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Journal articles on the topic 'Irreducible Holomorphic symplectic manifold'

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Published: 9 March 2023

Last updated: 10 March 2023

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1

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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2

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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3

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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4

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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5

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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6

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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7

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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8

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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9

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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10

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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11

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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12

Lo Bianco, Federico. "On the Primitivity of Birational Transformations of Irreducible Holomorphic Symplectic Manifolds." International Mathematics Research Notices 2019, no. 1 (June 16, 2017): 1–32. http://dx.doi.org/10.1093/imrn/rnx109.

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13

Iliev, Atanas, Grzegorz Kapustka, Michał Kapustka, and Kristian Ranestad. "EPW cubes." Journal für die reine und angewandte Mathematik (Crelles Journal) 2019, no. 748 (March 1, 2019): 241–68. http://dx.doi.org/10.1515/crelle-2016-0044.

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Abstract We construct a new 20-dimensional family of projective six-dimensional irreducible holomorphic symplectic manifolds. The elements of this family are deformation equivalent with the Hilbert scheme of three points on a K3 surface and are constructed as natural double covers of special codimension-three subvarieties of the Grassmannian G(3,6) . These codimension-three subvarieties are defined as Lagrangian degeneracy loci and their construction is parallel to that of EPW sextics, we call them the EPW cubes. As a consequence we prove that the moduli space of polarized IHS sixfolds of K3 -type, Beauville–Bogomolov degree 4 and divisibility 2 is unirational.

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14

Ouchi, Genki. "Lagrangian embeddings of cubic fourfolds containing a plane." Compositio Mathematica 153, no. 5 (March 23, 2017): 947–72. http://dx.doi.org/10.1112/s0010437x16008307.

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We prove that a very general smooth cubic fourfold containing a plane can be embedded into an irreducible holomorphic symplectic eightfold as a Lagrangian submanifold. We construct the desired irreducible holomorphic symplectic eightfold as a moduli space of Bridgeland stable objects in the derived category of the twisted K3 surface corresponding to the cubic fourfold containing a plane.

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15

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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16

Mongardi, Giovanni, and John Christian Ottem. "Curve classes on irreducible holomorphic symplectic varieties." Communications in Contemporary Mathematics 22, no. 07 (November 15, 2019): 1950078. http://dx.doi.org/10.1142/s0219199719500780.

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We prove that the integral Hodge conjecture holds for 1-cycles on irreducible holomorphic symplectic varieties of [Formula: see text]-type and of generalized Kummer type. As an application, we give a new proof of the integral Hodge conjecture for cubic fourfolds.

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17

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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18

Hassett, Brendan, and Yuri Tschinkel. "Flops on holomorphic symplectic fourfolds and determinantal cubic hypersurfaces." Journal of the Institute of Mathematics of Jussieu 9, no. 1 (August 11, 2009): 125–53. http://dx.doi.org/10.1017/s1474748009000140.

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AbstractWe study the birational geometry of irreducible holomorphic symplectic varieties arising as varieties of lines of general cubic fourfolds containing a cubic scroll. We compute the ample and moving cones, and exhibit a birational automorphism of infinite order explaining the chamber decomposition of the moving cone.

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19

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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20

Mongardi, Giovanni, and Claudio Onorati. "Birational geometry of irreducible holomorphic symplectic tenfolds of O’Grady type." Mathematische Zeitschrift 300, no. 4 (January 18, 2022): 3497–526. http://dx.doi.org/10.1007/s00209-021-02966-6.

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21

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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22

Matsushita, Daisuke. "ON FIBRE SPACE STRUCTURES OF A PROJECTIVE IRREDUCIBLE SYMPLECTIC MANIFOLD." Topology 38, no. 1 (January 1999): 79–83. http://dx.doi.org/10.1016/s0040-9383(98)00003-2.

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23

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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24

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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25

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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26

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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27

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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28

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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29

Lekili, Yankı, and James Pascaleff. "Floer cohomology of -equivariant Lagrangian branes." Compositio Mathematica 152, no. 5 (December 17, 2015): 1071–110. http://dx.doi.org/10.1112/s0010437x1500771x.

