# A global Torelli theorem for singular symplectic varieties

@article{Bakker2016AGT, title={A global Torelli theorem for singular symplectic varieties}, author={Benjamin Bakker and Christian Lehn}, journal={arXiv: Algebraic Geometry}, year={2016} }

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 hyperkahler manifolds to the context of singular…

## 25 Citations

The global moduli theory of symplectic varieties

- MathematicsJournal für die reine und angewandte Mathematik (Crelles Journal)
- 2022

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…

Deformations of rational curves on primitive symplectic varieties and applications

- Mathematics
- 2021

We study the deformation theory of rational curves on primitive symplectic varieties and show that if the rational curves cover a divisor, then, as in the smooth case, they deform along their Hodge…

Rational curves in holomorphic symplectic varieties and Gromov–Witten invariants

- MathematicsAdvances in Mathematics
- 2019

Global Torelli theorem for irreducible symplectic orbifolds

- MathematicsJournal de Mathématiques Pures et Appliquées
- 2020

Examples of Irreducible Symplectic Varieties

- MathematicsBirational Geometry and Moduli Spaces
- 2020

Irreducible symplectic manifolds are one of the three building blocks of compact Kahler manifolds with numerically trivial canonical bundle by the Beauville-Bogomolov decomposition theorem. There are…

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

- Mathematics
- 2020

We extend a result of Guan by showing that the second Betti number of a 4-dimensional primitively symplectic orbifold is at most 23 and there are at most 91 singular points. The maximal possibility…

Finiteness of polarized K3 surfaces and hyperk\"ahler manifolds

- Mathematics
- 2018

In the moduli space of polarized varieties the same unpolarized variety can occur multiple times However, for K3 surfaces, compact hyperk\"ahler manifolds, and abelian varieties the number is finite.…

On the Boucksom-Zariski decomposition for irreducible symplectic varieties and bounded negativity

- Mathematics
- 2019

Zariski decomposition plays an important role in the theory of algebraic surfaces due to many applications. Boucksom showed that it also holds for irreducible symplectic manifolds. Different variants…

On the dual positive cones and the algebraicity of a compact K\"ahler manifold.

- Mathematics
- 2020

We study the algebraicity of compact Kahler manifolds admitting a positive rational Hodge class of bidimension $(1,1)$. We show that if the dual pseudoeffective cone of a compact Kahler threefold $X$…

On the monodromy group of desingularised moduli spaces of sheaves on K3 surfaces

- MathematicsJournal of Algebraic Geometry
- 2022

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…

## References

SHOWING 1-10 OF 144 REFERENCES

The global moduli theory of symplectic varieties

- Mathematics
- 2018

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…

A survey of Torelli and monodromy results for holomorphic-symplectic varieties

- Mathematics
- 2011

We survey recent results about the Torelli question for holomorphicsymplectic varieties. Following are the main topics. A Hodge theoretic Torelli theorem. A study of the subgroup WExc, of the…

Deformations of singular symplectic varieties and termination of the log minimal model program

- Mathematics
- 2016

We generalize Huybrechts’ theorem on deformation equivalence of birational irreducible symplectic manifolds to the singular setting. More precisely, under suitable natural hypotheses, we show that…

On deformations of ℚ-factorial symplectic varieties

- Mathematics
- 2005

Abstract Our purpose is to give a positive answer to the following problem posed in [Namikawa, Y., Extension of 2-forms and symplectic varieties, J. reine angew. Math. 539 (2001), 123– 147.]:…

Period Mappings with Applications to Symplectic Complex Spaces

- Mathematics
- 2015

Extending Griffiths' classical theory of period mappings for compact Kahler manifolds, this book develops and applies a theory of period mappings of "Hodge-de Rham type" for families of open complex…

Families of rational curves on holomorphic symplectic varieties and applications to zero-cycles

- Mathematics
- 2014

We study families of rational curves on irreducible holomorphic symplectic varieties. We give a necessary and sufficient condition for a sufficiently ample linear system on a holomorphic symplectic…

Birational boundedness for holomorphic symplectic varieties, Zarhin's trick for $K3$ surfaces, and the Tate conjecture

- Mathematics
- 2014

We investigate boundedness results for families of holomorphic symplectic varieties up to birational equivalence. We prove the analogue of Zarhin's trick by for $K3$ surfaces by constructing big line…

Deformation theory of singular symplectic n-folds

- Mathematics
- 2000

By a symplectic manifold (or a symplectic n-fold) we mean a compact Kaehler manifold of even dimension n with a non-degenerate holomorphic 2form ω, i.e. ω is a nowhere-vanishing n-form. This notion…

Wall divisors and algebraically coisotropic subvarieties of irreducible holomorphic symplectic manifolds

- MathematicsTransactions of the American Mathematical Society
- 2018

Rational curves on Hilbert schemes of points on
K
3
K3
surfaces and generalised Kummer manifolds are constructed by using Brill–Noether theory on nodal curves on the underlying…