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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Hirzebruch-Riemann-Roch formulae on irreducible symplectic Kähler manifolds

Author: Marc A. Nieper
Journal: J. Algebraic Geom. 12 (2003), 715-739
Published electronically: June 26, 2003
MathSciNet review: 1993762
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Abstract | References | Additional Information

Abstract: In this article we investigate Hirzebruch-Riemann-Roch formulae for line bundles on irreducible symplectic Kähler manifolds. As Huybrechts has shown, for every irreducible symplectic Kähler manifold $X$ of dimension $2n$, there are numbers $a_0, a_2, \dots , a_{2n}$ such that \begin{equation*} \chi (L) = \sum _{k = 0}^n \frac {a_{2k}}{(2k)!} q_X(c_1(L))^k \end{equation*} for the Euler characteristic of a line bundle $L$, where $q_X: \mathrm {H}^2(X, \mathbb {C}) \to \mathbb {C}$ is the Beauville-Bogomolov quadratic form of $X$. Using Rozansky-Witten classes similarly to Hitchin and Sawon, we obtain a closed formula expressing the $a_{2k}$ in terms of Chern numbers of $X$.

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Additional Information

Marc A. Nieper
Affiliation: Mathematisches Institut der Univ. zu Köln, Weyertal 86–90, 50931 Köln, Germany
Address at time of publication: Schillingstr. 1, 50670 Köln. Germany

Received by editor(s): April 10, 2001
Received by editor(s) in revised form: August 22, 2001
Published electronically: June 26, 2003
Additional Notes: We are very grateful to Daniel Huybrechts for having carefully read preliminary versions of this paper, and to Michael Britze, Daniel Huybrechts, Manfred Lehn and many others for their support to us and helpful discussions about the subject