Recent zbMATH articles in MSC 32Ghttps://zbmath.org/atom/cc/32G2022-07-25T18:03:43.254055ZWerkzeugIntegral morphisms and log blow-upshttps://zbmath.org/1487.140052022-07-25T18:03:43.254055Z"Kato, Fumiharu"https://zbmath.org/authors/?q=ai:kato.fumiharuSummary: This paper is a revision of the author's old preprint ``Exactness, integrality, and log modifications''. We will prove that any quasi-compact morphism of fs log schemes can be modified locally on the base to an integral morphism by base change by fs log blow-ups.Categorical Saito theory. II: Landau-Ginzburg orbifoldshttps://zbmath.org/1487.140062022-07-25T18:03:43.254055Z"Tu, Junwu"https://zbmath.org/authors/?q=ai:tu.junwuThis paper is devoted to the study of equivariant matrix factorization category \(\mathrm{MF}_G(W)\) (under certain technical assumptions on \(G\) and \(W\)) and its associated variation of semi-infinite Hodge structures (VSHS). The author proves the existence of a canonical categorical primitive form, which defines a \(G\)-equivariant version of Saito's theory conjecturally mirror to the genus zero part of the FJRW theory. This mirror symmetry conjecture is verified for a quintic LG orbifold.
The contents in more detail:
Section 1 serves as an introduction to the main content of the paper, where the author gives backgrounds and motivations and also an overview of the paper.
In section 2 the author recalls some basic ingredients in non-commutative Hodge theory, most notably the VSHS structure on negative cyclic homology of saturated Calabi-Yau \(A_\infty\) categories.
In section 3 the author proves one of the main result in the paper: there exist a unique good splitting of the non-commutative Hodge filtration on \(\mathrm{MF}_G(W)\) which is equivariant under maximal diagonal symmetries, and the splitting is also compatible with the natural Calabi-Yau structure. A key idea is to reduce to the commutative case by localization formula of Hochschild invariants, and the commutative theory is handled by Saito's classical theory of primitive forms.
Section 4 is devoted to a (simplest) example given by the Fermat cubic with orbifold group \(G=\mathbb{Z}/3\mathbb{Z}\). Here all computations are done in a schematic and explicit way and the vanishing of higher order genus zero invariants are confirmed.
Section 5 is devoted to the example of quintic families. The restriction of the prepotential to the the marginal deformation parameter matches with the genus zero FJRW prepotential computed by \textit{A. Chiodo} and \textit{Y. Ruan} [Invent. Math. 182, No. 1, 117--165 (2010; Zbl 1197.14043)], proving the LG/LG mirror symmetry result for the quintic orbifold.
Reviewer: Kai Xu (Cambridge)Periods of complete intersection algebraic cycleshttps://zbmath.org/1487.140242022-07-25T18:03:43.254055Z"Villaflor Loyola, Roberto"https://zbmath.org/authors/?q=ai:villaflor-loyola.robertoLet \(X\) be a smooth, projective hypersurface in \(\mathbb{P}^{n+1}\) of degree \(d\) and \(n\) even. Consider a complete intersection subvariety \(Z\) in \(\mathbb{P}^{n+1}\) which is also contained in \(X\). The author computes the periods associated to \(Z\). Villaflor Loyola then uses this result to prove that algebraic cycles in \(X\) of the form \(\delta:= a[V_1]+b[V_2]\) where \(a, b \in \mathbb{Z}\), \(V_i\) are isomorphic to \(\mathbb{P}^{n/2}\) and \(V_1 \cap V_2 \cong \mathbb{P}^m\) for \(m<(n/2)- (d/(d-2))\) satisfy the variational Hodge conjecture i.e., for deformations of \(X\), the cohomology class of the algebraic cycle \(\delta\) remains of type \((n,n)\) if and only if \(\delta\) remains algebraic. The author also proves that the Hodge locus associated to \(\delta\) is smooth, reduced and coincides with the intersection of the Hodge loci associated to \([V_1]\) and \([V_2]\), in an open neighbourhood of the point, say \(o\), corresponding to the Fermat hypersurface. Additionally Villaflor Loyola shows, if one drops the condition on \(m\), then the dimension of the tangent space at \(o\) of the Hodge locus associated to \(\delta\) is of dimension strictly greater than the dimension of the intersection of the Hodge loci associated to \([V_1]\) and \([V_2]\).
