Geometric algebraicity of moduli spaces of compact Kähler symplectic manifolds.

1989 ◽  
Vol 1989 (397) ◽  
pp. 202-207 ◽  
Keyword(s):  
Moduli Spaces ◽  
Symplectic Manifolds

scholarly journals NON-SYMPLECTIC INVOLUTIONS ON MANIFOLDS OF -TYPE

2020 ◽  
pp. 1-25
Author(s):  
CHIARA CAMERE ◽  
ALBERTO CATTANEO ◽  
ANDREA CATTANEO
Keyword(s):  
Moduli Spaces ◽  
Quadratic Forms ◽  
Symplectic Manifolds ◽  
Hilbert Schemes ◽  
Small Rank ◽  
Twisted Sheaves ◽  
Formula Type

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.

scholarly journals Some remarks on moduli spaces of lattice polarized holomorphic symplectic manifolds

2018 ◽  
Vol 20 (04) ◽  
pp. 1750044 ◽  
Author(s):  
Chiara Camere
Keyword(s):  
Moduli Spaces ◽  
Symplectic Manifolds ◽  
Equivalence Classes ◽  
General Lattice ◽  
One Dimensional ◽  
Dimensional Boundary

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.

scholarly journals Verra Four-Folds, Twisted Sheaves, and the Last Involution

2018 ◽  
Vol 2019 (21) ◽  
pp. 6661-6710 ◽  
Author(s):  
Chiara Camere ◽  
Grzegorz Kapustka ◽  
Michał Kapustka ◽  
Giovanni Mongardi
Keyword(s):  
Moduli Spaces ◽  
Hilbert Scheme ◽  
K3 Surfaces ◽  
Symplectic Manifolds ◽  
K3 Surface ◽  
Enriques Surfaces ◽  
Twisted Sheaves

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.

Degenerating Kähler structures and geometric quantization

2014 ◽  
Vol 26 (09) ◽  
pp. 1430009 ◽  
Author(s):  
João P. Nunes
Keyword(s):  
Moduli Spaces ◽  
General Framework ◽  
Vector Bundles ◽  
Geometric Quantization ◽  
Theta Functions ◽  
Symplectic Manifolds ◽  
Important Class ◽  
General Philosophy ◽  
Cotangent Bundles ◽  
Kähler Structures

We review some recent results on the problem of the choice of polarization in geometric quantization. Specifically, we describe the general philosophy, developed by the author together with his collaborators, of treating real polarizations as limits of degenerating families of holomorphic polarizations. We first review briefly the general framework of geometric quantization, with a particular focus on the problem of the dependence of quantization on the choice of polarization. The problem of quantization in real polarizations is emphasized. We then describe the relation between quantization in real and Kähler polarizations in some families of symplectic manifolds, that can be explicitly quantized and that constitute an important class of examples: cotangent bundles of Lie groups, abelian varieties and toric varieties. Applications to theta functions and moduli spaces of vector bundles on curves are also reviewed.

scholarly journals A polyfold proof of the Arnold conjecture

2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Benjamin Filippenko ◽  
Katrin Wehrheim
Keyword(s):  
Field Theory ◽  
Moduli Spaces ◽  
Symplectic Manifolds ◽  
Joint Work ◽  
Pseudoholomorphic Curves ◽  
Countable Collection ◽  
Arnold Conjecture ◽  
Detailed Proof ◽  
Symplectic Field Theory

AbstractWe give a detailed proof of the homological Arnold conjecture for nondegenerate periodic Hamiltonians on general closed symplectic manifolds M via a direct Piunikhin–Salamon–Schwarz morphism. Our constructions are based on a coherent polyfold description for moduli spaces of pseudoholomorphic curves in a family of symplectic manifolds degenerating from $${{\mathbb {C}}{\mathbb {P}}}^1\times M$$ C P 1 × M to $${{\mathbb {C}}}^+ \times M$$ C + × M and $${{\mathbb {C}}}^-\times M$$ C - × M , as developed by Fish–Hofer–Wysocki–Zehnder as part of the Symplectic Field Theory package. To make the paper self-contained we include all polyfold assumptions, describe the coherent perturbation iteration in detail, and prove an abstract regularization theorem for moduli spaces with evaluation maps relative to a countable collection of submanifolds. The 2011 sketch of this proof was joint work with Peter Albers, Joel Fish.

scholarly journals Moduli spaces of irreducible symplectic manifolds

2010 ◽  
Vol 146 (2) ◽  
pp. 404-434 ◽  
Author(s):  
V. Gritsenko ◽  
K. Hulek ◽  
G. K. Sankaran
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
General Type ◽  
Symplectic Manifolds ◽  
Symmetric Varieties

AbstractWe study the moduli spaces of polarised irreducible symplectic manifolds. By a comparison with locally symmetric varieties of orthogonal type of dimension 20, we show that the moduli space of polarised deformation K3[2]manifolds with polarisation of degree 2dand split type is of general type ifd≥12.

Constructing symplectic manifolds

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Final Section ◽  
Symplectic Manifolds ◽  
Homotopy Equivalence ◽  
Codimension Two ◽  
Open Manifolds ◽  
Connected Sums ◽  
Symplectic Forms ◽  
Ambient Manifold ◽  
Blowing Up ◽  
Four Manifolds

This chapter examines various ways to construct symplectic manifolds and submanifolds. It begins by studying blowing up and down in both the complex and the symplectic contexts. The next section is devoted to a discussion of fibre connected sums and describes Gompf’s construction of symplectic four-manifolds with arbitrary fundamental group. The chapter also contains an exposition of Gromov’s telescope construction, which shows that for open manifolds the h-principle rules and the inclusion of the space of symplectic forms into the space of nondegenerate 2-forms is a homotopy equivalence. The final section outlines Donaldson’s construction of codimension two symplectic submanifolds and explains the associated decompositions of the ambient manifold.

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