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Journal articles on the topic 'Holomorphic symplectic manifold'

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Author: Grafiati

Published: 9 March 2023

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1

Mayrand, Maxence. "Hyperkähler metrics near Lagrangian submanifolds and symplectic groupoids." Journal für die reine und angewandte Mathematik (Crelles Journal) 2022, no. 782 (October 26, 2021): 197–218. http://dx.doi.org/10.1515/crelle-2021-0059.

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Abstract The first part of this paper is a generalization of the Feix–Kaledin theorem on the existence of a hyperkähler metric on a neighborhood of the zero section of the cotangent bundle of a Kähler manifold. We show that the problem of constructing a hyperkähler structure on a neighborhood of a complex Lagrangian submanifold in a holomorphic symplectic manifold reduces to the existence of certain deformations of holomorphic symplectic structures. The Feix–Kaledin structure is recovered from the twisted cotangent bundle. We then show that every holomorphic symplectic groupoid over a compact holomorphic Poisson surface of Kähler type has a hyperkähler structure on a neighborhood of its identity section. More generally, we reduce the existence of a hyperkähler structure on a symplectic realization of a holomorphic Poisson manifold of any dimension to the existence of certain deformations of holomorphic Poisson structures adapted from Hitchin’s unobstructedness theorem.

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2

Ali, Danish, Johann Davidov, and Oleg Mushkarov. "Holomorphic curvatures of twistor spaces." International Journal of Geometric Methods in Modern Physics 11, no. 03 (March 2014): 1450022. http://dx.doi.org/10.1142/s0219887814500224.

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We study the twistor spaces of oriented Riemannian 4-manifolds as a source of almost Hermitian 6-manifolds of constant or strictly positive holomorphic, Hermitian and orthogonal bisectional curvatures. In particular, we obtain explicit formulas for these curvatures in the case when the base manifold is Einstein and self-dual, and observe that the "squashed" metric on ℂℙ3 is a non-Kähler Hermitian–Einstein metric of positive holomorphic bisectional curvature. This shows that a recent result of Kalafat and Koca [M. Kalafat and C. Koca, Einstein–Hermitian 4-manifolds of positive bisectional curvature, preprint (2012), arXiv: 1206.3941v1 [math.DG]] in dimension four cannot be extended to higher dimensions. We prove that the Hermitian bisectional curvature of a non-Kähler Hermitian manifold is never a nonzero constant which gives a partial negative answer to a question of Balas and Gauduchon [A. Balas and P. Gauduchon, Any Hermitian metric of constant non-positive (Hermitian) holomorphic sectional curvature on a compact complex surface is Kähler, Math. Z.190 (1985) 39–43]. Finally, motivated by an integrability result of Vezzoni [L. Vezzoni, On the Hermitian curvature of symplectic manifolds, Adv. Geom.7 (2007) 207–214] for almost Kähler manifolds, we study the problem when the holomorphic and the Hermitian bisectional curvatures of an almost Hermitian manifold coincide. We extend the result of Vezzoni to a more general class of almost Hermitian manifolds and describe the twistor spaces having this curvature property.

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3

Biswas, Indranil, Tomás L. Gómez, and André Oliveira. "Complex Lagrangians in a hyperKähler manifold and the relative Albanese." Complex Manifolds 7, no. 1 (October 27, 2020): 230–40. http://dx.doi.org/10.1515/coma-2020-0106.

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AbstractLet M be the moduli space of complex Lagrangian submanifolds of a hyperKähler manifold X, and let ω̄ : 𝒜̂ → M be the relative Albanese over M. We prove that 𝒜̂ has a natural holomorphic symplectic structure. The projection ω̄ defines a completely integrable structure on the symplectic manifold 𝒜̂. In particular, the fibers of ω̄ are complex Lagrangians with respect to the symplectic form on 𝒜̂. We also prove analogous results for the relative Picard over M.

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4

Mongardi, Giovanni, Antonio Rapagnetta, and Giulia Saccà. "The Hodge diamond of O’Grady’s six-dimensional example." Compositio Mathematica 154, no. 5 (March 21, 2018): 984–1013. http://dx.doi.org/10.1112/s0010437x1700803x.

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We realize O’Grady’s six-dimensional example of an irreducible holomorphic symplectic (IHS) manifold as a quotient of an IHS manifold of$\text{K3}^{[3]}$type by a birational involution, thereby computing its Hodge numbers.

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5

Cho, Yunhyung. "Classification of six-dimensional monotone symplectic manifolds admitting semifree circle actions I." International Journal of Mathematics 30, no. 06 (June 2019): 1950032. http://dx.doi.org/10.1142/s0129167x19500320.