Reviewer: Ananyo Dan (Sheffield)Absolute Hodge and \(\ell\)-adic monodromyhttps://zbmath.org/1487.140322022-07-25T18:03:43.254055Z"Urbanik, David"https://zbmath.org/authors/?q=ai:urbanik.davidSummary: Let \(\mathbb{V}\) be a motivic variation of Hodge structure on a \(K\)-variety \(S\), let \(\mathcal{H}\) be the associated \(K\)-algebraic Hodge bundle, and let \(\sigma \in \text{Aut}(\mathbb{C}/K)\) be an automorphism. The absolute Hodge conjecture predicts that given a Hodge vector \(v \in \mathcal{H}_{\mathbb{C}, s}\) above \(s \in S(\mathbb{C})\) which lies inside \(\mathbb{V}_s \), the conjugate vector \(v_{\sigma } \in \mathcal{H}_{\mathbb{C}, s_{\sigma }}\) is Hodge and lies inside \(\mathbb{V}_{s_{\sigma }} \). We study this problem in the situation where we have an algebraic subvariety \(Z \subset S_{\mathbb{C}}\) containing \(s\) whose algebraic monodromy group \(\textbf{H}_Z\) fixes \(v\). Using relationships between \(\textbf{H}_Z\) and \(\textbf{H}_{Z_{\sigma }}\) coming from the theories of complex and \(\ell \)-adic local systems, we establish a criterion that implies the absolute Hodge conjecture for \(v\) subject to a group-theoretic condition on \(\textbf{H}_Z \). We then use our criterion to establish new cases of the absolute Hodge conjecture.Coordinates on the augmented moduli space of convex \(\mathbb{RP}^2\) structureshttps://zbmath.org/1487.140672022-07-25T18:03:43.254055Z"Loftin, John"https://zbmath.org/authors/?q=ai:loftin.john-c"Zhang, Tengren"https://zbmath.org/authors/?q=ai:zhang.tengrenLet \(S\) be a smooth, connected, oriented surface with negative Euler characteristic. A convex \(\mathbb{RP}^2\) structure \(\mu\) on \(S\) is given by a pair \((\phi, \rho)\) where \(\rho : \pi_1(S) \rightarrow \text{PGL}(3, \mathbb{R})\) is a representation and \(\phi : \tilde{S} \rightarrow \Omega\) is a \(\rho\)-equivariant diffeomorphism onto a properly convex domain \(\Omega \subset \mathbb{RP}^2\), called a developing map of \(\mu\). The deformation space of convex \(\mathbb{RP}^2\) structures on \(S\) is denoted by \(\mathcal{C}(S)\), and it consists of the equivalence classes of pairs \((f, \Sigma)\) where \(\Sigma\) is a convex \( \mathbb{RP}^2\) surface and \(f : S \rightarrow \Sigma\) is a diffeomorphism. In Section 2 of the paper under review, a topology in \(\mathcal{C}(S)\) is defined.
The goal of the article is to give a description of the topology of the augmented moduli space of convex real projective structures on \(S\). In order to do that, the authors consider a particular subset \(\mathcal{C}(S)^{\text{adm}} \subset \mathcal{C}(S)\) satisfying an admissibility condition defined on the punctures of \(\Sigma\). Then, the augmented deformation space of admissible convex \(\mathbb{RP}^2\) structures, \(\mathcal{C}(S)^{\text{aug}}\), is described in Subsection 2.3. The following step is to define a topology on \(\mathcal{C}(S)^{\text{aug}}\) (Subsection 2.4), by constructing a basis of the topology. A number of properties of this topology, is stated, for instance it is first countable, and is not locally compact at any point in \(\mathcal{C}(S)^{\text{aug}} \setminus \mathcal{C}(S)\). Then, Section 3 is devoted to describe a system of global coordinates on the image \(\text{hol}(\mathcal{C}(S))\) of the holonomy map from \(\mathcal{C}(S)\) to \(\mathcal{X}(\pi_1(S), \text{PGL}(3,\mathbb{R}))/\text{PGL}(3,\mathbb{R})\). The core of the paper is then Section 4, in which the main Theorem 1.1 is proved. This result states that the coordinates indicated above describe the topology on \(\mathcal{C}(S)^{\text{aug}}/\text{MCG}(S)\), where \(\text{MCG}(S)\) is the mapping class group of \(S\).
