Instanton invariants and flat connections on the Kummer surface
type: Item
instance of: scholarly article
title: Instanton invariants and flat connections on the Kummer surface
zbMATH Open document ID: 0754.57015
DOI: 10.1215/S0012-7094-91-06411-2
published in: Duke Mathematical Journal
publication date: 1992-09-27T00:00:00Z
review text: The author shows how to calculate a certain integer-valued instanton invariant for the \(K3\) surface \(X\) and any homotopy \(K3\) surface \(X(p)\) obtained by performing a logarithmic transform of order \(p\) on \(X\). He reproves the result of Friedman and Morgan that the \(X(p)\) are not diffeomorphic to \(X\) and in fact represent infinitely many distinct diffeomorphism types. The calculations neither use Donaldson's translation to stable bundles nor the \(C\)-infinity techniques developed by Gompf and Mrowka (in order to compute the same invariant), but draw on the fact that for certain nonsimply connected 4-manifolds or orbifolds the moduli space of flat connections has a real meaning for the theory. This moduli space can be understood purely topologically, which is the reason for the simplifications coming from this approach. In this paper the orbifold is the 4-torus divided by the involution \(t\to -t\), which gives the \(K3\) surface \(X\) after resolving the 16 double points with (\(-2\))-curves. The main idea now is that, though the manifolds \(X(p)\) also contain 16 (\(-2\))-curves, the fundamental groups of the complements (w.r.t. the standard embedding) are different, and through the instanton moduli space this difference can be detected in the manifolds \(X\) resp. \(X(p)\).
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 58J10
Mathematics Subject Classification ID: 57R55
Mathematics Subject Classification ID: 58E15
Mathematics Subject Classification ID: 32J15
zbMATH DE Number: 58545
zbMATH Keywords: Kummer surface
zbMATH Keywords: instanton invariant for the \(K3\) surface
zbMATH Keywords: homotopy K3- surface
zbMATH Keywords: logarithmic transform
zbMATH Keywords: diffeomorphism types
zbMATH Keywords: moduli space of flat connections
zbMATH Keywords: orbifold
author: P. B. Kronheimer
reviewed by: Peter Teichner
MaRDI profile type: Publication
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full work available at URL: s0012-7094-91-06411-2
OpenAlex ID: W1552530909
description: scientific article; zbMATH DE number 58545