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Let [Formula: see text] be a six-dimensional closed monotone symplectic manifold admitting an effective semifree Hamiltonian [Formula: see text]-action. We show that if the minimal (or maximal) fixed component of the action is an isolated point, then [Formula: see text] is [Formula: see text]-equivariantly symplectomorphic to some Kähler Fano manifold [Formula: see text] with a certain holomorphic [Formula: see text]-action. We also give a complete list of all such Fano manifolds and describe all semifree [Formula: see text]-actions on them specifically.

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6

Tardini, Nicoletta, and Adriano Tomassini. "On the cohomology of almost-complex and symplectic manifolds and proper surjective maps." International Journal of Mathematics 27, no. 12 (November 2016): 1650103. http://dx.doi.org/10.1142/s0129167x16501032.

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Let [Formula: see text] be an almost-complex manifold. In [Comparing tamed and compatible symplectic cones and cohomological properties of almost-complex manifolds, Comm. Anal. Geom. 17 (2009) 651–683], Li and Zhang introduce [Formula: see text] as the cohomology subgroups of the [Formula: see text]th de Rham cohomology group formed by classes represented by real pure-type forms. Given a proper, surjective, pseudo-holomorphic map between two almost-complex manifolds, we study the relationship among such cohomology groups. Similar results are proven in the symplectic setting for the cohomology groups introduced in [Cohomology and Hodge Theory on Symplectic manifolds: I, J. Differ. Geom. 91(3) (2012) 383–416] by Tseng and Yau and a new characterization of the hard Lefschetz condition in dimension [Formula: see text] is provided.

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7

Cahen, Michel, Thibaut Grouy, and Simone Gutt. "A possible symplectic framework for Radon-type transforms." International Journal of Geometric Methods in Modern Physics 13, Supp. 1 (October 2016): 1641002. http://dx.doi.org/10.1142/s0219887816410024.

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Our project is to define Radon-type transforms in symplectic geometry. The chosen framework consists of symplectic symmetric spaces whose canonical connection is of Ricci-type. They can be considered as symplectic analogues of the spaces of constant holomorphic curvature in Kählerian Geometry. They are characterized amongst a class of symplectic manifolds by the existence of many totally geodesic symplectic submanifolds. We present a particular class of Radon type transforms, associating to a smooth compactly supported function on a homogeneous manifold [Formula: see text], a function on a homogeneous space [Formula: see text] of totally geodesic submanifolds of [Formula: see text], and vice versa. We describe some spaces [Formula: see text] and [Formula: see text] in such Radon-type duality with [Formula: see text] a model of symplectic symmetric space with Ricci-type canonical connection and [Formula: see text] an orbit of totally geodesic symplectic submanifolds.

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8

DUISTERMAAT, J. J., and A. PELAYO. "COMPLEX STRUCTURES ON FOUR-MANIFOLDS WITH SYMPLECTIC TWO-TORUS ACTIONS." International Journal of Mathematics 22, no. 03 (March 2011): 449–63. http://dx.doi.org/10.1142/s0129167x11006854.

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We apply the general theory for symplectic torus actions with symplectic or coisotropic orbits to prove that a four-manifold with a symplectic two-torus action admits an invariant complex structure and give an identification of those that do not admit a Kähler structure with Kodaira's class of complex surfaces which admit a nowhere vanishing holomorphic (2,0)-form, but are not a torus nor a K3 surface.

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9

Solomon, Jake P., and Misha Verbitsky. "Locality in the Fukaya category of a hyperkähler manifold." Compositio Mathematica 155, no. 10 (September 6, 2019): 1924–58. http://dx.doi.org/10.1112/s0010437x1900753x.

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Let $(M,I,J,K,g)$ be a hyperkähler manifold. Then the complex manifold $(M,I)$ is holomorphic symplectic. We prove that for all real $x,y$, with $x^{2}+y^{2}=1$ except countably many, any finite-energy $(xJ+yK)$-holomorphic curve with boundary in a collection of $I$-holomorphic Lagrangians must be constant. By an argument based on the Łojasiewicz inequality, this result holds no matter how the Lagrangians intersect each other. It follows that one can choose perturbations such that the holomorphic polygons of the associated Fukaya category lie in an arbitrarily small neighborhood of the Lagrangians. That is, the Fukaya category is local. We show that holomorphic Lagrangians are tautologically unobstructed. Moreover, the Fukaya $A_{\infty }$ algebra of a holomorphic Lagrangian is formal. Our result also explains why the special Lagrangian condition holds without instanton corrections for holomorphic Lagrangians.

