HirzebruchRiemannRoch formulae on irreducible symplectic Kähler manifolds
Author:
Marc A. Nieper
Journal:
J. Algebraic Geom. 12 (2003), 715739
DOI:
https://doi.org/10.1090/S1056391103003254
Published electronically:
June 26, 2003
MathSciNet review:
1993762
Fulltext PDF
Abstract  References  Additional Information
Abstract: In this article we investigate HirzebruchRiemannRoch 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 BeauvilleBogomolov quadratic form of $X$. Using RozanskyWitten classes similarly to Hitchin and Sawon, we obtain a closed formula expressing the $a_{2k}$ in terms of Chern numbers of $X$.

atiyah67 Michael. F. Atiyah, Ktheory, W. A. Benjamin, Inc., 1967.
barnatan95 Dror BarNatan, On the Vassiliev knot invariants, Topology 34 (1995), no. 2, 423–472.
britze01 Michael Britze and Marc A. Nieper, HirzebruchRiemannRoch formulae on irreducible symplectic Kähler manifolds, arXiv:math.AG/0101062.
chmutov99 S. V. Chmutov and S. V. Duzhin, A lower bound for the number of Vassiliev knot invariants, Topology Appl. 92 (1999), no. 3, 201–223.
dasbach98 Oliver T. Dasbach, On the combinatorial structure of primitive Vassiliev invariants, II, J. Combin. Theory 81 (1998), no. 2, 127–139.
lehn99 Geir Ellingsrud, Lothar Göttsche, and Manfred Lehn, On the cobordism class of the Hilbert scheme of a surface, J. Algebraic Geom. 10 (2001), no. 1, 81–100.
hitchin99 Nigel Hitchin and Justin Sawon, Curvature and characteristic numbers of hyperKähler manifolds, Duke Math. J. 106 (2001), no. 3, 599–615.
huybrechts99 Daniel Huybrechts, Compact hyperKähler manifolds: basic results, Invent. Math. 135 (1999), no. 1, 63–113.
kapranov99 Mikhail Kapranov, RozanskyWitten invariants via Atiyah classes, Compos. Math. 115 (1999), no. 1, 71–113.
rozansky97 Lev Rozansky and Edward Witten, HyperKähler geometry and invariants of threemanifolds, Selecta Math. (N.S.) 3 (1997), no. 3, 401–458.
thurston00 Dylan P. Thurston, Wheeling: A diagrammatic analogue of the Duflo isomorphism, Ph.D. thesis, University of California at Berkeley, arXiv:math.QA/0006083, Spring 2000.
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
Email:
mail@marcnieper.de, marc@nieperwisskirchen.de
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