The Beauville-Bogomolov class as a characteristic class

Markman, Eyal

doi:10.1090/jag/750

Author: Eyal Markman
Journal: J. Algebraic Geom. 29 (2020), 199-245
DOI: https://doi.org/10.1090/jag/750
Published electronically: November 20, 2019
MathSciNet review: 4069649
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Abstract | References | Additional Information

Abstract:

Let $X$ be any compact Kähler manifold deformation equivalent to the Hilbert scheme of length $n$ subschemes on a $K3$ surface, $n\geq 2$. We construct over $X\times X$ a rank $2n-2$ reflexive twisted coherent sheaf $E$, which is locally free away from the diagonal. The characteristic classes $\kappa _i(E)\in H^{i,i}(X\times X,\mathbb {Q})$ of $E$ are invariant under the diagonal action of an index $2$ subgroup of the monodromy group of $X$. Given a point $x\in X$, the restriction $E_x$ of $E$ to $\{x\}\times X$ has the following properties.

The characteristic class $\kappa _i(E_x)\in H^{i,i}(X,\mathbb {Q})$ cannot be expressed as a polynomial in classes of lower degree if $2\leq i\leq n/2$.
The Beauville-Bogomolov class is equal to $c_2(TX)+2\kappa _2(E_x)$.

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Additional Information

Eyal Markman
Affiliation: Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003
MR Author ID: 355854
Email: markman@math.umass.edu

Received by editor(s): July 7, 2016
Published electronically: November 20, 2019
Article copyright: © Copyright 2019 University Press, Inc.