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Lagrangian fibrations of holomorphic-symplectic varieties of K3^[n]-type

Eyal Markman

Abstract

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 L be a nef line-bundle on X, such that the 2n-th power of c_1(L) vanishes and c_1(L) is primitive. Assume that the two dimensional subspace H^{2,0}(X) + H^{0,2}(X), of the second cohomology of X with complex coefficients, intersects trivially the integral cohomology. We prove that the linear system of L is base point free and it induces a Lagrangian fibration on X. In particular, the line-bundle L is effective. A determination of the semi-group of effective divisor classes on X follows, when X is projective. For a generic such pair (X,L), not necessarily projective, we show that X is bimeromorphic to a Tate-Shafarevich twist of a moduli space of stable torsion sheaves, each with pure one dimensional support, on a projective K3 surface.

Lagrangian fibrations of holomorphic-symplectic varieties of K3^[n]-type

Abstract

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 L be a nef line-bundle on X, such that the 2n-th power of c_1(L) vanishes and c_1(L) is primitive. Assume that the two dimensional subspace H^{2,0}(X) + H^{0,2}(X), of the second cohomology of X with complex coefficients, intersects trivially the integral cohomology. We prove that the linear system of L is base point free and it induces a Lagrangian fibration on X. In particular, the line-bundle L is effective. A determination of the semi-group of effective divisor classes on X follows, when X is projective. For a generic such pair (X,L), not necessarily projective, we show that X is bimeromorphic to a Tate-Shafarevich twist of a moduli space of stable torsion sheaves, each with pure one dimensional support, on a projective K3 surface.

Paper Structure

This paper contains 17 sections, 29 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Classification of primitive-isotropic classes
  3. An example of a Lagrangian fibration for each value of the monodromy invariants
  4. Period domains and period maps
  5. A projective $K3$ surface associated to an isotropic class
  6. A period domain as an affine line bundle over another
  7. The period map
  8. Density of periods of relative compactified Jacobians
  9. A period of a Lagrangian fibration in each Tate-Shafarevich line
  10. Monodromy equivariance of the fibration by Tate-Shafarevich lines
  11. Density
  12. Primitive isotropic classes and Lagrangian fibrations
  13. Tate-Shafarevich lines and twists
  14. The geometry of the universal curve
  15. A universal family of Tate-Shafarevich twists
  16. ...and 2 more sections

Key Result

Theorem 1.3

Let $X$ be an irreducible holomorphic symplectic manifold of $K3^{[n]}$-type and ${\mathcal{L}}$ a nef line-bundle, such that $c_1({\mathcal{L}})$ is primitive and isotropic with respect to the Beauville-Bogomolov-Fujiki pairing. Assume that $X$ is non-special. Then the line bundle ${\mathcal{L}}$ i

Theorems & Definitions (66)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.5
  • Corollary 1.6
  • Proposition 1.7
  • Remark 1.8
  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • ...and 56 more