The Beauville-Bogomolov class as a characteristic class

TL;DR

This paper constructs a special reflexive sheaf on certain hyperkähler manifolds and relates its characteristic classes to the Beauville-Bogomolov form, revealing new invariants in complex geometry.

Contribution

It introduces a rank 2n-2 reflexive sheaf on hyperkähler manifolds deformation equivalent to Hilbert schemes, linking its characteristic classes to the Beauville-Bogomolov class.

Findings

The characteristic class k_i(E_x) cannot be expressed as a polynomial in lower classes for certain i.

The Beauville-Bogomolov class equals c_2(TX)+2k_2(E_x).

The constructed sheaf's characteristic classes are monodromy invariant.

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).