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Witten triples and the Seiberg-Witten equations on a complex surface. (English) Zbl 1096.14038
The paper generalizes and unifies a type of Kobayashi-Hitchin correspondence that identifies irreducible Seiberg-Witten solutions on a complex surface with stable Witten triples. Previously perturbations of the forms \(2\pi i\beta^+, ({\overline{\eta}}-\eta)/2\) have been separately considered by Witten, Okonek-Teleman, Biquard et al, where \(\beta\) is a real 2-form of type \((1,1)\) and \(\eta\) is a holomorphic 2-form.
In the paper under review, the author combines them together and examines the Seiberg-Witten equations under a perturbation of the form \(2\pi i\beta^+ + ({\overline{\eta}}-\eta)/2\). Then an irreducible Seiberg-Witten solution, up to gauge equivalence, corresponds to a unique equivalence class of Witten (poly) stable triple consisting of a line bundle and two bundle homomorphisms. Moreover, if the metric on the complex surface is Kähler, the correspondence is is shown to be the set-theoretical support of an isomorphism of real analytic spaces.
MSC:
14J80 Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants)
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