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Wall-crossing formulas, Bott residue formula and the Donaldson invariants of rational surfaces

Geir Ellingsrud·Lothar Göttsche
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Abstract

We study the Donaldson invariants of rational surfaces and their dependence on the chambers in the ample cone. We build on a previous joint paper in which we have expressed the change of the Donaldson invariants on an algebraic surface $S$ under crossing a so-called good wall in terms of certain intersection numbers on Hilbert schemes of points on $S$. In the current paper we assume that $S$ is rational. Thish enables us to apply the Bott residue theorem to evaluate these intersection numbers with the help of a computer. Combining this with the blowup formulas we obtain an algorithm for computing the Donaldson invariants of all rational surfaces in almost all chambers. In particular we compute all the $SU(2)$- and $SO(3)$-Donaldson invariants of the projective plane of degree at most 50.

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BibTeXRIS

Geir Ellingsrud, Lothar Göttsche. 1995-07-15. Wall-crossing formulas, Bott residue formula and the Donaldson invariants of rational surfaces. https://arxiv.org/abs/alg-geom/9506019

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