The global moduli theory of symplectic varieties

Abstract

We develop the global moduli theory of symplectic varieties in the sense of Beauville. We prove a number of analogs of classical results from the smooth case, including a global Torelli theorem. In particular, this yields a new proof of Verbitsky's global Torelli theorem in the smooth case (assuming b2 ≥ 5) which does not use the existence of a hyperkähler metric or twistor deformations.

Numerical characterization of complex torus quotients

“…We will always normalize q X in such a way that it comes from an indivisible integral quadratic form H 2 (X, Z) → Z, cf. [BL18,Lemma 5.7]. With this convention, the BBF form is a topological invariant of X.…”

“…[O'G12, Proposition 2.2]. It has to be noticed that the proof given in [BL18,Proposition 5.20] has a different flavor. Also, the first result in this direction (under stronger assumptions) appears to be [Mat01, Lemma 2.4].…”

Deformations of rational curves on primitive symplectic varieties and applications

“…On the other hand, in many fundamental aspects of the theory (deformation theory, lattice structure, projectivity, Torelli theorems etc.) even the larger class of primitive symplectic varieties behaves very similarly to their smooth analogs, see [BL21,BL18,Men20]. It is worthwhile pointing out that both these classes of symplectic varieties coincide in the smooth case by a recent theorem of Schwald [Sch20b], see Section 2.1 for precise definitions and a more detailed discussion.…”

The dual Lagrangian fibration of known hyper-Kähler manifolds

“…To make our discussion less ambiguous, here we collect some definitions of singular hyper-Kähler varieties and compare them. Our main references are [BL18], [Sch20a] and [Men20].…”

“…• The local Torelli theorem holds for Def lt (X). In fact, global Torelli theorem holds in a suitable form [BL18]. We will use these facts in Section 6 and Appendix A, without mentioning them explicitly.…”