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10

Braverman, Maxim. "Symplectic cutting of Kähler manifolds." Journal für die reine und angewandte Mathematik (Crelles Journal) 1999, no. 508 (March 12, 1999): 85–98. http://dx.doi.org/10.1515/crll.1999.508.85.

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Abstract We obtain estimates on the character of the cohomology of an S1-equivariant holomorphic vector bundle over a Kähler manifold M in terms of the cohomology of the Lerman symplectic cuts and the symplectic reduction of M. In particular, we prove and extend inequalities conjectured by Wu and Zhang. The proof is based on constructing a flat family of complex spaces Mt (t ∈ ℂ) such that Mt is isomorphic to M for t ≠ 0, while M0 is a singular reducible complex space, whose irreducible components are the Lerman symplectic cuts.

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11

Amerik, Ekaterina, and Misha Verbitsky. "Construction of automorphisms of hyperkähler manifolds." Compositio Mathematica 153, no. 8 (May 31, 2017): 1610–21. http://dx.doi.org/10.1112/s0010437x17007138.

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Let $M$ be an irreducible holomorphic symplectic (hyperkähler) manifold. If $b_{2}(M)\geqslant 5$, we construct a deformation $M^{\prime }$ of $M$ which admits a symplectic automorphism of infinite order. This automorphism is hyperbolic, that is, its action on the space of real $(1,1)$-classes is hyperbolic. If $b_{2}(M)\geqslant 14$, similarly, we construct a deformation which admits a parabolic automorphism (and many other automorphisms as well).

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12

Ran, Ziv. "A BOGOMOLOV UNOBSTRUCTEDNESS THEOREM FOR LOG-SYMPLECTIC MANIFOLDS IN GENERAL POSITION." Journal of the Institute of Mathematics of Jussieu 19, no. 5 (November 9, 2018): 1509–19. http://dx.doi.org/10.1017/s1474748018000464.

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We consider compact Kählerian manifolds $X$ of even dimension 4 or more, endowed with a log-symplectic holomorphic Poisson structure $\unicode[STIX]{x1D6F1}$ which is sufficiently general, in a precise linear sense, with respect to its (normal-crossing) degeneracy divisor $D(\unicode[STIX]{x1D6F1})$. We prove that $(X,\unicode[STIX]{x1D6F1})$ has unobstructed deformations, that the tangent space to its deformation space can be identified in terms of the mixed Hodge structure on $H^{2}$ of the open symplectic manifold $X\setminus D(\unicode[STIX]{x1D6F1})$, and in fact coincides with this $H^{2}$ provided the Hodge number $h_{X}^{2,0}=0$, and finally that the degeneracy locus $D(\unicode[STIX]{x1D6F1})$ deforms locally trivially under deformations of $(X,\unicode[STIX]{x1D6F1})$.

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13

Jardim, Marcos, and Misha Verbitsky. "Trihyperkähler reduction and instanton bundles on." Compositio Mathematica 150, no. 11 (August 27, 2014): 1836–68. http://dx.doi.org/10.1112/s0010437x14007477.

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AbstractA trisymplectic structure on a complex $2n$-manifold is a three-dimensional space ${\rm\Omega}$ of closed holomorphic forms such that any element of ${\rm\Omega}$ has constant rank $2n$, $n$ or zero, and degenerate forms in ${\rm\Omega}$ belong to a non-degenerate quadric hypersurface. We show that a trisymplectic manifold is equipped with a holomorphic 3-web and the Chern connection of this 3-web is holomorphic, torsion-free, and preserves the three symplectic forms. We construct a trisymplectic structure on the moduli of regular rational curves in the twistor space of a hyperkähler manifold, and define a trisymplectic reduction of a trisymplectic manifold, which is a complexified form of a hyperkähler reduction. We prove that the trisymplectic reduction in the space of regular rational curves on the twistor space of a hyperkähler manifold $M$ is compatible with the hyperkähler reduction on $M$. As an application of these geometric ideas, we consider the ADHM construction of instantons and show that the moduli space of rank $r$, charge $c$ framed instanton bundles on $\mathbb{C}\mathbb{P}^{3}$ is a smooth trisymplectic manifold of complex dimension $4rc$. In particular, it follows that the moduli space of rank two, charge $c$ instanton bundles on $\mathbb{C}\mathbb{P}^{3}$ is a smooth complex manifold dimension $8c-3$, thus settling part of a 30-year-old conjecture.

