Recent zbMATH articles in MSC 53Dhttps://zbmath.org/atom/cc/53D2021-06-15T18:09:00+00:00WerkzeugOdd transgression for Courant algebroids.https://zbmath.org/1460.530702021-06-15T18:09:00+00:00"Bressler, Paul"https://zbmath.org/authors/?q=ai:bressler.paul"Rengifo, Camilo"https://zbmath.org/authors/?q=ai:rengifo.camiloSummary: The ``odd transgression'' introduced in [the authors, Lett. Math. Phys. 108, No. 9, 2099--2137 (2018; Zbl 1397.53092)] is applied to construct and study the inverse image functor in the theory of Courant algebroids.Virtual technique for orbifold Fredholm systems.https://zbmath.org/1460.530762021-06-15T18:09:00+00:00"Chen, Bohui"https://zbmath.org/authors/?q=ai:chen.bohui"Li, An-Min"https://zbmath.org/authors/?q=ai:li.an-min"Wang, Bai-Ling"https://zbmath.org/authors/?q=ai:wang.bailingSummary: In this paper, we review the the theory of virtual manifold/orbifolds developed by the first named author and Tian and develop the virtual technique for any orbfiold Fredholm system with compact moduli space \(\mathcal{M}\). This provides a description of \(\mathcal{M}\) terms of a virtual orbifold system \[\{(\mathcal{V}_I,E_I,\sigma_I)\}.\] Here \(\{\mathcal{V}_I\}\) is a virtual orbifold, and \(\{E_I\}\) is a finite rank virtual orbifold bundle with a virtual section \(\{\sigma_I\}\) such that the zero sets \(\{\sigma_I^{-1}(0)\}\) form a cover of the underlying moduli space \(\mathcal{M}\). A virtual orbifold system can be thought as a special class of Kuranishi structures on a moduli problem developed by Fukaya and Ono. Under some assumptions which guarantee the existence of a partition of unity and a virtual Euler form, we show that the virtual integration is well-defined for the resulting virtual orbifold system.
For the entire collection see [Zbl 1420.53004].Singularities, mirror symmetry, and the gauged linear sigma model. Summer school `Crossing the walls in enumerative geometry', Snowbird, UT, USA, May 21 -- June 1, 2018.https://zbmath.org/1460.140022021-06-15T18:09:00+00:00"Jarvis, Tyler J. (ed.)"https://zbmath.org/authors/?q=ai:jarvis.tyler-j"Priddis, Nathan (ed.)"https://zbmath.org/authors/?q=ai:priddis.nathanPublisher's description: This volume contains the proceedings of the workshop `Crossing the walls in enumerative geometry', held in May 2018 at Snowbird, Utah. It features a collection of both expository and research articles about mirror symmetry, quantized singularity theory (FJRW theory), and the gauged linear sigma model.
Most of the expository works are based on introductory lecture series given at the workshop and provide an approachable introduction for graduate students to some fundamental topics in mirror symmetry and singularity theory, including quasimaps, localization, the gauged linear sigma model (GLSM), virtual classes, cosection localization, \(p\)-fields, and Saito's primitive forms. These articles help readers bridge the gap from the standard graduate curriculum in algebraic geometry to exciting cutting-edge research in the field.
The volume also contains several research articles by leading researchers, showcasing new developments in the field.
The articles of this volume will be reviewed individually.Whiskered tori for presymplectic dynamical systems.https://zbmath.org/1460.370592021-06-15T18:09:00+00:00"de la Llave, Rafael"https://zbmath.org/authors/?q=ai:de-la-llave.rafael"Xu, Lu"https://zbmath.org/authors/?q=ai:xu.luThe main result of the paper is a persistence theorem for the whiskered tori in presymplectic systems. The authors formulate an invariance equation for an embedding and a provide a splitting of the tangent space at the range of the embedding in such a way that zeros of the resulting functional equation define whiskered tori with the corresponding stable and unstable splittings. The proof of the persistence result involves adjusting certain parameters, a common procedure in KAM theory. The method for the proof does not use transformation theory, thus allowing easier numerical implementations.
Reviewer: Mircea Crâşmăreanu (Iaşi)Bott-Samelson atlases, total positivity, and Poisson structures on some homogeneous spaces.https://zbmath.org/1460.530732021-06-15T18:09:00+00:00"Lu, Jiang-Hua"https://zbmath.org/authors/?q=ai:lu.jiang-hua|lu.jianghua"Yu, Shizhuo"https://zbmath.org/authors/?q=ai:yu.shizhuoSummary: Let \(G\) be a connected and simply connected complex semisimple Lie group. For a collection of homogeneous \(G\)-spaces \(G/Q\), we construct a finite atlas \(\mathcal{A}_{BS} (G/Q)\) on \(G/Q\), called the \textit{Bott-Samelson atlas}, and we prove that all of its coordinate functions are positive with respect to the Lusztig positive structure on \(G/Q\). We also show that the standard Poisson structure \(\pi_G/Q\) on \(G/Q\) is presented, in each of the coordinate charts of \(\mathcal{A}_{BS} (G/Q)\), as a symmetric Poisson CGL extension (or a certain localization thereof) in the sense of Goodearl-Yakimov, making \((G/Q, \pi_G/Q, \mathcal{A}_{BS} (G/Q))\) into a \textit{Poisson-Ore variety}. In addition, all coordinate functions in the Bott-Samelson atlas are shown to have complete Hamiltonian flows with respect to the Poisson structure \(\pi_G/Q\). Examples of \(G/Q\) include \(G\) itself, \(G/T\), \(G/B\), and \(G/N\), where \(T \subset G\) is a maximal torus, \(B \subset G\) a Borel subgroup, and \(N\) the uniradical of \(B\).The Kodaira dimension of contact 3-manifolds and geography of symplectic fillings.https://zbmath.org/1460.530752021-06-15T18:09:00+00:00"Li, Tian-Jun"https://zbmath.org/authors/?q=ai:li.tian-jun|li.tianjun"Mak, Cheuk Yu"https://zbmath.org/authors/?q=ai:mak.cheuk-yuIf \((Y,\xi)\) is a closed, connected, and co-oriented contact manifold, then a symplectic manifold \((N,\omega_N)\) with boundary \(\partial N\) and a locally defined outward Liouville vector field \(V\) along \(\partial N\) such that the induced contact manifold \((\partial N,\xi_N)\) is contactomorphic to \((Y,\xi)\) is called a symplectic filling of \((Y,\xi)\).
