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Seiberg-Witten invariants and rationality of complex surfaces. (English) Zbl 0883.57022
The paper begins with an easy introduction to $$\text{Spin}^c$$-structures on 4-manifolds and the associated Seiberg-Witten monopole equations. Then it explains the structure of the corresponding Seiberg-Witten invariants with special emphasis on the particular case of manifolds with $$b_+ =1$$. Next it is shown that monopoles on Kähler surfaces have a purely holomorphic interpretation as effective divisors. The present proof is a simplified version of the Kobayashi-Hitchin type correspondence which was obtained in [the authors, Int. J. Math. 6, No. 6, 893-910 (1995; Zbl 0846.57013)]. As an application, a simple and selfcontained proof is given for the fact that rationality of complex surfaces is a $${\mathcal C}^\infty$$-property.
Reviewer: C.Okonek (Zürich)

##### MSC:
 57N13 Topology of the Euclidean $$4$$-space, $$4$$-manifolds (MSC2010) 57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.) 57R57 Applications of global analysis to structures on manifolds
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