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14

DORFMEISTER, JOSEF G., and TIAN-JUN LI. "RELATIVE RUAN AND GROMOV–TAUBES INVARIANTS OF SYMPLECTIC 4-MANIFOLDS." Communications in Contemporary Mathematics 15, no. 01 (January 22, 2013): 1250062. http://dx.doi.org/10.1142/s0219199712500629.

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We define relative Ruan invariants that count embedded connected symplectic submanifolds which contact a fixed symplectic hypersurface V in a symplectic 4-manifold (X, ω) at prescribed points with prescribed contact orders (in addition to insertions on X\V). We obtain invariants of the deformation class of (X, V, ω). Two large issues must be tackled to define such invariants: (1) curves lying in the hypersurface V and (2) genericity results for almost complex structures constrained to make V pseudo-holomorphic (or almost complex). Moreover, these invariants are refined to take into account rim-tori decompositions. In the latter part of the paper, we extend the definition to disconnected submanifolds and construct relative Gromov–Taubes invariants.

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15

Yu, Tony Yue. "Gromov compactness in non-archimedean analytic geometry." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 741 (August 1, 2018): 179–210. http://dx.doi.org/10.1515/crelle-2015-0077.

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Abstract Gromov’s compactness theorem for pseudo-holomorphic curves is a foundational result in symplectic geometry. It controls the compactness of the moduli space of pseudo-holomorphic curves with bounded area in a symplectic manifold. In this paper, we prove the analog of Gromov’s compactness theorem in non-archimedean analytic geometry. We work in the framework of Berkovich spaces. First, we introduce a notion of Kähler structure in non-archimedean analytic geometry using metrizations of virtual line bundles. Second, we introduce formal stacks and non-archimedean analytic stacks. Then we construct the moduli stack of non-archimedean analytic stable maps using formal models, Artin’s representability criterion and the geometry of stable curves. Finally, we reduce the non-archimedean problem to the known compactness results in algebraic geometry. The motivation of this paper is to provide the foundations for non-archimedean enumerative geometry.

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16

Trautwein, Samuel. "Convergence of the Yang–Mills–Higgs flow on Gauged Holomorphic maps and applications." International Journal of Mathematics 29, no. 04 (April 2018): 1850024. http://dx.doi.org/10.1142/s0129167x18500246.

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The symplectic vortex equations admit a variational description as global minimum of the Yang–Mills–Higgs functional. We study its negative gradient flow on holomorphic pairs [Formula: see text] where [Formula: see text] is a connection on a principal [Formula: see text]-bundle [Formula: see text] over a closed Riemann surface [Formula: see text] and [Formula: see text] is an equivariant map into a Kähler Hamiltonian [Formula: see text]-manifold. The connection [Formula: see text] induces a holomorphic structure on the Kähler fibration [Formula: see text] and we require that [Formula: see text] descends to a holomorphic section of this fibration. We prove a Łojasiewicz type gradient inequality and show uniform convergence of the negative gradient flow in the [Formula: see text]-topology when [Formula: see text] is equivariantly convex at infinity with proper moment map, [Formula: see text] is holomorphically aspherical and its Kähler metric is analytic. As applications we establish several results inspired by finite dimensional GIT: First, we prove a certain uniqueness property for the critical points of the Yang–Mills–Higgs functional which is the analogue of the Ness uniqueness theorem. Second, we extend Mundet’s Kobayashi–Hitchin correspondence to the polystable and semistable case. The arguments for the polystable case lead to a new proof in the stable case. Third, in proving the semistable correspondence, we establish the moment–weight inequality for the vortex equation and prove the analogue of the Kempf existence and uniqueness theorem.

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17

BYTSENKO, A. A., M. CHAICHIAN, A. TUREANU, and F. L. WILLIAMS. "BRST-INVARIANT DEFORMATIONS OF GEOMETRIC STRUCTURES IN TOPOLOGICAL FIELD THEORIES." International Journal of Modern Physics A 28, no. 16 (June 28, 2013): 1350069. http://dx.doi.org/10.1142/s0217751x13500693.

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We study a Lie algebra of formal vector fields Wn with its application to the perturbative deformed holomorphic symplectic structure in the A-model, and a Calabi–Yau manifold with boundaries in the B-model. A relevant concept in the vertex operator algebra and the BRST cohomology is that of the elliptic genera (the one-loop string partition function). We show that the elliptic genera can be written in terms of spectral functions of the hyperbolic three-geometry (which inherits the cohomology structure of BRST-like operator). We show that equivalence classes of deformations are described by a Hochschild cohomology theory of the DG-algebra [Formula: see text], which is defined to be the cohomology of (-1)n Q + d Hoch . Here, [Formula: see text] is the initial nondeformed BRST operator while ∂ deform is the deformed part whose algebra is a Lie algebra of linear vector fields gl n. We discuss the identification of the harmonic structure (HT•(X);HΩ•(X)) of affine space X and the group [Formula: see text] (the HKR isomorphism), and bulk-boundary deformation pairing.

