A global Torelli theorem for singular symplectic varieties

Abstract

We systematically study the moduli theory of symplectic varieties (in the sense of Beauville) which admit a resolution by an irreducible symplectic manifold. In particular, we prove an analog of Verbitsky's global Torelli theorem for the locally trivial deformations of such varieties. Verbitsky's work on ergodic complex structures replaces twistor lines as the essential global input. In so doing we extend many of the local deformation-theoretic results known in the smooth case to such (not-necessarily-projective) symplectic varieties. We deduce a number of applications to the birational geometry of symplectic manifolds, including some results on the classification of birational contractions of K3 [n] -type varieties.

Algebraic approximation and the decomposition theorem for Kähler Calabi–Yau varieties

“…Corollary 2.28 Let X → S be a locally trivial family of normal varieties of Fujiki class C with rational singularities. Then for all p the function s → h 0 (X s , Proposition 2.26 in particular applies to crepant bimeromorphic morphisms of symplectic varieties; this greatly simplifies the approach of [11,Section 4] and yields a generalization of [11,Proposition 4.5]: 29 Let Y be a compact normal complex variety with a nowhere degenerate form σ ∈ H 0 (Y reg , 2 Y reg ) and let π : Y → X be a proper bimeromorphic map to a normal variety X with rational singularities. Then there is a locally trivial deformation Y → X of π over Def lt (X ), where X → Def lt (X ) is the miniversal locally trivial deformation of X .…”

“…S lift. For this we could appeal directly to the degeneration of reflexive Hodge-to-de Rham in low degrees [10,Lemma 2.4], but we also include a more direct argument using Corollary 2.27. Let π : Y → X be a simultaneous locally trivial resolution with special fiber π : Y → X .…”

“…We show that locally trivial deformations obtained by exponentiating the symplectic foliations of this splitting are always unobstructed. As in the proof of the corresponding result for symplectic varieties [10,Theorem 4.7], a crucial role is played by the degeneration of reflexive Hodge-to-de Rham in low degrees [11, Section 2]. The general results we prove about locally trivial deformations along foliations (Sect.…”

“…On the other hand, the proof of unobstructedness in the smooth case is fundamentally Hodge-theoretic, and from this perspective locally trivial deformations are more natural as they are topologically trivial (see [3, Proposition 6.1]). While some special cases have been established (see [10,11,46]), it is as yet unclear whether a locally trivial Bogomolov-Tian-Todorov theorem should hold.…”

On the Betti numbers of compact holomorphic symplectic orbifolds of dimension four

“…The limitedness of available examples suggests the possibility to bound or even classify IHS manifolds (see [28] for diffeomorphic types). As the second cohomology of an IHS manifold, together with the Beauville-Bogomolov quadratic form [2] and the weight-2 Hodge structure, controls most of its geometry [4,5,29,36,38,57,58], it is natural to ask the following question.…”