Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



The Beauville-Bogomolov class as a characteristic class

Author: Eyal Markman
Journal: J. Algebraic Geom. 29 (2020), 199-245
Published electronically: November 20, 2019
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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.

  1. 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$.
  2. The Beauville-Bogomolov class is equal to $ c_2(TX)+2\kappa _2(E_x)$.

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Eyal Markman
Affiliation: Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003

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