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Building on Seidel and Solomon’s fundamental work [Symplectic cohomology and$q$-intersection numbers, Geom. Funct. Anal. 22 (2012), 443–477], we define the notion of a $\mathfrak{g}$-equivariant Lagrangian brane in an exact symplectic manifold $M$, where $\mathfrak{g}\subset SH^{1}(M)$ is a sub-Lie algebra of the symplectic cohomology of $M$. When $M$ is a (symplectic) mirror to an (algebraic) homogeneous space $G/P$, homological mirror symmetry predicts that there is an embedding of $\mathfrak{g}$ in $SH^{1}(M)$. This allows us to study a mirror theory to classical constructions of Borel, Weil and Bott. We give explicit computations recovering all finite-dimensional irreducible representations of $\mathfrak{sl}_{2}$ as representations on the Floer cohomology of an $\mathfrak{sl}_{2}$-equivariant Lagrangian brane and discuss generalizations to arbitrary finite-dimensional semisimple Lie algebras.

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30

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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31

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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32

Matsushita, Daisuke. "On fibre space structures of a projective irreducible symplectic manifold, Topology 38, (1999) 79–83." Topology 40, no. 2 (March 2001): 431–32. http://dx.doi.org/10.1016/s0040-9383(99)00048-8.

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33

Liu, Baiying. "Genericity of Representations of p-Adic Sp2n and Local Langlands Parameters." Canadian Journal of Mathematics 63, no. 5 (October 18, 2011): 1107–36. http://dx.doi.org/10.4153/cjm-2011-017-2.

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Abstract Let G be the F-rational points of the symplectic group Sp2n, where F is a non-Archimedean local field of characteristic 0. Cogdell, Kim, Piatetski-Shapiro, and Shahidi constructed local Lang- lands functorial lifting from irreducible generic representations of G to irreducible representations of GL2n+1(F). Jiang and Soudry constructed the descent map from irreducible supercuspidal repre- sentations of GL2n+1(F) to those of G, showing that the local Langlands functorial lifting from the irreducible supercuspidal generic representations is surjective. In this paper, based on above results, using the same descent method of studying SO2n+1 as Jiang and Soudry, we will show the rest of local Langlands functorial lifting is also surjective, and for any local Langlands parameter , we construct a representation such that and ¾ have the same twisted local factors. As one application, we prove the G-case of a conjecture of Gross-Prasad and Rallis, that is, a local Langlands parameter is generic, i.e., the representation attached to is generic, if and only if the adjoint L-function of is holomorphic at s = 1. As another application, we prove for each Arthur parameter , and the corresponding local Langlands parameter , the representation attached to is generic if and only if is tempered.

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34

GADEA, P. M., A. MONTESINOS AMILIBIA, and J. MUÑOZ MASQUÉ. "Characterizing the complex hyperbolic space by Kähler homogeneous structures." Mathematical Proceedings of the Cambridge Philosophical Society 128, no. 1 (January 2000): 87–94. http://dx.doi.org/10.1017/s0305004199003825.

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The Kähler case of Riemannian homogeneous structures [3, 15, 18] has been studied in [1, 2, 6, 7, 13, 16], among other papers. Abbena and Garbiero [1] gave a classification of Kähler homogeneous structures, which has four primitive classes [Kscr ]1, …, [Kscr ]4 (see [6, theorem 5·1] for another proof and Section 2 below for the result). The purpose of the present paper is to prove the following result:THEOREM 1·1. A simply connected irreducible homogeneous Kähler manifold admits a nonvanishing Kähler homogeneous structure in Abbena–Garbiero's class [Kscr ]2 [oplus ] [Kscr ]4if and only if it is the complex hyperbolic space equipped with the Bergman metric of negative constant holomorphic sectional curvature.

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35

MOSSA, ROBERTO. "BALANCED METRICS ON HOMOGENEOUS VECTOR BUNDLES." International Journal of Geometric Methods in Modern Physics 08, no. 07 (November 2011): 1433–38. http://dx.doi.org/10.1142/s0219887811005841.