A symplectic capping, or cap, \((P,\omega_P)\) of \((Y,\xi)\) is defined in exactly the same way as a symplectic filling with the requirement that \(V\) points inward instead onward. The induced contact 1-form is denoted as \(\alpha_P\) and \((P,\omega_P,\alpha_P)\) forms a concave symplectic pair.
A Stein filling of \((Y, \xi)\) is a \(4\)-manifold \((N,\omega_N)\) that is a Stein domain such that \((Y, \xi)\) is contactomorhpic to \(\partial N\) with the contact structure induced by the complex tangencies. The characteristic numbers for \(N\) are the Betti number \(b_1(N)\), signature \(\sigma(N)\), and the Euler characteristic \(\chi(N)\).
If \((X,\omega)\) is a closed symplectic 4-manifold and \(D\) is a smooth symplectic surface in \(X\), then \(D\) is called maximal if any symplectic exceptional class in \((X,\omega)\) pairs positively with \([D]\).
If \((P,\omega_P)\) is a concave symplectic manifold and \(\alpha_P\) is a contact 1-form on \(\partial P\) induced by an inward pointing Liouville vector field, then \((P,\omega_P,\alpha_P)\) is called a concave symplectic pair with rational period if \(\frac1{2\pi}[(\omega_P,\alpha_P)]\in H^2(P,\partial P;\mathbb{Q})\).
If \((P,\omega_P,\alpha_P)\) is a concave symplectic pair with rational period, then a closed symplectic hypersurface \(D\) is called a Donaldson hypersurface of \((P,\omega_P,\alpha_P)\) if it is Lefschetz dual to an integral multiple of \(\frac1{2\pi}[(\omega_P,\alpha_P)]\). A symplectic cap is called a Donaldson cap if it admits a Donaldson hypersurface, and a Donaldson cap with a chosen Donaldson hypersurface is called a polarized cap.
If \(X\) is a closed, oriented smooth 4-manifold and \(\mathcal{E}_X\) is the set of cohomology classes whose Poincaré duals are represented by smoothly embedded spheres of self-intersection \(-1\), then \(X\) is said to be minimal if \(\mathcal{E}_X\) is the empty set.
A compact-oriented manifold \(W\) such that \(\partial W=Y_+\cup (-Y_-)\) is called an oriented cobordism from a closed oriented manifold \(Y_-\) to another \(Y_+\). An oriented cobordism \(W\) with a symplectic structure \(\omega\) compatible with the orientation is called a symplectic cobordism if \((W,\omega)\) is symplectic concave at \(Y_-\) and convex at \(Y_+\). A symplectic cobordism \((W,\omega)\) is called exact if the locally defined Liouville vector fields near \(Y_+\) and \(Y_-\) extend to a global Liouville vector field.
In [Mich. Math. J. 51, No. 2, 327--337 (2003; Zbl 1043.53066)], \textit{A. Stipsicz} showed that the set \(\{2\chi(N)+3\sigma(N)\}\subset\mathbb{Z}\) is bounded from below for any Stein filling \((N,\omega_N)\) of \((Y,\xi)\). As a nice corollary, they showed that if \(b_2^+=0\) for any Stein filling \((N,\omega_N)\) of \((Y,\xi)\), then the set \(\{(b_1(N),\chi(N),\sigma(N))\}\) is finite.
In [Proc. Lond. Math. Soc. (3) 114, No. 1, 159--187 (2017; Zbl 1373.57046)], the present authors and \textit{K. Yasui} introduced three types of caps, namely, Calabi-Yau caps, uniruled caps, and adjunction caps for contact manifolds.
A Calabi-Yau cap of a contact \(3\)-manifold \((Y,\xi)\) is a compact symplectic manifold \((P,\omega_P)\) which is a strong concave filling of \((Y,\xi)\) such that \(c_1(P)\) is torsion.
An uniruled cap of a contact \(3\)-manifold \((Y,\xi)\) is a symplectic concave filling \((P,\omega_P)\) of \((Y,\xi)\) such that \(c_1(P)\cdot[(\omega_P,\alpha_P)]>0\) for some Liouville one-form \(\alpha_P\), where \([(\omega_P,\alpha_P)]\) is a relative cohomology class in \(H^2(P,\partial P,\mathbb R)\).
Another type of caps called adjunction caps is based on an observation that existence of a smoothly embedded surface in a closed symplectic manifold with sufficiently large self-intersection number relative to the genus implies that the symplectic manifold is uniruled.
These authors also proved that if a contact \(3\)-manifold \((Y,\xi)\) admits a Calabi-Yau cap, or a uniruled cap, or an adjunction cap \((P,\omega_P)\), then the set \(\{(b_1(P),\chi(P),\sigma(P))\}\) is finite.
If, moreover, a Calabi-Yau cap \((P,\omega_P)\) cannot be embedded in an uniruled manifold, then all exact fillings of \((Y,\xi)\) have torsion first Chern class.
In this paper, the authors introduce the Kodaira dimension of contact 3-manifolds, establish the geography of various kinds of symplectic fillings, find a lower bound of \(2\chi+3\sigma\) for all of its minimal symplectic fillings, and discuss various aspects of exact self-cobordisms of fillable contact 3-manifolds.
The Kodaira dimension of a contact 3-manifold \((Y,\xi)\) is defined as follows
\[\mathrm{Kod}(Y,\xi)=\left\{
\begin{array}{rl}
-\infty & \text{if it admits a uniruled cap} \\
0 & \text{if it admits a Calabi-Yau cap but admits no uniruled caps} \\
1 & \text{if it admits neither a Calabi-Yau cap nor a uniruled cap}
\end{array}
\right.\]
The authors show that the symplectic filling version of Stipsicz's result holds for any contact 3-manifold with \(\mathrm{Kod}=-\infty\), and the exact filling version of Stipsicz's result holds for any contact 3-manifold with \(\mathrm{Kod}=0\). Moreover, explicit homological bounds can be obtained in the case \(\mathrm{Kod}=-\infty\) given a uniruled cap and in the case \(\mathrm{Kod}=0\) given a Calabi-Yau cap. There are many contact 3-manifolds with \(\mathrm{Kod}=1\).