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18

BYTSENKO, A. A. "BRST-INVARIANT DEFORMATIONS OF GEOMETRIC STRUCTURES IN SIGMA MODELS." International Journal of Modern Physics A 26, no. 22 (September 10, 2011): 3769–80. http://dx.doi.org/10.1142/s0217751x11054231.

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The closed string correlators can be constructed from the open ones using topological string theories as a model. The space of physical closed string states is isomorphic to the Hochschild cohomology of (A,Q) (operator Q of ghost number one), - this statement has been verified by means of computation of the Hochschild cohomology of the category of D -branes. We study a Lie algebra of formal vector fields Wn with its application to the perturbative deformed holomorphic symplectic structure in the A -model, and a Calabi-Yau manifold with boundaries in the B -model. We show that equivalent classes of deformations are describing by a Hochschild cohomology theory of the DG-algebra, [Formula: see text], [Formula: see text], which is defined to be the cohomology of (-1)nQ+d Hoch . Here [Formula: see text] is the initial non-deformed BRST operator while ∂ deform is the deformed part whose algebra is a Lie algebra of linear vector fields gl n. We assume that if in the theory exists a single D -brane then all the information associated with deformations is encoded in an associative algebra A equipped with a differential [Formula: see text]. In addition equivalence classes of deformations of these data are described by a Hochschild cohomology of (A,Q), an important geometric invariant of the (anti)holomorphic structure on X. We also discuss the identification of the harmonic structure (HT•(X); HΩ•(X)) of affine space X and the group [Formula: see text] (the HKR isomorphism), and bulk-boundary deformation pairing.

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19

BYTSENKO, A. A. "BRST-INVARIANT DEFORMATIONS OF GEOMETRIC STRUCTURES IN SIGMA MODELS." International Journal of Modern Physics: Conference Series 03 (January 2011): 75–86. http://dx.doi.org/10.1142/s2010194511001164.

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The closed string correlators can be constructed from the open ones using topological string theories as a model. The space of physical closed string states is isomorphic to the Hochschild cohomology of (A, Q) (operator Q of ghost number one), - this statement has been verified by means of computation of the Hochschild cohomology of the category of D-branes. We study a Lie algebra of formal vector fields Wn with its application to the perturbative deformed holomorphic symplectic structure in the A-model, and a Calabi-Yau manifold with boundaries in the B-model. We show that equivalent classes of deformations are describing by a Hochschild cohomology theory of the DG-algebra [Formula: see text], [Formula: see text], which is defined to be the cohomology of (-1)n Q + d Hoch . Here [Formula: see text] is the initial non-deformed BRST operator while ∂deform is the deformed part whose algebra is a Lie algebra of linear vector fields gl n. We assume that if in the theory exists a single D-brane then all the information associated with deformations is encoded in an associative algebra A equipped with a differential [Formula: see text]. In addition equivalence classes of deformations of these data are described by a Hochschild cohomology of (A, Q), an important geometric invariant of the (anti)holomorphic structure on X. We also discuss the identification of the harmonic structure (HT•(X); HΩ•(X)) of affine space X and the group [Formula: see text] (the HKR isomorphism), and bulk-boundary deformation pairing.

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20

Zehmisch, Kai. "Holomorphic jets in symplectic manifolds." Journal of Fixed Point Theory and Applications 17, no. 2 (August 5, 2014): 379–402. http://dx.doi.org/10.1007/s11784-014-0178-z.

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21

Polterovich, Leonid. "Symplectic displacement energy for Lagrangian submanifolds." Ergodic Theory and Dynamical Systems 13, no. 2 (June 1993): 357–67. http://dx.doi.org/10.1017/s0143385700007410.

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AbstractRecently H. Hofer defined a new symplectic invariant which has a beautiful dynamical meaning. In the present paper we study this invariant for Lagrangian submanifolds of symplectic manifolds. Our approach is based on Gromov's theory of pseudo-holomorphic curves.

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22

HIND, R. "STEIN FILLINGS OF LENS SPACES." Communications in Contemporary Mathematics 05, no. 06 (December 2003): 967–82. http://dx.doi.org/10.1142/s0219199703001178.

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We describe a foliation by finite energy holomorphic curves of some symplectic manifolds which are constructed from Stein manifolds with Lens space boundaries. One application is that all such Stein manifolds bounded by the same contact Lens space are equivalent up to Stein homotopy.