The article ends with Section 5, in which the topology on \(\mathcal{C}(S)^{\text{aug}}/\text{MCG}(S)\) is related to regular cubic differentials, proving that there is a natural homeomorphism between \(\mathcal{C}(S)^{\text{aug}}/\text{MCG}(S)\) and \(\mathcal{K}(S)^{\text{reg}}\), the orbifold vector bundle of regular cubic differentials over the Deligne-Mumford compactification of the moduli space of Riemann surfaces homeomorphic to \(S\), \(\overline{\mathcal{M}(S)}\). Finally, in an Appendix, it is studied how the image of the developing map of two regular convex \(\mathbb{RP}^2\) structures on \(S\) with the same holonomy can differ.
All the constructions are quite complicated, but the article is very carefully written, and provides all the details necessary to the understanding.
Reviewer: José Javier Etayo (Madrid)MBM classes and contraction loci on low-dimensional hyperkähler manifolds of \(K3^{[n]}\) typehttps://zbmath.org/1487.140872022-07-25T18:03:43.254055Z"Amerik, Ekaterina"https://zbmath.org/authors/?q=ai:amerik.ekaterina-yu"Verbitsky, Misha"https://zbmath.org/authors/?q=ai:verbitsky.mishaSummary: We describe the extremal rays and the exceptional loci of extremal contractions on a hyperkähler manifold of \(K3^{[n]}\) type for small \(n\) by deforming to the Hilbert scheme of a non-algebraic \(K3\) surface.Characterizing symplectic Grassmannians by varieties of minimal rational tangentshttps://zbmath.org/1487.141112022-07-25T18:03:43.254055Z"Hwang, Jun-Muk"https://zbmath.org/authors/?q=ai:hwang.jun-muk"Li, Qifeng"https://zbmath.org/authors/?q=ai:li.qifengThe main result of this article is the following theorem:
Let \(S\) be a symplectic or an odd-symplectic Grassmannian. Let \(X\) be a uniruled projective manifold with a family \(\mathcal K\) of rational curves of minimal degree. Assume that the variety of minimal rational tangents -- VMRT -- \(\mathcal C_x\subset \mathbb P(T_xX)\) of \(\mathcal K\) at a general point \(x\) in \(X\) is projectively equivalent to the VMRT \(\mathcal C_s\subset \mathbb P(T_sS)\) of lines at a general point \(s\) in \(S\). Then some Euclidean neighborhood of a general member of \(\mathcal K\) is biholomorphic to a Euclidean neighborhood of a general line in one of the presymplectic Grassmannians.
This theorem can be seen as an extension of a result of Mok in the case of an irreducible Hermitian symmetric space \(S\) [\textit{N. Mok}, AMS/IP Stud. Adv. Math. 42, 41--61 (2008; Zbl 1182.14042)]. The result of Mok was generalized by J. Hong and J.M. Hwang to the case where \(S\) is a homogeneous space \(G/P\) of a complex simple Lie group \(G\) with a maximal parabolic subgroup \(P\) associated to a long root. Also in this article \(S\) is of the form \(S/P\) but it is associated to a short root. This implies that the VMRT \(\mathcal C_s\) is not a homogeneous variety. Moreover the techniques of Mok based on the differential geometric machinery of Cartan connections constructed by Tanaka cannot be used in this context. The Authors overcome this difficulty by constructing a Cartan connection associated to a fairly general setting, a \(G_0\)-structure on a complex manifold equipped with a filtration; this is done in Section 2 of the article.