Making use of the notions of a maximal surface and a Donaldson cap, the authors show in the main result of the paper that for any contact 3-manifold \((Y,\xi)\), the set \(\{2\chi(N)+3\sigma(N)\}\subset\mathbb{Z}\) is bounded from below when \((N,\omega_N)\) is a minimal symplectic filling of \((Y,\xi)\). Moreover, they explicitly calculate the lower bound for a polarized symplectic cap.
This result, together with the above results for \(\mathrm{Kod}=-\infty\) and \(\mathrm{Kod}=0\) contact 3-manifolds, provides a comprehensive geography picture for various fillings of contact 3-manifolds with a fixed Kodaira dimension.
The authors also prove that for any contact 3-manifold \((Y,\xi)\), there exists a symplectic cap \((P,\omega_P)\) of \((Y,\xi)\) such that for any minimal symplectic filling \((N,\omega_N)\) of \((Y,\xi)\), the glued symplectic manifold \((N\cup P,\omega)\) is minimal. In particular, any minimal convex 4-manifold embeds into a minimal closed symplectic 4-manifold.
By combining this result with the previous one, the authors show that for any fillable contact 3-manifold \((Y,\xi)\), the set \(
\{2\chi(W)+3\sigma(W)\}\subset\mathbb{Z}\) is bounded below by \(0\) when \((W,\omega_W)\) is an exact cobordism from \((Y,\xi)\) to itself. In particular, if it is also bounded above, then the set is \(\{0\}\).
Reviewer: Andrew Bucki (Edmond)Infinitesimal automorphisms of VB-groupoids and algebroids.https://zbmath.org/1460.530712021-06-15T18:09:00+00:00"Esposito, Chiara"https://zbmath.org/authors/?q=ai:esposito.chiara"Vitagliano, Luca"https://zbmath.org/authors/?q=ai:vitagliano.luca"Tortorella, Alfonso Giuseppe"https://zbmath.org/authors/?q=ai:tortorella.alfonso-giuseppeSummary: VB-groupoids and algebroids are \textit{vector bundle objects} in the categories of Lie groupoids and Lie algebroids, respectively, and they are related via the \textit{Lie functor}. VB-groupoids and algebroids play a prominent role in Poisson and related geometries. Additionally, they can be seen as models for vector bundles over singular spaces. In this paper we study their infinitesimal automorphisms, i.e. vector fields on them generating a flow by diffeomorphisms preserving both the linear and the groupoid/algebroid structures. For a special class of VB-groupoids/algebroids coming from representations of Lie groupoids/algebroids, we prove that infinitesimal automorphisms are the same as multiplicative sections of a certain derivation VB-groupoid/algebroid.Geodesic flows on real forms of complex semi-simple Lie groups of rigid body type.https://zbmath.org/1460.530742021-06-15T18:09:00+00:00"Ratiu, Tudor S."https://zbmath.org/authors/?q=ai:ratiu.tudor-stefan"Tarama, Daisuke"https://zbmath.org/authors/?q=ai:tarama.daisukeFree rigid body dynamics has been generalized to semi-simple Lie groups in [\textit{A. S. Mishchenko} and \textit{A. T. Fomenko}, Math. USSR, Izv. 12, 371--389 (1978; Zbl 0405.58031)] and [\textit{A. S. Mishchenko} and \textit{A. T. Fomenko}, Tr. Semin. Vektorn. Tenzorn. Anal. 19, 3--94 (1979; Zbl 0452.58015)].
The motions of the generalized free rigid bodies can be formulated as geodesic flows for certain left-invariant metrics on semi-simple Lie groups, called of rigid body type.
The Williamson type of an isolated equilibrium point for a Hamiltonian system is a triple giving the numbers of elliptic, hyperbolic, and focus-focus components.
The main result of the article under review characterizes the Williamson types of the isolated relative equilibria on generic adjoint orbits in terms of the root system of the complexified Lie-algebra.
Reviewer: Miguel Paternain (Montevideo)Real hypersurfaces in \(\mathbb{C}P^2\) with constant Reeb sectional curvature.https://zbmath.org/1460.530202021-06-15T18:09:00+00:00"Wang, Yaning"https://zbmath.org/authors/?q=ai:wang.yaningSummary: We prove that the structure Jacobi operator \(R_\xi\) on a real hypersurface \(M\) in the complex projective space \(\mathbb{C}P^2\) satisfies \(R_\xi+\kappa\phi^2=0\) with \(\kappa\in\mathbb{R}^\ast\) (or equivalently, constant Reeb sectional curvature) if and only if either \(M\) is of type \((A)\) or \(M\) is locally congruent to a non-homogeneous Hopf hypersurface with vanishing Hopf principal curvature.Five lectures on topological field theory.https://zbmath.org/1460.810942021-06-15T18:09:00+00:00"Teleman, Constantin"https://zbmath.org/authors/?q=ai:teleman.constantinSummary: Topological quantum field theories -- TQFTs -- arose in physics as the baby (zero energy) sector of honest quantum field theories, which showed an unexpected dependence on the large scale topology of space-time. The zero energy part of the Hilbert space of states does not evolve in time, as by definition it is killed by the Hamiltonian; so, at first sight, its physics appears to be uninteresting. But this argument fails to consider a space-time with interesting topology.
In mathematics, TQFT emerged as an intriguing organizing structure for certain brave new topological or differential invariants of manifolds, which could not be captured by standard techniques of algebraic topology. (We will see a reason for that.)
