Donaldson's polynomials for \(K3\) surfaces
The aim of the paper is to compute Donaldson's polynomial \(\gamma_ c(S)\) when \(S\) is a K3 surface and \(c\) is odd. Since it is known that \(\gamma_ c(S)=a_ cq_ S^{2c-3}\) for some number \(a_ c\), where \(q_ S\) is the intersection form on \(S\), it is sufficient to compute the value \(\gamma_ c(\alpha)\) on a homology class such that \(q_ S(\alpha)\neq 0\). This is done as follows. Let \(H\) be a polarization of \(S\). The author defines a polynomial \(\delta_ c(S,H)\) on the second homology of \(S\) by mimicking the definition of \(\gamma_ c(S)\) in the context of algebraic geometry. It is showed that \(\delta_ c(S,H)=\gamma_ c(S)\). The restriction of \(\delta_ c(S,H)\) to \(D=(H^{2,0}(S)\oplus H^{0,2}(S))\cap H^ 2(S;{\mathbb{R}})\) is then computed by integrating a top differential form on the moduli space of \(H\)-stable rank-two vector bundles on \(S\) with \(c_ 1=0\), \(c_ 2=c\). Since the intersection form is positive definite on \(D\) this determines \(a_ c\).
scientific article; zbMATH DE number 4215956