As consequence they are able to characterize symplectic and odd-symplectic Grassmannians among Fano manifolds of Picard number \(1\) by their VMRT at a general point. They also prove their rigidity under global Kahler deformation.
Reviewer: Emilia Mezzetti (Trieste)Deep learning Gauss-Manin connectionshttps://zbmath.org/1487.141282022-07-25T18:03:43.254055Z"Heal, Kathryn"https://zbmath.org/authors/?q=ai:heal.kathryn"Kulkarni, Avinash"https://zbmath.org/authors/?q=ai:kulkarni.avinash"Sertöz, Emre Can"https://zbmath.org/authors/?q=ai:sertoz.emre-cThere has been much activity in the last few years on machine-learning applied to understanding pure mathematics, in order to formulate new conjectures, verify formulae and speed up computations.
The current paper is very interesting: it applies a deep neural networks to the computation of Gauss-Manin connections, exemplified by families of hypersurfaces as algebraic varieties. The methods are novel and it demonstrates a significant speed-up in the usually difficult computation of connections on manifolds. It shows that there must be some underlying pattern in connections on which the neural network is picking up.
Reviewer: Yang-Hui He (London)Deformations of higher-page analogues of \(\partial{\bar{\partial}} \)-manifoldshttps://zbmath.org/1487.320632022-07-25T18:03:43.254055Z"Popovici, Dan"https://zbmath.org/authors/?q=ai:popovici.dan"Stelzig, Jonas"https://zbmath.org/authors/?q=ai:stelzig.jonas-robin"Ugarte, Luis"https://zbmath.org/authors/?q=ai:ugarte.luisSummary: We extend the notion of small essential deformations of Calabi-Yau complex structures from the case of the Iwasawa manifold, for which they were introduced recently by the first-named author, to the general case of page-\(1- \partial{{\bar{\partial }}} \)-manifolds that were jointly introduced very recently by all three authors. We go on to obtain an analogue of the unobstructedness theorem of Bogomolov, Tian and Todorov for Calabi-Yau page-\(1- \partial{{\bar{\partial }}} \)-manifolds. As applications of this discussion, we study the small deformations of certain Nakamura solvmanifolds and reinterpret the cases of the Iwasawa manifold and its 5-dimensional analogue from this standpoint.The Kuranishi map for vector bundles on certain products of curveshttps://zbmath.org/1487.320642022-07-25T18:03:43.254055Z"Ballico, E."https://zbmath.org/authors/?q=ai:ballico.edoardo"Gasparim, E."https://zbmath.org/authors/?q=ai:gasparim.elizabeth"Rubilar, F."https://zbmath.org/authors/?q=ai:rubilar.francisco"Suzuki, B."https://zbmath.org/authors/?q=ai:suzuki.brunoSummary: We describe deformations of vector bundles on surfaces that are a product of two smooth projective curves. We explicitly describe the Kuranishi map around unstable vector bundles and compare the homologies of the Kuranishi spaces of stable and unstable deformations.Strongly homotopy Lie algebras and deformations of calibrated submanifoldshttps://zbmath.org/1487.320652022-07-25T18:03:43.254055Z"Fiorenza, Domenico"https://zbmath.org/authors/?q=ai:fiorenza.domenico"Lê, Hông Vân"https://zbmath.org/authors/?q=ai:le-hong-van."Schwachhöfer, Lorenz"https://zbmath.org/authors/?q=ai:schwachhofer.lorenz-j"Vitagliano, Luca"https://zbmath.org/authors/?q=ai:vitagliano.lucaSummary: For an element \(\Psi\) in the graded vector space \(\Omega^\ast (M,TM)\) of tangent bundle valued forms on a smooth manifold \(M\), a \(\Psi \)-submanifold is defined as a submanifold \(N\) of \(M\) such that \(\Psi_{\vert N} \in \Omega^\ast (N,TN)\). The class of \(\Psi \)-submanifolds encompasses calibrated submanifolds, complex submanifolds and all Lie subgroups in compact Lie groups. The graded vector space \(\Omega^\ast (M,TM)\) carries a natural graded Lie algebra structure, given by the Frölicher-Nijenhuis bracket \([-,-]^{FN}\). When \(\Psi\) is an odd degree element with \([\Psi,\Psi]^{FN}=0\), we associate to a \(\Psi \)-submanifold \(N\) a strongly homotopy Lie algebra, which