For the entire collection see [Zbl 1362.14002].On the integrability conditions and operators of the \(F((K+1),(K-1))\)-structure satisfying \(F^{K+1}+F^{K-1}=0\), \((F\neq 0, K\eqslantgtr 2)\) on cotangent bundle and tangent bundle.https://zbmath.org/1460.530322021-06-15T18:09:00+00:00"Das, Lovejoy"https://zbmath.org/authors/?q=ai:das.lovejoy-s-k"Çayir, Haşim"https://zbmath.org/authors/?q=ai:cayir.hasimSummary: This paper consists of two main sections. In the first part, we find the integrability conditions of the horizontal lifts of \(F((K+1),(K-1))-\) structure satisfying \(F^{K+1}+F^{K-1}=0\), \((F\neq 0, K\eqslantgtr 2)\). Later, we get the results of Tachibana operators applied to vector and covector fields according to the horizontal lifts of \(F((K+1),(K-1))\)-structure in cotangent bundle \(T^{\ast }(M^n)\). Finally, we have studied the purity conditions of Sasakian metric with respect to the horizontal lifts of the structure. In the second part, all results obtained in the first section were obtained according to the complete and horizontal lifts of the structure in tangent bundle \(T(M^n)\).Enumeration of holomorphic cylinders in log Calabi-Yau surfaces. II: Positivity, integrality and the gluing formula.https://zbmath.org/1460.141252021-06-15T18:09:00+00:00"Yu, Tony Yue"https://zbmath.org/authors/?q=ai:yu.tony-yueSummary: We prove three fundamental properties of counting holomorphic cylinders in log Calabi-Yau surfaces: positivity, integrality and the gluing formula. Positivity and integrality assert that the numbers of cylinders, defined via virtual techniques, are in fact nonnegative integers. The gluing formula roughly says that cylinders can be glued together to form longer cylinders, and the number of longer cylinders equals the product of the numbers of shorter cylinders. Our approach uses Berkovich geometry, tropical geometry, deformation theory and the ideas in the proof of associativity relations of Gromov-Witten invariants by Maxim Kontsevich [\textit{M. Kontsevich} and \textit{Yu. Manin}, Commun. Math. Phys. 164, No. 3, 525--562 (1994; Zbl 0853.14020)]. These three properties provide evidence for a conjectural relation between counting cylinders and the broken lines of Gross, Hacking and Keel [\textit{M. Gross} et al., Publ. Math., Inst. Hautes Étud. Sci. 122, 65--168 (2015; Zbl 1351.14024)].
For part I see [the author, Math. Ann. 366, No. 3--4, 1649--1675 (2016; Zbl 1375.14186)].Dynamics and topology of conformally Anosov contact 3-manifolds.https://zbmath.org/1460.530662021-06-15T18:09:00+00:00"Hozoori, Surena"https://zbmath.org/authors/?q=ai:hozoori.surenaSummary: We provide obstructions to the existence of conformally Anosov Reeb flows on a 3-manifold that partially generalize similar obstructions to Anosov Reeb flows. In particular, we show \(\mathbb{S}^3\) does not admit conformally Anosov Reeb flows. We also give a Riemannian geometric condition on a metric compatible with a contact structure implying that a Reeb field is Anosov. From this we can give curvature conditions on a metric compatible with a contact structure that implies universal tightness of the contact structure among other things.On transitive contact and CR algebras.https://zbmath.org/1460.320832021-06-15T18:09:00+00:00"Marini, Stefano"https://zbmath.org/authors/?q=ai:marini.stefano"Medori, Costantino"https://zbmath.org/authors/?q=ai:medori.costantino"Nacinovich, Mauro"https://zbmath.org/authors/?q=ai:nacinovich.mauro"Spiro, Andrea"https://zbmath.org/authors/?q=ai:spiro.andrea-fSummary: We consider locally homogeneous \(CR\) manifolds and show that, under a condition only depending on their underlying contact structure, their \(CR\) automorphisms form a finite dimensional Lie group.Quantisation of extremal Kähler metrics.https://zbmath.org/1460.320482021-06-15T18:09:00+00:00"Hashimoto, Yoshinori"https://zbmath.org/authors/?q=ai:hashimoto.yoshinori|hashimoto.yoshinori.1Summary: Suppose that a polarised Kähler manifold \((X, L)\) admits an extremal metric \(\omega\). We prove that there exists a sequence of Kähler metrics \(\{\omega_k\}_k\), converging to \(\omega\) as \(k\rightarrow\infty\), each of which satisfies \(\bar{\partial}\text{grad}^{1,0}_{\omega_k}\rho_k (\omega_k)=0\); the (1, 0)-part of the gradient of the Bergman function is a holomorphic vector field.On open book embedding of contact manifolds in the standard contact sphere.https://zbmath.org/1460.530682021-06-15T18:09:00+00:00"Saha, Kuldeep"https://zbmath.org/authors/?q=ai:saha.kuldeepSummary: We prove some open book embedding results in the contact category with a constructive approach. As a consequence, we give an alternative proof of a theorem of Etnyre and Lekili that produces a large class of contact 3-manifolds admitting contact open book embeddings in the standard contact 5-sphere. We also show that all the Ustilovsky \((4m+1)\)-spheres contact open book embed in the standard contact \((4m+3)\)-sphere.Mini-workshop: superpotentials in algebra and geometry. Abstracts from the mini-workshop held February 23--29, 2020.https://zbmath.org/1460.000352021-06-15T18:09:00+00:00"Bossinger, Lara (ed.)"https://zbmath.org/authors/?q=ai:bossinger.lara"González, Eduardo (ed.)"https://zbmath.org/authors/?q=ai:gonzalez.eduardo-h-a|gonzalez.eduardo|gonzalez.eduardo-gutierrez"Rietsch, Konstanze (ed.)"https://zbmath.org/authors/?q=ai:rietsch.konstanze"Williams, Lauren K. (ed.)"https://zbmath.org/authors/?q=ai:williams.lauren-kSummary: Mirror symmetry has been at the epicenter of many mathematical discoveries in the past twenty years. It was discovered by physicists in the setting of super conformal field theories (SCFTs) associated to closed string theory, mathematically described by \(\sigma\)-models. These \(\sigma\)-models turn out in two different ways: the A-model and the B-model. Physical considerations predict that deformations of the SCFT of either \(\sigma\)-model should be isomorphic. Thus the mirror symmetry conjecture states that the A-model of a particular Calabi-Yau space \(X\) must be isomorphic to the B-model of its mirror \(\check{X}\). Mirror symmetry has been extended beyond the Calabi-Yau setting, in particular to Fano varieties, using the so called Landau-Ginzburg models. That is a non-compact manifold equipped with a complex valued function called the \textit{superpotential}.
