associated with extended areas of the material hypersurface, and their properties, such as charge and mass, with topological and geometric properties of this hypersurface. The central place in the mathematical apparatus for describing the material hypersurface itself and matter particles is played by marker fields, which are similar in essence to hydrodynamic markers used in classical hydrodynamics. Based on the theory of marker fields, questions of the topological structure of particles and connection between the electric charge and the topology of a material hypersurface are discussed. The mass of particles is represented as a property of the material hypersurface itself and has the meaning of gravitational and inertial mass at the same time. The fields, gravitational and electromagnetic, are properties of the material hypersurface geometry expressed in terms of marker fields. To describe the dynamics of particles, the geometric principle of averaging is introduced, which, as a result, leads to the equations of Newtonian mechanics and quantum theory.