×

Seiberg-Witten invariants for 4-manifolds with \(b_+=0\). (English) Zbl 1034.53091

Peternell, Thomas (ed.) et al., Complex analysis and algebraic geometry. A volume in memory of Michael Schneider. Berlin: Walter de Gruyter (ISBN 3-11-016204-0/hbk). 347-357 (2000).
As one might guess from the title, this paper defines Seiberg-Witten invariants for \(4\text{-manifolds}\) with \(b_+=0\). It also contains other related results. The Seiberg-Witten equations depend on parameters. The set of parameter values for which the Seiberg-Witten moduli space contains reducible solutions has codimension \(b_+\) in the parameter space. Thus, when \(b_+>1\) the moduli space for a generic parameter is a manifold and the moduli spaces associated to any pair of generic parameters are connected by a path missing the codiminsional \(b_+\) subspace, therefore cobordant. When \(b_+=1\), there is a codimension 1 “wall” in the parameter space. This discrimant or wall cuts the parameter space into a collection of components, and there is an invariant associated to each component. Furthermore, the invariants associated to adjacent components differ by plus or minus one and there is a wall crossing formula, giving the sign. When \(b_+=0\) every moduli space contains reducibles.
The authors define an invariant by counting the irreducible solutions in the zero dimensional moduli spaces. They give an analytic description of the wall and prove a wall crossing formula. In addition, this paper contains a formula describing the Seiberg-Witten invariant of a \(3\text{-manifold}\), \(M\), in terms of the Seiberg-Witten invariant of \(M\times S^1\). As a by-product, they obtain a wall crossing formula for the \(b_1=0\) \(3\text{-manifold}\) invariant. The final result in this nice paper is a formula for the \(b_+=0\) invariant of Hermitian surfaces.
For the entire collection see [Zbl 0933.00031].

MSC:

53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
32L05 Holomorphic bundles and generalizations
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
57R57 Applications of global analysis to structures on manifolds
PDFBibTeX XMLCite
Full Text: arXiv