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23

Beauville, Arnaud. "Antisymplectic involutions of holomorphic symplectic manifolds." Journal of Topology 4, no. 2 (2011): 300–304. http://dx.doi.org/10.1112/jtopol/jtr002.

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24

Gromov, M. "Pseudo holomorphic curves in symplectic manifolds." Inventiones Mathematicae 82, no. 2 (June 1985): 307–47. http://dx.doi.org/10.1007/bf01388806.

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25

Camere, Chiara. "Lattice polarized irreducible holomorphic symplectic manifolds." Annales de l’institut Fourier 66, no. 2 (2016): 687–709. http://dx.doi.org/10.5802/aif.3022.

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26

Broka, Damien, and Ping Xu. "Symplectic realizations of holomorphic Poisson manifolds." Mathematical Research Letters 29, no. 4 (2022): 903–44. http://dx.doi.org/10.4310/mrl.2022.v29.n4.a1.

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27

Camere, Chiara. "Some remarks on moduli spaces of lattice polarized holomorphic symplectic manifolds." Communications in Contemporary Mathematics 20, no. 04 (May 20, 2018): 1750044. http://dx.doi.org/10.1142/s0219199717500444.

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We construct quasi-projective moduli spaces of [Formula: see text]-general lattice polarized irreducible holomorphic symplectic manifolds. Moreover, we study their Baily–Borel compactification and investigate a relation between one-dimensional boundary components and equivalence classes of rational Lagrangian fibrations defined on mirror manifolds.

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28

Brecan, Ana-Maria, Tim Kirschner, and Martin Schwald. "Unobstructedness of hyperkähler twistor spaces." Mathematische Zeitschrift 300, no. 3 (October 6, 2021): 2485–517. http://dx.doi.org/10.1007/s00209-021-02841-4.

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AbstractA family of irreducible holomorphic symplectic (ihs) manifolds over the complex projective line has unobstructed deformations if its period map is an embedding. This applies in particular to twistor spaces of ihs manifolds. Moreover, a family of ihs manifolds over a subspace of the period domain extends to a universal family over an open neighborhood in the period domain.

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29

GEIGES, HANSJÖRG, and KAI ZEHMISCH. "Symplectic cobordisms and the strong Weinstein conjecture." Mathematical Proceedings of the Cambridge Philosophical Society 153, no. 2 (February 28, 2012): 261–79. http://dx.doi.org/10.1017/s0305004112000163.

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AbstractWe study holomorphic spheres in certain symplectic cobordisms and derive information about periodic Reeb orbits in the concave end of these cobordisms from the non-compactness of the relevant moduli spaces. We use this to confirm the strong Weinstein conjecture (predicting the existence of null-homologous Reeb links) for various higher-dimensional contact manifolds, including contact type hypersurfaces in subcritical Stein manifolds and in some cotangent bundles. The quantitative character of this result leads to the definition of a symplectic capacity.

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30

Boissière, Samuel, Marc Nieper-Wißkirchen, and Alessandra Sarti. "Smith theory and irreducible holomorphic symplectic manifolds." Journal of Topology 6, no. 2 (February 13, 2013): 361–90. http://dx.doi.org/10.1112/jtopol/jtt002.

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31

Franco, Emilio, Marcos Jardim, and Grégoire Menet. "Brane involutions on irreducible holomorphic symplectic manifolds." Kyoto Journal of Mathematics 59, no. 1 (April 2019): 195–235. http://dx.doi.org/10.1215/21562261-2018-0009.

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32

Ran, Ziv. "Deformations of holomorphic pseudo-symplectic Poisson manifolds." Advances in Mathematics 304 (January 2017): 1156–75. http://dx.doi.org/10.1016/j.aim.2016.09.016.

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33

Camere, Chiara, Grzegorz Kapustka, Michał Kapustka, and Giovanni Mongardi. "Verra Four-Folds, Twisted Sheaves, and the Last Involution." International Mathematics Research Notices 2019, no. 21 (February 1, 2018): 6661–710. http://dx.doi.org/10.1093/imrn/rnx327.

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Abstract We study the geometry of some moduli spaces of twisted sheaves on K3 surfaces. In particular we introduce induced automorphisms from a K3 surface on moduli spaces of twisted sheaves on this K3 surface. As an application we prove the unirationality of moduli spaces of irreducible holomorphic symplectic manifolds of K3[2]-type admitting non-symplectic involutions with invariant lattices U(2) ⊕ D4(−1) or U(2) ⊕ E8(−2). This complements the results obtained in [43], [13], and the results from [29] about the geometry of irreducible holomorphic symplectic (IHS) four-folds constructed using the Hilbert scheme of (1, 1) conics on Verra four-folds. As a byproduct we find that IHS four-folds of K3[2]-type with Picard lattice U(2) ⊕ E8(−2) naturally contain non-nodal Enriques surfaces.

