×

Grafting Seiberg-Witten monopoles. (English) Zbl 1027.57030

Let \((M,\omega)\) be a symplectic 4-manifold and \(E_0\), \(E_1\) two complex line bundles over \(M\). Being given solutions \((A,\psi_i)\) of the Seiberg-Witten equations for the \(\text{Spin}^c\)-structures associated to \(E_i\), this paper produces a solution of the Seiberg-Witten equation for the \(\text{Spin}^c\)-structure associated to the tensor product \(E_0\otimes E_1\).

MSC:

57R57 Applications of global analysis to structures on manifolds
53C27 Spin and Spin\({}^c\) geometry
53D99 Symplectic geometry, contact geometry
58J05 Elliptic equations on manifolds, general theory
57R17 Symplectic and contact topology in high or arbitrary dimension
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI arXiv EuDML EMIS

References:

[1] S Donaldson, I Smith, Lefschetz pencils and the canonical class for symplectic four-manifolds, Topology 42 (2003) 743 · Zbl 1012.57040
[2] D McDuff, The local behaviour of holomorphic curves in almost complex 4-manifolds, J. Differential Geom. 34 (1991) 143 · Zbl 0736.53038
[3] J W Morgan, Z Szab√≥, C H Taubes, A product formula for the Seiberg-Witten invariants and the generalized Thom conjecture, J. Differential Geom. 44 (1996) 706 · Zbl 0974.53063
[4] C H Taubes, \(\mathrm{SW}{\Rightarrow}\mathrm{Gr}\): from the Seiberg-Witten equations to pseudo-holomorphic curves, J. Amer. Math. Soc. 9 (1996) 845 · Zbl 0867.53025
[5] C H Taubes, \(\mathrm{Gr}{\Rightarrow}\mathrm{SW}\): from pseudo-holomorphic curves to Seiberg-Witten solutions, J. Differential Geom. 51 (1999) 203 · Zbl 1036.53066
[6] C H Taubes, \(\mathrm{GR}=\mathrm{SW}\): counting curves and connections, J. Differential Geom. 52 (1999) 453 · Zbl 1040.53096
[7] C H Taubes, Counting pseudo-holomorphic submanifolds in dimension 4, J. Differential Geom. 44 (1996) 818 · Zbl 0883.57020
[8] C H Taubes, The Seiberg-Witten invariants and symplectic forms, Math. Res. Lett. 1 (1994) 809 · Zbl 0853.57019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.