On the Chern numbers of the generalised Kummer varieties

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 ${\int }_X{c}_1(X{)}^2\ne 0$, 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 $(\IP^2)^{[k]}$ of points on the projective plane. Since $\int_{\IP^2} c_1(\IP^2)^2 = 9 \neq 0$ 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 $n\le 8$ is included.

9 pages