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34

Golla, Marco, and Laura Starkston. "The symplectic isotopy problem for rational cuspidal curves." Compositio Mathematica 158, no. 7 (July 2022): 1595–682. http://dx.doi.org/10.1112/s0010437x2200762x.

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We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex projective plane. We prove that every such curve is isotopic to a complex curve in degrees up to five, and for curves with one singularity whose link is a torus knot. Classification results of symplectic isotopy classes rely on pseudo-holomorphic curves together with a symplectic version of birational geometry of log pairs and techniques from four-dimensional topology.

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Fu, Lie, and Grégoire Menet. "On the Betti numbers of compact holomorphic symplectic orbifolds of dimension four." Mathematische Zeitschrift 299, no. 1-2 (January 12, 2021): 203–31. http://dx.doi.org/10.1007/s00209-020-02682-7.

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AbstractWe extend a result of Guan by showing that the second Betti number of a 4-dimensional primitively symplectic orbifold is at most 23 and there are at most 91 singular points. The maximal possibility 23 can only occur in the smooth case. In addition to the known smooth examples with second Betti numbers 7 and 23, we provide examples of such orbifolds with second Betti numbers 3, 5, 6, 8, 9, 10, 11, 14 and 16. In an appendix, we extend Salamon’s relation among Betti/Hodge numbers of symplectic manifolds to symplectic orbifolds.

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Ye, Rugang. "Filling by holomorphic curves in symplectic 4-manifolds." Transactions of the American Mathematical Society 350, no. 1 (1998): 213–50. http://dx.doi.org/10.1090/s0002-9947-98-01970-9.

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O’Grady, K. G. "Involutions and linear systems on holomorphic symplectic manifolds." GAFA Geometric And Functional Analysis 15, no. 6 (November 25, 2005): 1223–74. http://dx.doi.org/10.1007/s00039-005-0538-3.

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Matsushita, Daisuke. "Equidimensionality of Lagrangian fibrations on holomorphic symplectic manifolds." Mathematical Research Letters 7, no. 4 (2000): 389–91. http://dx.doi.org/10.4310/mrl.2000.v7.n4.a4.

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Lehn, Christian. "Deformations of Lagrangian subvarieties of holomorphic symplectic manifolds." Mathematical Research Letters 23, no. 2 (2016): 473–97. http://dx.doi.org/10.4310/mrl.2016.v23.n2.a9.

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Chiang, River, and Liat Kessler. "Homologically trivial symplectic cyclic actions need not extend to Hamiltonian circle actions." Journal of Topology and Analysis 12, no. 04 (January 3, 2019): 1047–71. http://dx.doi.org/10.1142/s1793525319500742.

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We give examples of symplectic actions of a cyclic group, inducing a trivial action on homology, on four-manifolds that admit Hamiltonian circle actions, and show that they do not extend to Hamiltonian circle actions. Our work applies holomorphic methods to extend combinatorial tools developed for circle actions to study cyclic actions.

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Pym, Brent, and Pavel Safronov. "Shifted Symplectic Lie Algebroids." International Mathematics Research Notices 2020, no. 21 (September 7, 2018): 7489–557. http://dx.doi.org/10.1093/imrn/rny215.

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Abstract Shifted symplectic Lie and $L_{\infty }$ algebroids model formal neighborhoods of manifolds in shifted symplectic stacks and serve as target spaces for twisted variants of the classical topological field theory defined by Alexandrov--Kontsevich--Schwarz--Zaboronsky. In this paper, we classify zero-, one-, and two-shifted symplectic algebroids and their higher gauge symmetries, in terms of classical geometric “higher structures”, such as Courant algebroids twisted by $\Omega ^{2}$-gerbes. As applications, we produce new examples of twisted Courant algebroids from codimension-two cycles, and we give symplectic interpretations for several well-known features of higher structures (such as twists, Pontryagin classes, and tensor products). The proofs are valid in the $C^{\infty }$, holomorphic, and algebraic settings and are based on a number of technical results on the homotopy theory of $L_{\infty }$ algebroids and their differential forms, which may be of independent interest.

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CROOKS, PETER. "AN EQUIVARIANT DESCRIPTION OF CERTAIN HOLOMORPHIC SYMPLECTIC VARIETIES." Bulletin of the Australian Mathematical Society 97, no. 2 (February 20, 2018): 207–14. http://dx.doi.org/10.1017/s0004972717001095.

