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

Abstract

We survey recent results about the Torelli question for holomorphic-symplectic varieties. Following are the main topics. A Hodge theoretic Torelli theorem. A study of the subgroup W, of the isometry group of the weight 2 Hodge structure, generated by reflection with respect to exceptional divisors. A description of the birational Kahler cone as a fundamental domain for the W-action on the positive cone. A proof of a weak version of Morrison's movable cone conjecture. A description of the moduli spaces of polarized holomorphic symplectic varieties as monodromy quotients of period domains of type IV.

46 pages. v3: The previous attempt to simplify the statement of Conjecture 10.6 resulted in a false statement. The statement of the conjecture is corrected

References in corpus (2)

• Hyper-Kähler fourfolds and Grassmann geometry
• Higher-dimensional analogues of K3 surfaces

Cited by in corpus (78)

• Mapping class group and global Torelli theorem for hyperkahler manifolds: an erratum
• The K3 category of a cubic fourfold
• Bridgeland's stability and the positive cone of the moduli spaces of stable objects on an abelian surface
• Classification of automorphisms on a deformation family of hyperkähler fourfolds by p-elementary lattices
• Stability conditions in families
• Induced Automorphisms on Irreducible Symplectic Manifolds
• Cremona transformations and derived equivalences of K3 surfaces
• Birational Geometry of Singular Moduli Spaces of O'Grady Type
• Moduli spaces of torsion sheaves on K3 surfaces and derived equivalences
• Symplectic involutions on deformations of K3^[2]
• Projectivity and Birational Geometry of Bridgeland moduli spaces
• A remark on the Torelli theorem for cubic fourfolds
• Hyperkähler fourfolds and Kummer surfaces
• On the monodromy group of desingularised moduli spaces of sheaves on K3 surfaces
• Families of rational curves on holomorphic symplectic varieties and applications to zero-cycles
• Birational geometry of the intermediate Jacobian fibration of a cubic fourfold
• Automorphism groups of Calabi-Yau manifolds of Picard number two
• Non-symplectic involutions of irreducible symplectic manifolds of K3^[n]-type
• Non-natural non-symplectic involutions on symplectic manifolds of K3^{[2]}-type
• Mori cones of holomorphic symplectic varieties of K3 type
• Teichmuller space for hyperkahler and symplectic structures
• On the existence of universal families of marked irreducible holomorphic symplectic manifolds
• Contraction centers in families of hyperkahler manifolds
• Monodromy Invariants and Polarization Types of Generalized Kummer Fibrations
• Moduli of double EPW-sextics
• Tautological classes on moduli space of hyperkähler manifolds
• Finiteness of Klein actions and real structures on compact hyperkähler manifolds
• Derived equivalent Hilbert schemes of points on K3 surfaces which are not birational
• Lattice polarized irreducible holomorphic symplectic manifolds
• A classification of Lagrangian planes in holomorphic symplectic varieties
• A global Torelli theorem for hyperkaehler manifolds (after Verbitsky)
• Collections of parabolic orbits in homogeneous spaces, homogeneous dynamics and hyperkahler geometry
• On the Betti numbers of compact holomorphic symplectic orbifolds of dimension four
• Symplectic birational transformations of finite order on O'Grady's sixfolds
• On the double EPW sextic associated to a Gushel-Mukai fourfold
• Ergodic complex structures on hyperkahler manifolds
• Higher-dimensional analogues of K3 surfaces
• Induced birational transformations on O'Grady's sixfolds
• On the nonsymplectic involutions of the Hilbert square of a K3 surface
• Hochschild cohomology versus the Jacobian ring, and the Torelli theorem for cubic fourfolds
• Automorphisms of generalised Kummer fourfolds
• Construction of automorphisms of hyperkähler manifolds
• Character Formulas on Cohomology of Deformations of Hilbert Schemes of K3 Surfaces
• Extremal rays and automorphisms of holomorphic symplectic varieties
• Prime order automorphisms of abelian surfaces: a lattice-theoretic point of view
• Symplectic involutions of $K{3}^{[n]}$ type and Kummer $n$ type manifolds
• On polarization types of Lagrangian fibrations
• On Irreducible Symplectic Varieties of ${K3}^{[n]}$-type in Positive Characteristic
• On the period of Lehn, Lehn, Sorger, and van Straten's symplectic eightfold
• On the image of the period map for polarized hyperkähler manifolds
• Brane involutions on irreducible holomorphic symplectic manifolds
• Gopakumar-Vafa type invariants of holomorphic symplectic 4-folds
• Ergodic complex structures on hyperkahler manifolds: an erratum
• Two polarized K3 surfaces associated to the same cubic fourfold
• Symplectic rigidity of O'Grady's tenfolds
• Rational Hodge isometries of hyper-Kahler varieties of K3[n]-type are algebraic
• Boucksom-Zariski and Weyl chambers on irreducible holomorphic symplectic manifolds
• A local Torelli theorem for log symplectic manifolds
• A global Torelli theorem for rigid hyperholomorphic sheaves
• Every rational Hodge isometry between two K3 surfaces is algebraic
• Rational curves in holomorphic symplectic varieties and Gromov-Witten invariants
• Strongly ambiguous Hilbert squares of projective K3 surfaces with Picard number one
• The second integral cohomology of moduli spaces of sheaves on K3 and Abelian surfaces
• Supersingular O'Grady varieties of dimension six
• Computing Riemann-Roch polynomials and classifying hyper-Kähler fourfolds
• Around and beyond the canonical class
• On birational transformations of Hilbert schemes of points on K3 surfaces
• Galois representations on the cohomology of hyper-Kähler varieties
• Regenerations and applications
• The geometry of antisymplectic involutions, I
• Rigid hyperholomorphic sheaves remain rigid along twistor deformations of the underlying hyparkahler manifold
• On symplectic birational self-maps of projective hyperkähler manifolds of K3${}^{[n]}$-type
• Kodaira dimension of universal holomorphic symplectic varieties
• On the irrationality of moduli spaces of projective hyperkähler manifolds
• An elementary description of nef cone for irreducible holomorphic symplectic manifolds
• A cone conjecture for log Calabi-Yau surfaces
• On subgroups of an automorphism group of an irreducible symplectic manifold
• Wall-Crossing implies Brill-Noether. Applications of stability conditions on surfaces