A global Torelli theorem for singular symplectic varieties

Abstract

We systematically study the moduli theory of singular symplectic varieties which have a resolution by an irreducible symplectic manifold and prove an analog of Verbitsky's global Torelli theorem. In place of twistor lines, Verbitsky's work on ergodic complex structures provides the essential global input. On the one hand, our deformation theoretic results are a further generalization of Huybrechts' theorem on deformation equivalence of birational hyperkähler manifolds to the context of singular symplectic varieties. On the other hand, our global moduli theory provides a framework for understanding and classifying the symplectic singularities that arise from birational contractions of irreducible symplectic manifolds, and there are a number of applications to K3 [n] -type varieties.

The global moduli theory of symplectic varieties

“…In section 6 we prove a (weak) singular analog of the Demailly-Pȃun theorem and apply it to deduce the projectivity criterion, Theorem 1.3 (see Theorem 6.12). We also prove analogs of results of Huybrechts [Huy99] and [BL16] on the inseparability of bimeromorphic symplectic varieties in moduli, including part (3) of Theorems 1.1 and 1.2 (see Theorem 6.16 and Corollary 6.18). In section 7 we indicate the necessary changes to [KLSV18] to show the existence of limits of projective families for which the period does not degenerate in the Q-factorial terminal setting.…”

“…It is not hard now to deduce a local Torelli theorem for locally trivial deformations. Preliminary versions have been established by Namikawa [Nam01a], Kirschner [Kir15, Theorem 3.4.12], Matsushita [Mat15], and the authors [BL16].…”

“…In the general setting, one could try to reduce to Theorem 1.1 by passing to a simultaneous Q-factorial terminalization, but in this case a new strategy is needed as Verbitsky's proof (and even Huybrechts' proof of the surjectivity of the period map [Huy99, Theorem 8.1]) fundamentally uses the existence of hyperkähler metrics and twistor deformations. Instead, we prove Theorem 1.2 directly using Verbitsky's classification [Ver15,Theorem 1.16] (see also [BL16,Remark 5.7]) of orbit closures of the monodromy action on the period domain via Ratner theory, which is responsible for the assumption b 2 (X) ≥ 5 in the theorem. The surjectivity in Theorem 1.1 then follows from a slight generalization to the Q-factorial terminal case of work of Kollár-Laza-Saccà-Voisin [KLSV18] on projective degenerations using MMP techniques.…”

“…In fact, there is another problem with the naive generalization of the argument of [BL16]: the replacement of a crepant resolution should be a Q-factorial terminalization, but this is not guaranteed to exist. In the projective case the existence of a Q-factorial terminalization is a consequence of deep results of Birkar-Cascini-Hacon-M c Kernan [BCHM10] on the termination of an appropriate version of the MMP, but it is not even clear a priori that a symplectic variety can be deformed to a projective one (although Namikawa [Nam02] has results in this direction).…”

“…Previous work. In [BL16] the authors extended many of the classical results about compact irreducible symplectic manifolds to primitive symplectic varieties admitting a crepant resolution through the study of their locally trivial deformations. Menet [Men18] has proven a version of the global Torelli theorem for certain primitive symplectic varieties with orbifold singularities using twistor deformations.…”