governs the formal and (under certain assumptions) smooth deformations of \(N\) as a \(\Psi \)-submanifold, and we show that under certain assumptions these deformations form an analytic variety. As an application we revisit formal and smooth deformation theory of complex closed submanifolds and of \(\varphi \)-calibrated closed submanifolds, where \(\varphi\) is a parallel form in a real analytic Riemannian manifold.Deformation theory of holomorphic Cartan geometries. II.https://zbmath.org/1487.320662022-07-25T18:03:43.254055Z"Biswas, Indranil"https://zbmath.org/authors/?q=ai:biswas.indranil"Dumitrescu, Sorin"https://zbmath.org/authors/?q=ai:dumitrescu.sorin"Schumacher, Georg"https://zbmath.org/authors/?q=ai:schumacher.georgSummary: In this continuation of [the authors, Indag. Math., New Ser. 31, No. 3, 512--524 (2020; Zbl 1439.32030)], we investigate the deformations of holomorphic Cartan geometries where the underlying complex manifold is allowed to move. The space of infinitesimal deformations of a flat holomorphic Cartan geometry is computed. We show that the natural forgetful map, from the infinitesimal deformations of a flat holomorphic Cartan geometry to the infinitesimal deformations of the underlying flat principal bundle on the topological manifold, is an isomorphism.An analytic application of Geometric Invariant Theory. II: Coarse moduli spaceshttps://zbmath.org/1487.320672022-07-25T18:03:43.254055Z"Buchdahl, Nicholas"https://zbmath.org/authors/?q=ai:buchdahl.nicholas-p"Schumacher, Georg"https://zbmath.org/authors/?q=ai:schumacher.georgIn an earlier companion of this article [the authors, J. Geom. Phys. 165, Article ID 104237, 14 p. (2021; Zbl 1472.14047)], geometric invariant theory is applied to construct quotients of a compact Kähler with the aim at obtaining local models for a classifying space of (poly)stable holomorphic vector bundles, and containing the coarse moduli space of stable bundles as an open subspace. The article under review shows that this classifying space, considered in the weakly normal category, is a coarse moduli space in the sense of complex geometry when the topology is fixed as induced by the space of Hermite-Einstein connections modulo the group of unitary gauge transformations.
Reviewer: Mihai Putinar (Santa Barbara)Currents, systoles, and compactifications of character varietieshttps://zbmath.org/1487.320682022-07-25T18:03:43.254055Z"Burger, M."https://zbmath.org/authors/?q=ai:burger.marc"Iozzi, A."https://zbmath.org/authors/?q=ai:iozzi.alessandra"Parreau, A."https://zbmath.org/authors/?q=ai:parreau.anne"Pozzetti, M. B."https://zbmath.org/authors/?q=ai:pozzetti.maria-beatriceSummary: We study the Weyl chamber length compactification both of the Hitchin and of the maximal character varieties and determine therein an open set of discontinuity for the action of the mapping class group. This result is obtained as a consequence of a canonical decomposition of a geodesic current on a surface of finite type arising from a topological decomposition of the surface along special geodesics. We show that each component either is associated to a measured lamination or has positive systole. For a current with positive systole, we show that the intersection function on the set of closed curves is bilipschitz equivalent to the length function with respect to a hyperbolic metric.Uniformizing surfaces via discrete harmonic mapshttps://zbmath.org/1487.320692022-07-25T18:03:43.254055Z"Kajigaya, Toru"https://zbmath.org/authors/?q=ai:kajigaya.toru"Tanaka, Ryokichi"https://zbmath.org/authors/?q=ai:tanaka.ryokichiSummary: We show that for any closed surface of genus greater than one and for any finite weighted graph filling the surface, there exists a hyperbolic metric which realizes the least Dirichlet energy harmonic embedding of the graph among a fixed homotopy class and all hyperbolic metrics on the surface. We give explicit examples of such hyperbolic surfaces through a new interpretation