Papers › On the Chern numbers of the generalised Kummer varieties
On the Chern numbers of the generalised Kummer varieties
Marc Arnold Nieper-Wisskirchen
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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 ∫_X c₁(X)² ≠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 (IP²)^([k]) of points on the projective plane. Since ∫_(IP²) c₁(IP²)² = 9 ≠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 ≤8 is included.
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