In general, there is no clear recipe to construct the mirror for a given variety which demonstrates the need of joining mathematical forces from a wide range. The main aim of this Mini-Workshop was to bring together experts from the different communities (such as symplectic geometry and topology, the theory of cluster varieties, Lie theory and algebraic combinatorics) and to share the state of the art on superpotentials and explore connections between different constructions.Hessenberg varieties, Slodowy slices, and integrable systems.https://zbmath.org/1460.141022021-06-15T18:09:00+00:00"Abe, Hiraku"https://zbmath.org/authors/?q=ai:abe.hiraku"Crooks, Peter"https://zbmath.org/authors/?q=ai:crooks.peterLet \(G\) be a simply-connected semisimple complex algebraic group of rank \(r\) with centre \(Z\) and fix a pair of opposite Borel subgroups \(B\) and \(B_{-}\) of \(G\); let \(\mathfrak{g}\), \(\mathfrak{b}\) and \(\mathfrak{b}_{-}\) be their Lie algebras, \(\mathfrak{t} := \mathfrak{b}\cap \mathfrak{b}_{-}\), a Cartan subalgebra of \(\mathfrak{g}\), and let \(\Delta\) be the corresponding set of roots. Now the choice of \(B\) determines subsets of simple, respectively positive, roots \(\Pi\), respectively \(\Delta_+\), so that \(\mathfrak{b} = \oplus_{\alpha \in \Delta_+} \mathfrak{g}_{\alpha}\), \(\mathfrak{b}_{-} = \bigoplus_{\alpha \in \Delta_+} \mathfrak{g}_{-\alpha}\). Set \(\mathfrak{g}_{\alpha}^{\times} = \mathfrak{g}_{\alpha} \backslash 0\) for \(\alpha \in \Delta\). The main character of the paper under review is the quasi-affine variety \[ \mathcal O_{\text{Toda}} := \mathfrak{t}+ \sum_{\alpha \in \Pi} \mathfrak{g}_{-\alpha}^{\times} \] which is isomorphic to a coadjoint orbit of \(B\) and therefore bears a symplectic structure given by the Kostant-Kirillov-Souriau form. Now by the well-known Chevalley restriction theorem, the algebra \(\mathbb C[\mathfrak{g}]^G\) of \(G\)-invariant polynomial functions on \(\mathfrak{g}\) is a polynomial algebra itself. Choose homogeneous algebraically independent generators \(f_1, \dots, f_r\) of \(\mathbb C[\mathfrak{g}]^G\). Pick also \(e_{\alpha} \in \mathfrak{g}_{\alpha}^{\times}\) for \(\alpha \in \Pi\) and set \(\zeta = -\sum_{\alpha \in \Pi} e_{\alpha}\). A celebrated theorem of Kostant states that the restrictions of the translations by \(\zeta\) of \(f_1, \dots, f_r\) form a completely integrable system on \(\mathcal O_{\text{Toda}}\) (with respect to the Kostant-Kirillov-Souriau form) called the \emph{Toda lattice}. See [\textit{B. Kostant}, Sel. Math., New Ser. 2, No. 1, 43--91 (1996; Zbl 0868.14024)]. The purpose of this article is to extend the Toda lattice to a family of Hessenberg varieties. Towards this goal the authors proceed in two steps.
First they consider the Slodowy slice \(S_{\text{reg}} := \xi + \ker \text{ad}_{\eta}\) associated to a suitable (principal) \(\mathfrak{sl}_2\)-triple \((\xi, h, \eta)\). It is known that \(G \times S_{\text{reg}}\) can be identified with a symplectic subvariety of the cotangent bundle of \(G\) and that \(G/Z \times S_{\text{reg}}\) is a symplectic quotient of \(G \times S_{\text{reg}}\). Furthermore the invariant polynomials \(f_1, \dots, f_r\) together with \(\zeta\) above give rise to the so-called Mishchenko-Fomenko polynomials. Pulling back these polynomials along the moment map \(\mu: G /Z \times S_{\text{reg}} \to \mathfrak{g}\) one gets a completely integrable system on \(G /Z \times S_{\text{reg}}\), see [\textit{P. Crooks} and \textit{S. Rayan}, Math. Res. Lett. 26, No. 1, 9--33 (2019; Zbl 1421.32029)]. The main achievement of the first step is the construction of a natural embedding of symplectic varieties \(\kappa : \mathcal{O}_{\text{Toda}} \hookrightarrow G /Z \times S_{\text{reg}}\) relating the Toda lattice with the mentioned completely integrable system.
Second, the \(B\)-submodule \(H_{0} := \mathfrak{b}+ \sum_{\alpha \in \Pi} \mathfrak{g}_{-\alpha}\) of \(\mathfrak{g}\) determines a \(G\)-equivariant vector bundle on \(G/B\) with total space \(X(H_0) := G\times_{B} H_0\). Let \(\mu_0: X(H_{0}) \to \mathfrak{g}\) be the morphism given by \(\mu_0\left([g, x]\right) = \operatorname{Ad}_g (x)\). The fibres \(X(x, H_{0}) := \mu_0^{-1}(x)\) of \(\mu_0\) are the Hessenberg varieties associated to \(H_0\). Set \(H_{0}^{\times} := \mathfrak{b}+ \sum_{\alpha \in \Pi} \mathfrak{g}_{-\alpha}^{\times}\). The authors prove that \(X(H_0)\) carries a natural Poisson structure having a unique open (and dense) symplectic leaf which is \(X(H_0^{\times}) = G\times_{B} H_0^{\times}\). Then they prove that there exists an open immersion \(\varphi: G /Z \times S_{\text{reg}} \hookrightarrow X(H_0)\) such that \(\mu_0\varphi = \mu\). The image of \(\varphi\) is \(X(H_0^{\times})\) which is isomorphic to \( G /Z \times S_{\text{reg}}\) as a symplectic variety via \(\varphi\).
Now pulling back the Mishchenko-Fomenko polynomials along the map \(\mu_0\) one gets a completely integrable system on \(X(H_0)\). In conclusion, the composition \(\varphi\kappa \) is an embedding of completely integrable systems from the Toda lattice on \(\mathcal{O}_{\text{Toda}}\) into that one arising from the Mishchenko-Fomenko polynomials on the family \(X(H_0)\) of Hessenberg varieties. Finally the authors discuss briefly some applications to the geometry of Hessenberg varieties.