of the Nielsen realization problem for the mapping class groups.Gromov-Witten invariants of \(\mathbb{P}^1\) coupled to a KdV tau functionhttps://zbmath.org/1487.320702022-07-25T18:03:43.254055Z"Norbury, Paul"https://zbmath.org/authors/?q=ai:norbury.paul-tSummary: We consider the pull-back of a natural sequence of cohomology classes \(\Theta_{g, n} \in H^{2 (2 g - 2+ n)}(\overline{\mathcal{M}}_{g, n}, \mathbb{Q})\) to the moduli space of stable maps \(\overline{\mathcal{M}}_{g, n}(\mathbb{P}^1, d)\). These classes are related to the Brézin-Gross-Witten tau function of the KdV hierarchy via \(Z^{B G W}(\hbar, t_0, t_1, \dots) = \exp \sum \frac{ \hbar^{2 g - 2}}{ n !} \int_{\overline{\mathcal{M}}_{g, n}} \Theta_{g, n} \cdot \prod_{j = 1}^n \psi_j^{k_j} \prod t_{k_j} \). Insertions of the pull-backs of the classes \(\Theta_{g, n}\) into the integrals defining Gromov-Witten invariants define new invariants which we show in the case of target \(\mathbb{P}^1\) are given by a random matrix integral and satisfy the Toda equation.On convergence of random walks on moduli spacehttps://zbmath.org/1487.320712022-07-25T18:03:43.254055Z"Prohaska, Roland"https://zbmath.org/authors/?q=ai:prohaska.rolandSummary: The purpose of this paper is to establish convergence of random walks on the moduli space of abelian differentials on compact Riemann surfaces in two different modes: convergence of the \(n\)-step distributions from almost every starting point in an affine invariant submanifold toward the associated affine invariant measure, and almost sure pathwise equidistribution toward the affine invariant measure on the \(\text{SL}_2 (\mathbb{R})\)-orbit closure of an arbitrary starting point. These are analogues to previous results for random walks on homogeneous spaces.Random hyperbolic surfaces of large genus have first eigenvalues greater than \(\frac{3}{16}-\epsilon\)https://zbmath.org/1487.320722022-07-25T18:03:43.254055Z"Wu, Yunhui"https://zbmath.org/authors/?q=ai:wu.yunhui"Xue, Yuhao"https://zbmath.org/authors/?q=ai:xue.yuhaoSummary: Let \(\mathcal{M}_g\) be the moduli space of hyperbolic surfaces of genus \(g\) endowed with the Weil-Petersson metric. In this paper, we show that for any \(\epsilon >0\), as genus \(g\) goes to infinity, a generic surface \(X\in \mathcal{M}_g\) satisfies that the first eigenvalue \(\lambda_1(X)>\frac{3}{16}-\epsilon\). As an application, we also show that a generic surface \(X\in \mathcal{M}_g\) satisfies that the diameter \(\mathrm{diam}(X)<(4+\epsilon)\ln (g)\) for large genus.Generators of the cohomology ring, after Newsteadhttps://zbmath.org/1487.320732022-07-25T18:03:43.254055Z"Basu, Suratno"https://zbmath.org/authors/?q=ai:basu.suratno"Dan, Ananyo"https://zbmath.org/authors/?q=ai:dan.ananyo"Kaur, Inder"https://zbmath.org/authors/?q=ai:kaur.inderSummary: The moduli space of rank \(2\) stable, locally-free sheaves with fixed odd degree determinant over a smooth, projective curve is a classical object. In the early 1970s, Newstead gave the generators of the cohomology ring of this moduli space. There are generalizations of this result to higher rank, but nothing is known for the case when the underlying curve is singular. In this article, we generalize Newstead's result to the case when the underlying curve is irreducible, nodal. We show that the generators of the cohomology ring in the nodal curve case arise naturally as degeneration of Newstead's generators in the smooth curve case.Logarithmic foliationshttps://zbmath.org/1487.321212022-07-25T18:03:43.254055Z"Cerveau, Dominique"https://zbmath.org/authors/?q=ai:cerveau.dominique"Lins Neto, Alcides"https://zbmath.org/authors/?q=ai:lins-neto.alcidesThe authors study singular holomorphic foliations of arbitrary codimension on projective spaces defined by logarithmic forms. Here a logarithmic form on a complex manifold \(M\) is a meromorphic \(q\)-form \(\eta\) on \(M\) such that the pole divisors of \(\eta\) and \(d\eta\) are reduced.