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Varieties of the form$G\times S_{\!\text{reg}}$, where$G$is a complex semisimple group and$S_{\!\text{reg}}$is a regular Slodowy slice in the Lie algebra of$G$, arise naturally in hyperkähler geometry, theoretical physics and the theory of abstract integrable systems. Crooks and Rayan [‘Abstract integrable systems on hyperkähler manifolds arising from Slodowy slices’,Math. Res. Let., to appear] use a Hamiltonian$G$-action to endow$G\times S_{\!\text{reg}}$with a canonical abstract integrable system. To understand examples of abstract integrable systems arising from Hamiltonian$G$-actions, we consider a holomorphic symplectic variety$X$carrying an abstract integrable system induced by a Hamiltonian$G$-action. Under certain hypotheses, we show that there must exist a$G$-equivariant variety isomorphism$X\cong G\times S_{\!\text{reg}}$.

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Jabuka, Stanislav. "Symplectic surfaces and generic $J$-holomorphic structures on 4-manifolds." Illinois Journal of Mathematics 48, no. 2 (April 2004): 675–85. http://dx.doi.org/10.1215/ijm/1258138406.

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Szab�, Zolt�n, and Peter Ozsv�th. "Holomorphic triangle invariants and the topology of symplectic four-manifolds." Duke Mathematical Journal 121, no. 1 (January 2004): 1–34. http://dx.doi.org/10.1215/s0012-7094-04-12111-6.

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Matsushita, Daisuke. "On singular fibres of Lagrangian fibrations over holomorphic symplectic manifolds." Mathematische Annalen 321, no. 4 (December 1, 2001): 755–73. http://dx.doi.org/10.1007/s002080100251.

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Müller, Stefan. "C0-characterization of symplectic and contact embeddings and Lagrangian rigidity." International Journal of Mathematics 30, no. 09 (August 2019): 1950035. http://dx.doi.org/10.1142/s0129167x19500356.

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We present a novel [Formula: see text]-characterization of symplectic embeddings and diffeomorphisms in terms of Lagrangian embeddings. Our approach is based on the shape invariant, which was discovered by Sikorav and Eliashberg, intersection theory and the displacement energy of Lagrangian submanifolds, and the fact that non-Lagrangian submanifolds can be displaced immediately. This characterization gives rise to a new proof of [Formula: see text]-rigidity of symplectic embeddings and diffeomorphisms. The various manifestations of Lagrangian rigidity that are used in our arguments come from [Formula: see text]-holomorphic curve methods. An advantage of our techniques is that they can be adapted to a [Formula: see text]-characterization of contact embeddings and diffeomorphisms in terms of coisotropic (or pre-Lagrangian) embeddings, which in turn leads to a proof of [Formula: see text]-rigidity of contact embeddings and diffeomorphisms. We give a detailed treatment of the shape invariants of symplectic and contact manifolds, and demonstrate that shape is often a natural language in symplectic and contact topology. We consider homeomorphisms that preserve shape, and propose a hierarchy of notions of Lagrangian topological submanifold. Moreover, we discuss shape-related necessary and sufficient conditions for symplectic and contact embeddings, and define a symplectic capacity from the shape.

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LU, Guangcun. "THE WEINSTEIN CONJECTURE ON SOME SYMPLECTIC MANIFOLDS CONTAINING THE HOLOMORPHIC SPHERES." Kyushu Journal of Mathematics 52, no. 2 (1998): 331–51. http://dx.doi.org/10.2206/kyushujm.52.331.

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Boissière, Samuel, and Alessandra Sarti. "A note on automorphisms and birational transformations of holomorphic symplectic manifolds." Proceedings of the American Mathematical Society 140, no. 12 (December 1, 2012): 4053–62. http://dx.doi.org/10.1090/s0002-9939-2012-11277-8.

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GUAN, DANIEL. "EXAMPLES OF COMPACT HOLOMORPHIC SYMPLECTIC MANIFOLDS WHICH ARE NOT KÄHLERIAN III." International Journal of Mathematics 06, no. 05 (October 1995): 709–18. http://dx.doi.org/10.1142/s0129167x95000298.

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Knutsen, Andreas Leopold, Margherita Lelli-Chiesa, and Giovanni Mongardi. "Wall divisors and algebraically coisotropic subvarieties of irreducible holomorphic symplectic manifolds." Transactions of the American Mathematical Society 371, no. 2 (September 20, 2018): 1403–38. http://dx.doi.org/10.1090/tran/7340.

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