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Let E → M be a holomorphic vector bundle over a compact Kähler manifold (M, ω) and let E = E1 ⊕ ⋯ ⊕ Em → M be its decomposition into irreducible factors. Suppose that each Ej admits a ω-balanced metric in Donaldson–Wang terminology. In this paper we prove that E admits a unique ω-balanced metric if and only if [Formula: see text] for all j, k = 1,…, m, where rj denotes the rank of Ej and Nj = dim H0(M, Ej). We apply our result to the case of homogeneous vector bundles over a rational homogeneous variety (M, ω) and we show the existence and rigidity of balanced Kähler embedding from (M, ω) into Grassmannians.

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36

Cahen, Benjamin. "Berezin quantization on generalized flag manifolds." MATHEMATICA SCANDINAVICA 105, no. 1 (September 1, 2009): 66. http://dx.doi.org/10.7146/math.scand.a-15106.

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Let $M=G/H$ be a generalized flag manifold where $G$ is a compact, connected, simply-connected Lie group with Lie algebra $\mathfrak{g}$ and $H$ is the centralizer of a torus. Let $\pi$ be a unitary irreducible representation of $G$ which is holomorphically induced from a character of $H$. Using a complex parametrization of a dense open subset of $M$, we realize $\pi$ on a Hilbert space of holomorphic functions. We give explicit expressions for the differential $d\pi$ of $\pi$ and for the Berezin symbols of $\pi (g)$ ($g\in G$) and $d\pi (X)$ ($X\in \mathfrak{g}$). In particular, we recover some results of S. Berceanu and we partially generalize a result of K. H. Neeb.

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37

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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38

BISWAS, INDRANIL. "ON THE ALGEBRAIC HOLONOMY OF STABLE PRINCIPAL BUNDLES." International Journal of Mathematics 22, no. 06 (June 2011): 775–87. http://dx.doi.org/10.1142/s0129167x11007033.

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Let EG be a stable principal G–bundle over a compact connected Kähler manifold, where G is a connected reductive linear algebraic group defined over ℂ. Let H ⊂ G be a complex reductive subgroup which is not necessarily connected, and let EH ⊂ EG be a holomorphic reduction of structure group to H. We prove that EH is preserved by the Einstein–Hermitian connection on EG. Using this we show that if EH is a minimal reductive reduction (which means that there is no complex reductive proper subgroup of H to which EH admits a holomorphic reduction of structure group), then EH is unique in the following sense: For any other minimal reduction of structure group (H′, EH′) of EG to some reductive subgroup H′, there is some element g ∈ G such that H′ = g-1Hg and EH′ = EHg. As an application, we show the following: Let M be a simply connected, irreducible smooth complex projective variety of dimension n such that the Picard number of M is one. If the canonical line bundle KM is ample, then the algebraic holonomy of the holomorphic tangent bundle T1, 0M is GL (n, ℂ). If [Formula: see text] is ample, the rank of the Picard group of M is one, the biholomorphic automorphism group of M is finite, and M admits a Kähler–Einstein metric, then the algebraic holonomy of T1, 0M is GL (n, ℂ). These answer some questions posed in V. Balaji and J. Kollár, Publ. Res. Inst. Math. Sci.44 (2008) 183–211.

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39

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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40

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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41

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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42

Bogomolov, Fedor, Nikon Kurnosov, Alexandra Kuznetsova, and Egor Yasinsky. "Geometry and Automorphisms of Non-Kähler Holomorphic Symplectic Manifolds." International Mathematics Research Notices, April 19, 2021. http://dx.doi.org/10.1093/imrn/rnab043.

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Abstract We consider the only one known class of non-Kähler irreducible holomorphic symplectic manifolds, described in the works by D. Guan and the 1st author. Any such manifold $Q$ of dimension $2n-2$ is obtained as a finite degree $n^2$ cover of some non-Kähler manifold $W_F$, which we call the base of $Q$. We show that the algebraic reduction of $Q$ and its base is the projective space of dimension $n-1$. Besides, we give a partial classification of submanifolds in $Q$, describe the degeneracy locus of its algebraic reduction and prove that the automorphism group of $Q$ satisfies the Jordan property.