One of the main results of the paper (Theorem 2.1) concerns normal forms for closed logarithmic forms: \\
Let \(\eta\) be a germ at \(0\in\mathbb{C}^n\) of a closed logarithmic \(p\)-form with poles along a hypersurface \(X = (f_1 . . . f_r = 0)\) with strictly ordinary singularities outside \(0\in\mathbb{C}^n\). Assume that \(n > p + 2\). Then:\\
\begin{enumerate}
\item[(a)] If \(r < p\) then \(\eta\) is exact; \(\eta = d\Theta\), where \(\Theta\) is logarithmic non-closed and has the same pole divisor as \(\eta\).
\item [(b)] If \(r \geq p\), then \[\eta = \sum \lambda_I \frac{df_{i_1}}{f_{i_1}}\wedge\ldots\wedge\frac{df_{i_p}}{f_{i_p}} + d\Theta,\] where, either \(\Theta =0 \), or \(\Theta\) is logarithmic non-closed and has pole divisor contained in \(X\).
\end{enumerate}
As a consequence of Theorem 2.1, in the general case one gets normal forms in the case of logarithmic \(p\)-forms on \(\mathbb{CP}^n\) (Corollary 1.3).
A natural question one might ask concerning the structure of logarithmic foliations is the following (Problem 1.9): Does a foliation on \(\mathbb{CP}^n\), defined by a logarithmic \(p\)-form, \(2 \leq p < n\), is an intersection of \(p\) codimension-one logarithmic foliations? A partial answer to this question is given by Theorem 1.10 in the case of strictly ordinary singularities.
Reviewer: Judith Brinkschulte (Leipzig)Dolbeault and Bott-Chern formalities: deformations and \(\partial \overline{\partial}\)-lemmahttps://zbmath.org/1487.321552022-07-25T18:03:43.254055Z"Sferruzza, Tommaso"https://zbmath.org/authors/?q=ai:sferruzza.tommaso"Tomassini, Adriano"https://zbmath.org/authors/?q=ai:tomassini.adrianoSummary: It is proved that the properties of being Dolbeault formal and geometrically-Bott-Chern-formal are not closed under holomorphic deformations of the complex structure. Further, we construct a compact complex manifold which satisfies the \(\partial \overline{\partial}\)-lemma but admits a non vanishing Aeppli-Bott-Chern-Massey product.Partially integrable almost CR structureshttps://zbmath.org/1487.321822022-07-25T18:03:43.254055Z"Akahori, Takao"https://zbmath.org/authors/?q=ai:akahori.takaoSummary: Let \((M, D)\) be a compact contact manifold with \(\dim_{\mathbf{R}} M = 2 n-1 \geq 5\). This means that: \(M\) is a \(C^\infty\) differential manifold with \(\dim_{\mathbf{R}} M = 2n - 1 \geq 5\). And \(D\) is a subbundle of the tangent bundle \(TM\) which satisfying; there is a real one form \(\theta\) such that \(D = \{X : X \in TM, \theta (X) = 0\}\), and \(\theta\wedge \bigwedge^{n-1}(d\theta) \neq 0\) at every point of \(p\) of \(M\). Especially, we assume that our \(D\) admits almost CR structure, \((M, S)\). In this paper, inspired by the work of \textit{Y. Matsumoto} [Differ. Geom. Appl. 45, 78--114 (2016; Zbl 1350.32033)], we study the difference of partially integrable almost CR structures from actual CR structures. And we discuss partially integrable almost CR structures from the point of view of the deformation theory of CR structures [the author, Invent. Math. 63, 311--334 (1981; Zbl 0496.32015); the author et al., Mich. Math. J. 50, No. 3, 517--549 (2002; Zbl 1065.32018)].A stability theorem for projective CR manifoldshttps://zbmath.org/1487.321892022-07-25T18:03:43.254055Z"Brinkschulte, Judith"https://zbmath.org/authors/?q=ai:brinkschulte.judith"Hill, C. Denson"https://zbmath.org/authors/?q=ai:hill.c-denson"Nacinovich, Mauro"https://zbmath.org/authors/?q=ai:nacinovich.mauroSummary: We consider smooth deformations of the \textit{CR} structure of a smooth 2-pseudoconcave compact \textit{CR} submanifold \(\operatorname{M}\) of a reduced complex analytic variety \(\operatorname{X}\) outside the intersection \(D \cap \operatorname{M}\) with the support \(D\) of a Cartier divisor of a positive line bundle \(\mathtt{F}_{\operatorname{X}}\). We show that nearby structures still admit projective \textit{CR} embeddings. Special results are obtained under the additional assumptions that \(\operatorname{X}\) is a projective space or a Fano variety.The \(\mathbb{F}_p\)-Selberg integral of type \(A_n\)https://zbmath.org/1487.330182022-07-25T18:03:43.254055Z"Rimányi, Richárd"https://zbmath.org/authors/?q=ai:rimanyi.richard"Varchenko, Alexander"https://zbmath.org/authors/?q=ai:varchenko.alexander-nThe paper represents the Selberg Integral formula in the form of notation An where p is an odd prime number. The authors claimed that they have already established and proved the formula $A_1$ type in a previous paper. The $A_2$ type formula is proved in this paper and also sketch the proof the $A_n$ type formula for $n>2$. The paper contains useful collections of facts such as Cancellation of factorials, Wilson theorem, Dyson theorem, \(\mathbb{F}_p\) integrals and \(\mathbb{F}_p\) Beta integral to establish some main results, related theorems and Lemmas. These theorems are $A_n$ type \(\mathbb{F}_p\)-Selberg integral and present its evaluation formula, Selberg formula for $n=2$, transition form of the $A_{n-1}$ type formula to the $A_n$ type formula, in particular this transition form is known as $A_1$ type to the $A_2$ type Selberg formula. The theorems, Lemmas and relationships established here are quite interesting and useful for further research.
Reviewer: Vijay Yadav (Virar)Coarse density of subsets of moduli spacehttps://zbmath.org/1487.570272022-07-25T18:03:43.254055Z"Dozier, Benjamin"https://zbmath.org/authors/?q=ai:dozier.benjamin"Sapir, Jenya"https://zbmath.org/authors/?q=ai:sapir.jenyaIn this paper, a subset of a (non-compact) metric space is coarsely dense if there exists a constant \(K\) such that every point in the larger space is within distance \(K\) from some point of the subset.
In the paper under review, the authors prove that an algebraic subvariety of the moduli space of closed orientable surfaces is coarsely dense with respect to the Teichmüller metric if and only if the dimension of the subvariety coincides with that of the moduli space. They also prove a related result if the Teichmüller metric is replaced by the Thurston metric (with some appropriate sense for coarse density in this nonsymmetric setting): In this case the algebraic subvariety is coarsely dense if and only if the subvariety coincides with that of the moduli space. The proofs involve the Deligne-Mumford compactification of moduli spaces by stable Riemann surfaces. The authors apply this result to determine which strata of abelian differentials have a coarsely dense projection in moduli space. They prove a result on coarse density of projections of \(\mathrm{GL}(2,\mathbb{R})\)-orbit closures in the space of abelian differentials.
Reviewer: Athanase Papadopoulos (Strasbourg)