On the Chern numbers of generalised Kummer varieties

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 $\int_X c_1(X)^2 \neq 0$, 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 $X^{[k]}$. In~\cite{ellingsrud87} and~\cite{ellingsrud96} it is shown 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 $n \leq 8$ is included.

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Published 1 January 2002