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43

Schwald, Martin. "On the Definition of Irreducible Holomorphic Symplectic Manifolds and Their Singular Analogs." International Mathematics Research Notices, April 19, 2021. http://dx.doi.org/10.1093/imrn/rnab032.

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Abstract In the definition of irreducible holomorphic symplectic manifolds the condition of being simply connected can be replaced by vanishing irregularity. We discuss holomorphic symplectic, finite quotients of complex tori with ${\operatorname{h}}^0(X,\,\Omega ^{[2]}_X)=1$ and their Lagrangian fibrations. Neither $X$ nor the base can be smooth unless $X$ is a $2$-torus.

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44

Lehn, Christian, Manfred Lehn, Christoph Sorger, and Duco van Straten. "Twisted cubics on cubic fourfolds." Journal für die reine und angewandte Mathematik (Crelles Journal) 2017, no. 731 (January 1, 2017). http://dx.doi.org/10.1515/crelle-2014-0144.

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45

Onorati, Claudio. "On the monodromy group of desingularised moduli spaces of sheaves on K3 surfaces." Journal of Algebraic Geometry, March 16, 2022. http://dx.doi.org/10.1090/jag/802.

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In this paper we prove a conjecture of Markman about the shape of the monodromy group of irreducible holomorphic symplectic manifolds of OG10 type. As a corollary, we also compute the locally trivial monodromy group of the underlying singular symplectic variety.

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46

Brandhorst, Simon, and Alberto Cattaneo. "Prime Order Isometries of Unimodular Lattices and Automorphisms of IHS Manifolds." International Mathematics Research Notices, October 13, 2022. http://dx.doi.org/10.1093/imrn/rnac279.

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Abstract We characterize conjugacy classes of isometries of odd prime order in unimodular ${\mathbb {Z}}$-lattices. This is applied to give a complete classification of odd prime order non-symplectic automorphisms of the known deformation types of irreducible holomorphic symplectic manifolds up to deformation and birational conjugacy.

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47

CAMERE, CHIARA, ALBERTO CATTANEO, and ANDREA CATTANEO. "NON-SYMPLECTIC INVOLUTIONS ON MANIFOLDS OF -TYPE." Nagoya Mathematical Journal, February 27, 2020, 1–25. http://dx.doi.org/10.1017/nmj.2019.43.

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We study irreducible holomorphic symplectic manifolds deformation equivalent to Hilbert schemes of points on a $K3$ surface and admitting a non-symplectic involution. We classify the possible discriminant quadratic forms of the invariant and coinvariant lattice for the action of the involution on cohomology and explicitly describe the lattices in the cases where the invariant lattice has small rank. We also give a modular description of all $d$ -dimensional families of manifolds of $K3^{[n]}$ -type with a non-symplectic involution for $d\geqslant 19$ and $n\leqslant 5$ and provide examples arising as moduli spaces of twisted sheaves on a $K3$ surface.

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48

Markman, Eyal. "On the existence of universal families of marked irreducible holomorphic symplectic manifolds." Kyoto Journal of Mathematics, November 2020. http://dx.doi.org/10.1215/21562261-2019-0075.

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49

Addington, Nicolas, and Manfred Lehn. "On the symplectic eightfold associated to a Pfaffian cubic fourfold." Journal für die reine und angewandte Mathematik (Crelles Journal) 2017, no. 731 (January 1, 2017). http://dx.doi.org/10.1515/crelle-2014-0145.

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50

HUANG, CHE-HUNG. "A RIGIDITY PROPERTY OF PLURIHARMONIC MAPS FROM PROJECTIVE MANIFOLDS." Bulletin of the Australian Mathematical Society, October 13, 2022, 1–3. http://dx.doi.org/10.1017/s0004972722001113.

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Abstract Suppose M is a complex projective manifold of dimension $\geq 2$ , V is the support of an ample divisor in M and U is an open set in M that intersects each irreducible component of V. We show that a pluriharmonic map $f:M\to N$ into a Kähler manifold N is holomorphic whenever $f\vert _{V\,\cap \, U}$ is holomorphic.

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