2001/01/08 by Michael Britze, Britze, Michael, Marc A. Nieper +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AG
paper · pdf · doi:10.48550/arxiv.math/0101062
24 pages, includes dbnsymb font
arxiv created 2001/01/08 · openalex publication_date 2001/01/08 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article we investigate Hirzebruch-Riemann-Roch formulae for line bundles on irreducible symplectic Kähler manifolds. As Huybrechts has shown, for every irreducible complex Kähler manifold X of dimension 2n, there are numbers a0, a2, ..., a2n such that χ(L) = ∑k = 0n a2k/(2k)! qX(c1(L))k for the Euler characteristic of a line bundle L, where qX: H2(X, \mathbbm C) → \mathbbm C is the Beauville-Bogomolov quadratic form of X. Using Rozansky-Witten classes similar to Hitchin and Sawon, we obtain a formula expressing the a2k in terms of Chern numbers of X. Furthermore, for the n-th generalized Kummer variety \KA n, we prove χ(L) = (n + 1) \binomq(c1(L)) / 2 + n n by purely algebro-geometric methods, where q is the form qX up to a positive rational constant. A similar formula is already known for the Hilbert scheme of zero-dimensional subschemes of length n on a K3-surface. Using our results, we are able to calculate all Chern numbers of the generalized Kummer varieties \KA n for n ≤ 5. For n ≤ 4 these results were previously obtained by Sawon.