On the Chern numbers of the generalised Kummer varieties

arXivmath/0204197MaRDI QIDQ6471804

Marc A. Nieper-Wisskirchen

Publication date: 15 April 2002

Abstract: Let $A^{[[n]]}$ denote the $2(n-1)$-dimensional generalised Kummer variety constructed from the abelian surface $A$. Further, let $X$ be an arbitrary smooth projective surface with $in{t}_X{c}_1(X{)}^{2}eq0$, and $X^{[k]}$ the Hilbert scheme of zero-dimensional subschemes of $X$ of length $k$. We give a formula which expresses the value of any complex genus on $A^{[[n]]}$ in terms of Chern numbers of the varieties $X^{[k]}$. It is shown by Ellingsrud and Stroemme how to use Bott's residue formula to effectively calculate the Chern numbers of the Hilbert schemes $(I{P}^2{)}^{[k]}$ of points on the projective plane. Since $in{t}_{I{P}^2}{c}_1(I{P}^2{)}^2=9eq0$ we can use these numbers and our formula to calculate the Chern numbers of the generalised Kummer varieties. A table with all Chern numbers of the generalised Kummer varieties $A^{[[n]]}$ for $nleq8$ is included.

Has companion code repository: https://github.com/8d1h/bott

Mathematics Subject Classification ID

Parametrization (Chow and Hilbert schemes) (14C05)

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