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The Beauville-Bogomolov class as a characteristic class

Eyal Markman

Abstract

Let X be any compact Kahler manifold deformation equivalent to the Hilbert scheme of length n subschemes on a K3 surface, n>1. We construct over XxX a rank 2n-2 reflexive twisted sheaf E, which is locally free away from the diagonal. The characteristic classes of E are invariant under the diagonal action of an index two subgroup of the monodromy group. Given a point x in X, the restriction E_x of E to {x}xX has the following properties. (1) The characteristic class k_i(E_x) in H^{i,i}(X,Q) can not be expressed as a polynomial in classes of lower degree, if 1<i<(n+1)/2. (2) The Beauville-Bogomolov class is equal to c_2(TX)+2k_2(E_x).

The Beauville-Bogomolov class as a characteristic class

Abstract

Let X be any compact Kahler manifold deformation equivalent to the Hilbert scheme of length n subschemes on a K3 surface, n>1. We construct over XxX a rank 2n-2 reflexive twisted sheaf E, which is locally free away from the diagonal. The characteristic classes of E are invariant under the diagonal action of an index two subgroup of the monodromy group. Given a point x in X, the restriction E_x of E to {x}xX has the following properties. (1) The characteristic class k_i(E_x) in H^{i,i}(X,Q) can not be expressed as a polynomial in classes of lower degree, if 1<i<(n+1)/2. (2) The Beauville-Bogomolov class is equal to c_2(TX)+2k_2(E_x).

Paper Structure

This paper contains 22 sections, 36 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. The main results
  3. Notation
  4. Characteristic classes of projective bundles and twisted sheaves
  5. Characteristic classes and Brauer classes of projective bundles
  6. Twisted sheaves
  7. Sheaves of Azumaya algebras and their characteristic classes
  8. Monodromy invariant classes
  9. The rational Hodge classes $\kappa_i(X)$
  10. Monodromy invariant classes $\kappa_i({\mathcal{F}})$ over ${\mathcal{M}}(v)\times {\mathcal{M}}(v)$
  11. Proof of the monodromy invariance of $\kappa_i(X)$ and $\kappa_i({\mathcal{F}})$
  12. A reflexive sheaf over ${\mathcal{M}}(v)\times {\mathcal{M}}(v)$
  13. Lifting deformations of a moduli space ${\mathcal{M}}$ to deformation of the pair $({\mathcal{M}},E)$
  14. Hyperholomorphic sheaves
  15. Twistor deformations of pairs
  16. ...and 7 more sections

Key Result

Proposition 1.2

Theorems & Definitions (83)

  • Definition 1.1
  • Proposition 1.2
  • Remark 1.3
  • Theorem 1.4
  • Lemma 1.5
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • ...and 73 more