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3-dimensional Seiberg-Witten invariants and non-Kählerian geometry. (English) Zbl 0909.57010

The paper is concerned with Seiberg-Witten invariants of closed \(3\)-manifolds and their relation to the \(4\)-dimensional invariants. One of the main results is a comparison theorem which identifies – as analytic spaces – the irreducible part of Seiberg-Witten moduli space of a \(3\)-manifold \(M\) with the irreducible part of the corresponding Seiberg-Witten moduli space of \(S^1\times M\). Then the authors consider the complex integrable case when the \(4\)-dimensional Seiberg-Witten invariants are effectively computable. Since the Hermitian complex structures on \(S^1\times M\) are, generally, non-Kählerian, the authors use a formula of Gauduchon for the Dirac operator. Then, in the case where \(b_1(M)\) is even, some problems arise in the definition of the natural complex orientation of the line bundle \(\text{det(index}(d^*+d^+))\). The manifolds \((M,g)\) for which the method can be used effectively are Riemannian manifolds \((M,g)\) which admit a Riemannian foliation by geodesics. The method is illustrated in the case of \(S^3\).
Reviewer: V.Oproiu (Iaşi)

MSC:

57R57 Applications of global analysis to structures on manifolds
32Q99 Complex manifolds
53C12 Foliations (differential geometric aspects)
57N10 Topology of general \(3\)-manifolds (MSC2010)
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