Reviewer: Nicolás Andruskiewitsch (Córdoba)Complex structures, moment maps, and the Ricci form.https://zbmath.org/1460.320582021-06-15T18:09:00+00:00"García-Prada, Oscar"https://zbmath.org/authors/?q=ai:garcia-prada.oscar"Salamon, Dietmar A."https://zbmath.org/authors/?q=ai:salamon.dietmar-a"Trautwein, Samuel"https://zbmath.org/authors/?q=ai:trautwein.samuelSummary: The Ricci form is a moment map for the action of the group of exact volume preserving diffeomorphisms on the space of almost complex structures. This observation yields a new approach to the Weil-Petersson symplectic form on the Teichmüller space of isotopy classes of complex structures with real first Chern class zero and nonempty Kähler cone.A linear algebraic setting for Jacobi structures.https://zbmath.org/1460.530652021-06-15T18:09:00+00:00"Cioroianu, Eugen-Mihăiţă"https://zbmath.org/authors/?q=ai:cioroianu.eugen-mihaita"Vizman, Cornelia"https://zbmath.org/authors/?q=ai:vizman.corneliaThe properties that are of linear algebraic nature of Poisson, Jacobi and contact geometries are considered.
The theory of the linear version of so-called contact dual pairs is developed. In particular, it is proved that in a full linear contact dual pair, the characteristic subspace of the two Jacobi vector spaces are either both odd-dimensional contact, or both even-dimensional locally conformally symplectic.
Reviewer: Mihail Banaru (Smolensk)Asymptotics of eigenfunctions of the bouncing ball type of the operator \(\nabla D(x)\nabla\) in a domain bounded by semirigid walls.https://zbmath.org/1460.810322021-06-15T18:09:00+00:00"Klevin, A. I."https://zbmath.org/authors/?q=ai:klevin.a-iSummary: We consider the problem on the semiclassical spectrum of the operator \(\nabla D(x)\nabla\) with Bessel-type degeneration on the boundary of a two-dimensional domain (semirigid walls). It is well known that the asymptotic eigenfunctions associated with Lagrangian manifolds can be constructed using a modification of the Maslov canonical operator. We obtain asymptotic eigenfunctions associated with the simplest periodic trajectories of the corresponding Hamiltonian system with reflections on the domain boundary.A global version of Günther's polysymplectic formalism using vertical projections.https://zbmath.org/1460.530672021-06-15T18:09:00+00:00"McClain, Tom"https://zbmath.org/authors/?q=ai:mcclain.tomSummary: I construct a global version of the local polysymplectic approach to covariant Hamiltonian field theory pioneered by C. Günther. Beginning with the geometric framework of the theory, I specialize to vertical vector fields to construct the (poly)symplectic structures, derive Hamilton's field equations, and construct a generalized Poisson structure. I then examine a few key examples to determine the nature of the necessary vertical projections and find that the theory seems to provide the geometric analog of the canonical transformation approach to covariant Hamiltonian field theory advanced by Struckmeier and Redelbach. I conclude with a few remarks about possible applications of this framework to the geometric quantization of classical field theories.Lattices for Landau-Ginzburg orbifolds.https://zbmath.org/1460.320652021-06-15T18:09:00+00:00"Ebeling, Wolfgang"https://zbmath.org/authors/?q=ai:ebeling.wolfgang"Takahashi, Atsushi"https://zbmath.org/authors/?q=ai:takahashi.atsushi.3A Landau-Ginzburg orbifold is a pair \(( f , G)\) where \(f\) is an invertible polynomial (a quasi-homogeneous polynomial with as many terms as variables) and \(G\) a finite abelian group of symmetries of \(f\). For such orbifolds there is a Berglund-Hübsch-Henningson duality. In joint work of the authors with \textit{S. M. Gusein-Zade} [J. Geom. Phys. 106, 184--191 (2016; Zbl 1379.32025)] a symmetry property of the orbifold E-functions of dual pairs has been established. It is used here to show symmetry of the orbifoldized elliptic genera.
Motivated by a formula proved here for the signature of the Milnor fibre in terms of the elliptic genus \(Z(\tau, z)\) in the non orbifold case, a definition is given for the orbifoldized signature. This raises the question of the existence of a lattice with this signature. For \(n = 3\), two lattices are defined, one for the A-model when \(G\subset \text{SL}(3;\mathbb{C}) \cap G_f\), the other one for the dual B-model.
The first uses the crepant resolution \(Y\) of the ambient space \(\mathbb{C}^3/G\). Let \(Z\) be the inverse image of the zero set of \(f\). The lattice is the free part of the image of \(H^3(Y , Z;\mathbb{Z})\) in \(H^2(Z;\mathbb{Z})\). For the trivial case \(G=\{\text{id}\}\) this is the usual Milnor lattice. The lattice has the correct rank and signature. It is described in detail for \(14+8\) pairs \(( f , G)\).
The other lattice comes from the numerical Grothendieck group of the stable homotopy category of \(L_{\widehat G}\)-graded matrix factorisations of \(f\), where \(L_{\widehat G}\) is related to the maximal grading of \(f\). The rank of this lattice is equal to the rank of the first lattice for the dual pair. For \(f\) defining an ADE singularity is the root lattice of the corresponding type. The Authors conjecture that these lattices are interchanged under the duality of pairs.
\textit{S. M. Gusein-Zade} and the first author [Manuscr. Math. 155, No. 3--4, 335--353 (2018; Zbl 1393.14056)] also defined an orbifold version of the Milnor lattice for a pair \(( f , G)\). It is not known whether that lattice coincides with one of the lattices here.
Reviewer: Jan Stevens (Göteborg)Homotopy algebras, deformation theory and quantization. Selected papers based on the presentations at the conference, September 16 and 22, 2018, Poznań, Poland.https://zbmath.org/1460.170022021-06-15T18:09:00+00:00"Fialowski, Alice (ed.)"https://zbmath.org/authors/?q=ai:fialowski.alice"Grabowska, Katarzyna (ed.)"https://zbmath.org/authors/?q=ai:grabowska.katarzyna"Grabowski, Janusz (ed.)"https://zbmath.org/authors/?q=ai:grabowski.janusz"Schlichenmaier, Martin (ed.)"https://zbmath.org/authors/?q=ai:schlichenmaier.martinThe articles of this volume will be reviewed individually.
