A CLASSIFICATION OF LAGRANGIAN PLANES IN HOLOMORPHIC SYMPLECTIC VARIETIES

@article{Bakker2015ACO,
  title={A CLASSIFICATION OF LAGRANGIAN PLANES IN HOLOMORPHIC SYMPLECTIC VARIETIES},
  author={Benjamin Bakker},
  journal={Journal of the Institute of Mathematics of Jussieu},
  year={2015},
  volume={16},
  pages={859 - 877}
}
  • Benjamin Bakker
  • Published 23 October 2013
  • Mathematics
  • Journal of the Institute of Mathematics of Jussieu
Classically, an indecomposable class $R$ in the cone of effective curves on a K3 surface $X$ is representable by a smooth rational curve if and only if $R^{2}=-2$ . We prove a higher-dimensional generalization conjectured by Hassett and Tschinkel: for a holomorphic symplectic variety $M$ deformation equivalent to a Hilbert scheme of $n$ points on a K3 surface, an extremal curve class $R\in H_{2}(M,\mathbb{Z})$ in the Mori cone is the line in a Lagrangian $n$ -plane $\mathbb{P}^{n}\subset M$ if… 
MBM loci in families of hyperkahler manifolds and centers of birational contractions
An MBM class on a hyperkahler manifold M is a second cohomology class such that its orthogonal complement in H^2(M) contains a maximal dimensional face of the boundary of the Kahler cone for some
A global Torelli theorem for singular symplectic varieties
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
Lagrangian planes in hyperk\"ahler varieties of $K3^{[n]}$-type
Jieao Song recently conjectured a formula for the class of a Lagrangian plane on a hyperk¨ahler variety of K 3 [ n ] -type in terms of the class of a line on it. We give a proof of this conjecture if
Hodge classes of type (2, 2) on Hilbert squares of projective K3 surfaces
We give a basis for the vector space generated by rational Hodge classes of type (2, 2) on the Hilbert square of a projective K3 surface, which is a subspace of the singular cohomology ring with
Derived categories of hyper-K\"ahler manifolds: extended Mukai vector and integral structure
. We introduce a linearised form of the square root of the Todd class inside the Verbitsky component of a hyper-Kähler manifold using the extended Mukai lattice. This enables us to define a Mukai
A G ] 6 M ay 2 02 1 DERIVED CATEGORIES OF HYPER-KÄHLER MANIFOLDS : EXTENDED MUKAI VECTOR AND INTEGRAL STRUCTURE
We introduce a linearised form of the square root of the Todd class inside the Verbitsky component of a hyper-Kähler manifold using the extended Mukai lattice. This enables us to define a Mukai
The cotangent bundle of K3 surfaces of degree two
. K3 surfaces have been studied from many points of view, but the positivity of the cotangent bundle is not well understood. In this paper we explore the surprisingly rich geometry of the
Extremal rays and automorphisms of holomorphic symplectic varieties
We survey recent results on ample cones and birational contractions of holomorphic symplectic varieties of K3 type, focusing on explicit constructions and concrete examples.
Nef cycles on some hyperkahler fourfolds
We study the cones of surfaces on varieties of lines on cubic fourfolds and Hilbert schemes of points on K3 surfaces. From this we obtain new examples of nef cycles which fail to be pseudoeffective.
Character formulas on cohomology of deformations of Hilbert schemes of K3 surfaces
TLDR
The graded character formula of the generic Mumford-Tate group representation on the cohomology ring of X is computed, and a generating series is derived for deducing the number of canonical Hodge classes on X.
...
...

References

SHOWING 1-10 OF 55 REFERENCES
Lagrangian hyperplanes in holomorphic symplectic varieties
Lagrangian Fibrations of Holomorphic-Symplectic Varieties of K3[n]-Type
Let X be a compact Kahler holomorphic-symplectic manifold, which is deformation equivalent to the Hilbert scheme of length n subschemes of a K3 surface. Let \(\mathcal{L}\) be a nef line-bundle on X,
Lagrangian 4-planes in holomorphic symplectic varieties of K3[4]-type
We classify the cohomology classes of Lagrangian 4-planes ℙ4 in a smooth manifold X deformation equivalent to a Hilbert scheme of four points on a K3 surface, up to the monodromy action. Classically,
Characterizations of Projective Space and Applications to Complex Symplectic Manifolds
We obtain new criteria for a normal projective variety to be projective $n$-space. Our main result asserts that a normal projective variety which carries a closed, doubly-dominant, unsplitting family
A global Torelli theorem for hyperkahler manifolds
A mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies. We compute a mapping class group of a hypekahler manifold $M$, showing that it is
MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations
We use wall-crossing with respect to Bridgeland stability conditions to systematically study the birational geometry of a moduli space $$M$$M of stable sheaves on a K3 surface $$X$$X: (a) We describe
Ergodic complex structures on hyperkähler manifolds
Let M be a compact complex manifold. The corresponding Teichmüller space Teich is the space of all complex structures on M up to the action of the group $${{\rm Diff}_0(M)}$$Diff0(M) of isotopies.
Intersection Numbers of Extremal Rays on Holomorphic Symplectic Varieties
Suppose X is a smooth projective complex variety. Let N1(X, Z) ⊂ H2(X, Z) and N 1 (X, Z) ⊂ H 2 (X, Z) denote the group of curve classes modulo homological equivalence and the Neron-Severi group
Brill-Noether duality for moduli spaces of sheaves on K3 surfaces
Components of the Moduli space of sheaves on a K3 surface are parametrized by a lattice; the (algebraic) Mukai lattice. Isometries of the Mukai lattice often lift to symplectic birational
Projectivity and birational geometry of Bridgeland moduli spaces
We construct a family of nef divisor classes on every moduli space of stable complexes in the sense of Bridgeland. This divisor class varies naturally with the Bridgeland stability condition. For a
...
...