O’Grady, Kieran G. Donaldson’s polynomials for \(K3\) surfaces. (English) Zbl 0735.57011 J. Differ. Geom. 35, No. 2, 415-428 (1992). 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 \(a_ c\). Reviewer: K.G.O’Grady Cited in 4 Documents MSC: 57R20 Characteristic classes and numbers in differential topology Keywords:Donaldson’s polynomial; K3 surface; moduli space; rank-two vector bundles; intersection form PDFBibTeX XMLCite \textit{K. G. O'Grady}, J. Differ. Geom. 35, No. 2, 415--428 (1992; Zbl 0735.57011) Full Text: DOI