Indexed articles:
\textit{Felder, Giovanni}, Derived representation schemes and supersymmetric gauge theory, 9-35 [Zbl 07350407]
\textit{Gutt, Simone}, Group actions in deformation quantization, 37-62 [Zbl 07350408]
\textit{Stiénon, Mathieu; Xu, Ping}, Atiyah classes and Kontsevich-Duflo type theorem for dg manifolds, 63-110 [Zbl 07350409]
\textit{Sheinman, Oleg}, Quantization of Lax integrable systems and conformal field theory, 111-122 [Zbl 07350410]
\textit{Kiselev, Arthemy V.; Buring, Ricardo}, The Kontsevich graph orientation morphism revisited, 123-139 [Zbl 07350411]
\textit{Ecker, Jill; Schlichenmaier, Martin}, The low-dimensional algebraic cohomology of the Witt and the Virasoro algebra with values in natural modules, 141-174 [Zbl 07350412]
\textit{Fialowski, Alice; Iohara, Kenji}, On Lie algebras of generalized Jacobi matrices, 175-186 [Zbl 07350413]BV and BFV for the H-twisted Poisson sigma model.https://zbmath.org/1460.810512021-06-15T18:09:00+00:00"Ikeda, Noriaki"https://zbmath.org/authors/?q=ai:ikeda.noriaki"Strobl, Thomas"https://zbmath.org/authors/?q=ai:strobl.thomasSummary: We present the BFV and the BV extension of the Poisson sigma model (PSM) twisted by a closed 3-form \(H\). There exist superfield versions of these functionals such as for the PSM and, more generally, for the AKSZ sigma models. However, in contrast to those theories, they depend on the Euler vector field of the source manifold and contain terms mixing data from the source and the target manifold. Using an auxiliary connection \(\nabla\) on the target manifold \(M\), we obtain alternative, purely geometrical expressions without the use of superfields, which are new also for the ordinary PSM and promise adaptations to other Lie algebroid-based gauge theories: The BV functional, in particular, is the sum of the classical action, the Hamiltonian lift of the (only on-shell nilpotent) BRST differential, and a term quadratic in the antifields which is essentially the basic curvature and measures the compatibility of \(\nabla\) with the Lie algebroid structure on \(T^*M\). We finally construct a \(\operatorname{Diff}(M)\)-equivariant isomorphism between the two BV formulations.Remarks on the homological mirror symmetry for tori.https://zbmath.org/1460.140882021-06-15T18:09:00+00:00"Kobayashi, Kazushi"https://zbmath.org/authors/?q=ai:kobayashi.kazushiSummary: Let us consider an \(n\)-dimensional complex torus \(T_{J=T}^{2 n}:=\mathbb{C}^n \slash 2\pi(\mathbb{Z}^n\oplus T\mathbb{Z}^n)\). Here, \(T\) is a complex matrix of order \(n\) whose imaginary part is positive definite. In particular, when we consider the case \(n=1\), the complexified symplectic form of a mirror partner of \(T_{J=T}^2\) is defined by using \(-\frac{1}{T}\) or \(T\). However, if we assume \(n\geq 2\) and that \(T\) is a singular matrix, we cannot define a mirror partner of \(T_{J=T}^{2n}\) as a natural generalization of the case \(n=1\) to the higher dimensional case. In this paper, we propose a way to avoid this problem, and discuss the homological mirror symmetry.Sharp systolic inequalities for Riemannian and Finsler spheres of revolution.https://zbmath.org/1460.530372021-06-15T18:09:00+00:00"Abbondandolo, Alberto"https://zbmath.org/authors/?q=ai:abbondandolo.alberto"Bramham, Barney"https://zbmath.org/authors/?q=ai:bramham.barney"Hryniewicz, Umberto L."https://zbmath.org/authors/?q=ai:hryniewicz.umberto-l"Salomão, Pedro A. S."https://zbmath.org/authors/?q=ai:salomao.pedro-a-sThe systolic ratio of a two-dimensional Riemannian sphere \(S\) is the positive number
\[
\rho_{\mathrm{sys}}(S)=\frac{l_{\min}(S)^2}{\mathrm{area}(S)},
\]
where \(l_{\min}(S)\) denotes the length of the shortest non-constant closed geodesic on \(S\) and \(\mathrm{area}(S)\) is the Riemannian area of \(S\). This number is invariant by isometries and rescaling. The authors prove that the systolic ratio of a sphere of revolution \(S\) does not exceed \(\pi\) and equals \(\pi\) if and only if \(S\) is Zoll, i.e., all of its geodesics are closed and have the same length. By the term, sphere of
revolution, the authors mean a smooth surface \(S\) in \(\mathbb{R}^3\) which is diffeomorphic to a sphere and is invariant with respect to the rotations around the \(z\)-axis. More generally, the authors show that in the rotationally symmetric Finsler metrics on a sphere of revolution, the systolic ratio of a sphere of revolution \(S\) does not exceed \(\pi\) and equals \(\pi\) if and only if the metric is Riemannian and Zoll.
Reviewer: Chandan Kumar Mondal (Durgapur)Characteristic Jacobi operator on almost cosymplectic 3-manifolds.https://zbmath.org/1460.530352021-06-15T18:09:00+00:00"Inoguchi, Jun-Ichi"https://zbmath.org/authors/?q=ai:inoguchi.jun-ichiSummary: The Ricci tensor, \(\varphi\)-Ricci tensor and the characteristic Jacobi operator on cosymplectic 3-manifolds are investigated.Classification of flat Lagrangian H-umbilical submanifolds in indefinite complex Euclidean spaces.https://zbmath.org/1460.530692021-06-15T18:09:00+00:00"Deng, Shangrong"https://zbmath.org/authors/?q=ai:deng.shangrongSummary: In this article, we completely characterize flat Lagrangian H-umbilical submanifolds in the indefinite complex Euclidean spaces \(C^n_s\). Consequently, in conjunction with a result from [\textit{B.-Y. Chen}, Tohoku Math. J. (2) 51, No. 1, 13--20 (1999; Zbl 0986.53020)], Lagrangian \(H\)-umbilical submanifolds in the indefinite complex Euclidean \(n\)-space \(C^n_s\) with \(n > 2\) are completely classified.On quasi-Sasakian \(3\)-manifolds with respect to the Schouten-Van Kampen connection.https://zbmath.org/1460.530472021-06-15T18:09:00+00:00"Perktaş, Selcen Yüksel"https://zbmath.org/authors/?q=ai:perktas.selcen-yuksel"Yildiz, Ahmet"https://zbmath.org/authors/?q=ai:yildiz.ahmetSummary: In this paper we study some soliton types on a quasi-Sasakian 3-manifold with respect to the Schouten-van Kampen connection.Vafa-Witten invariants for projective surfaces. I: Stable case.https://zbmath.org/1460.530272021-06-15T18:09:00+00:00"Tanaka, Yuuji"https://zbmath.org/authors/?q=ai:tanaka.yuuji"Thomas, Richard P."https://zbmath.org/authors/?q=ai:thomas.richard-pThe theory of Vafa-Witten invariants on complex (Kähler) projective surfaces is created.
