Comparison of the Donaldson polynomial invariants with their algebro- geometric analogues
type: Item
instance of: scholarly article
title: Comparison of the Donaldson polynomial invariants with their algebro- geometric analogues
zbMATH Open document ID: 0801.57014
DOI: 10.1016/0040-9383(93)90001-C
author: John W. Morgan
published in: Topology
publication date: 1994-11-29T00:00:00Z
review text: When \(M\) is a closed simply connected smooth 4-manifold with \(b_ 2^ + > 1\) and odd, and \(\beta\) is an orientation for \(H^ 2_ +(M,\mathbb{R})\), Donaldson has defined polynomial invariants \(\gamma_ c(M,\beta)\). These are diffeomorphism type invariants and provide a major technique for understanding differential structures on 4-manifolds; much current research is devoted to their structure and computation. Their definition uses the Uhlenbeck compactification \(\overline{\mathcal M}(P_ c,g)\) of the moduli space \({\mathcal M}(P_ c,g)\) of \(g\)-ASD connections of a principal \(SU(2)\)-bundle \(P_ c\) over \(M\) with \(c_ 2(P_ c) = c\) and a map \(\mu: H_ 2(M) \to H^ 2(\overline{\mathcal M}(P_ c,g))\). For \(S\) a simply connected algebraic surface with \(p_ g(S) > 0\) with polarization \(\mathcal L\), \textit{K. G. O'Grady} [Invent. Math. 107, No. 2, 351-395 (1992; Zbl 0769.14008)] has defined analogues \(\delta_ c(S,{\mathcal L})\) of these polynomials using the Gieseker compactification \(\overline{\mathcal M}^ G_ c(S,{\mathcal L})\) of the moduli space of \(\mathcal L\)-slope stable vector bundles within the Gieseker moduli space \(\widehat{\mathcal M}^ G_ c(S,{\mathcal L})\) and a map \(\nu: H_ 2(S) \to H^ 2(\widehat{\mathcal M}^ G_ c(S,{\mathcal L}))\) similar to \(\mu\). Since the algebro-geometric polynomials \(\delta_ c(S,{\mathcal L})\) depend on the complex structure and the polarization, it is unclear whether they are in fact diffeomorphism type invariants. However, their definition allows their computation via algebraic geometric techniques, which have provided the most powerful computational techniques in the gauge theory of 4-manifolds until fairly recently. \textit{S. K. Donaldson} [Proc. Lond. Math. Soc., III. Ser. 50, 1- 26 (1985; Zbl 0547.53019)] had shown there is a homeomorphism between the moduli space of \(\mathcal L\)-slope stable holomorphic vector bundles with \(c_ 1 = 0\) and \(c_ 2 = c\) and the moduli space \({\mathcal M}(P_ c,g_{\…
reviewed by: Terry Lawson
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 57R57
Mathematics Subject Classification ID: 32L10
zbMATH DE Number: 488107
zbMATH Keywords: moduli space of \(g-\)ASD connections
zbMATH Keywords: moduli space of stable vector bundles
zbMATH Keywords: Donaldson polynomials
zbMATH Keywords: closed simply connected smooth 4- manifold
zbMATH Keywords: differential structures on 4-manifolds
zbMATH Keywords: Uhlenbeck compactification
zbMATH Keywords: Gieseker compactification
zbMATH Keywords: algebro-geometric polynomials
zbMATH Keywords: simply connected algebraic surface
zbMATH Keywords: polarization
zbMATH Keywords: canonical class
zbMATH Keywords: complete intersection
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 488107