Grafting Seiberg-Witten monopoles. (English) Zbl 1027.57030

Jabuka, Stanislav

Algebr. Geom. Topol. 3, 155-185 (2003).

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 \(E_0\otimes E_1\).

Reviewer: R.E.Stong (Charlottesville)

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)

Keywords:

symplectic 4-manifolds; Seiberg-Witten gauge theory; \(J\)-holomorphic curves

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 Szabó, 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