The startpoint is the consideration of the solutions of the Vafa-Witten equations on such surfaces.
This theory includes, in particular, the construction of \(\operatorname{U}(r)\) and \(\operatorname{SU}(r)\) Vafa-Witten invariants.
Note that Vafa-Witten invariants are related to the deformation theory of Higgs spaces.
Reviewer: Mihail Banaru (Smolensk)A summary on symmetries and conserved quantities of autonomous Hamiltonian systems.https://zbmath.org/1460.370522021-06-15T18:09:00+00:00"Román-Roy, Narciso"https://zbmath.org/authors/?q=ai:roman-roy.narcisoThe existence of symmetries of Hamiltonian and Lagrangian systems is related to the existence of conserved quantities. As it is well known, the standard procedure to obtain conserved quantities consists in introducing the so-called Noether symmetries, and then use the Noether theorem. However, these kinds of symmetries do not exhaust the set of symmetries. As it is known, there are symmetries which are not of Noether type, but they also generate conserved quantities and they are sometimes called hidden symmetries. In this paper the author establishes a complete scheme of classification of all the different kinds of symmetries of Hamiltonian systems, explaining how to obtain the associated conserved quantities in each case. The author follows the same lines of argument as in the analysis made in [\textit{W. Sarlet} and \textit{F. Cantrijn}, J. Phys. A, Math. Gen. 14, 479--492 (1981; Zbl 0464.58010)] for nonautonomous Lagrangian systems, where the authors obtained conserved quantities for different kinds of symmetries that do not leave the Poincaré-Cartan form invariant.
Reviewer: Nicolai K. Smolentsev (Kemerovo)Poisson-Lie algebras and singular symplectic forms associated to corank 1 type singularities.https://zbmath.org/1460.530722021-06-15T18:09:00+00:00"Fukuda, T."https://zbmath.org/authors/?q=ai:fukuda.takuo"Janeczko, S."https://zbmath.org/authors/?q=ai:janeczko.stanislawSummary: We show that there exists a natural Poisson-Lie algebra associated to a singular symplectic structure \(\omega \). We construct Poisson-Lie algebras for the Martinet and Roussarie types of singularities. In the special case when the singular symplectic structure is given by the pullback from the Darboux form, \( \omega=F^*\omega_0\), this Poisson-Lie algebra is a basic symplectic invariant of the singularity of the smooth mapping \(F\) into the symplectic space \((\mathbb{R}^{2n},\omega_0)\). The case of \(A_k\) singularities of pullbacks is considered, and Poisson-Lie algebras for \(\Sigma_{2,0}\), \(\Sigma_{2,2,0}^{\mathrm{e}}\) and \(\Sigma_{2,2,0}^{\mathrm{h}}\) stable singularities of 2-forms are calculated.Characterization of toric systems via transport costs.https://zbmath.org/1460.370532021-06-15T18:09:00+00:00"Hohloch, Sonja"https://zbmath.org/authors/?q=ai:hohloch.sonjaThe author investigates completely integrable Hamiltonian systems that induce effective torus actions. Delzant's symplectic classification of completely integrable systems on compact manifolds in [\textit{T. Delzant}, Bull. Soc. Math. Fr. 116, No. 3, 315--339 (1988; Zbl 0676.58029)] describes those that induce effective Hamiltonian torus actions, called compact toric actions. The invariant that characterizes a toric system is the image of its momentum map. This is a convex polytope, and this class of convex polytopes represents all possible compact toric systems up to equivariant symplectomorphism. Consequently a toric system is determined by a finite set of data given by its momentum polytope.
This paper presents a characterization of toric systems using transport costs. Transport costs measure how toric or non-toric a system is. Toric systems on \(2n\)-dimensional manifolds are those systems that have zero transport costs with respect to the time-\(T\) map, where \(T \in \mathbb{R}^n\) is the period of the acting \(n\)-torus.
The setting is as follows. \((M, \omega)\) is a \(2n\)-dimensional compact symplectic manifold with momentum map \(h = (h_1, h_2, \dots, h_n) : M \rightarrow \mathbb{R}^n\) and \(\varphi^h : \mathbb{R} \times M \rightarrow M\) is the flow associated with \(h\). A system is then called toric if the action generated by the flow \(\varphi^h\) is an effective \(\mathbb{T}^n\)-action and \(2\pi\) is the minimum common period of the \(n\) Hamiltonian circle actions induced by \(h_1, \dots, h_n : M \rightarrow \mathbb{R}\).
The author's main result uses a cost functional \(C^h_t (U, c) = \int_U c(x, \varphi_t^h (x)) \,d\mu_\omega\), where \(c: M \times M \rightarrow \mathbb{R}^+\) is a continuous metric-like cost function, \(U \subseteq M \) is open, and \(d \mu_\omega\) is the measure associated with the volume form \(\omega^n\). The \(T\)-periodicity cost of \((M,\omega,h)\) with respect to the cost function \(c\) is \(C_T^h (M,c)\). The main result is then that \((M,\omega,h)\) is toric if and only if \(C^h_{(2\pi, \dots, 2\pi)}(M,c) = 0\) and \(C_s^h (M,c) > 0\) for all \(s \in \mathbb{R}^n\) except for \(\{2\pi k, \dots, 2\pi k\}\) where \(k\) is an integer.
Reviewer: William J. Satzer Jr. (St. Paul)