Taubes, Clifford Henry The Seiberg-Witten and Gromov invariants. (English) Zbl 0854.57020 Math. Res. Lett. 2, No. 2, 221-238 (1995). This paper is a very good survey of Seiberg-Witten and Gromov invariants for symplectic 4-manifolds. It contains all relevant definitions and many outlines of proofs. The main result equates the invariants from the title (up to sign). Seiberg-Witten invariants are defined for any closed oriented 4-manifold with \(b^+_2 \geq 2\) whereas Gromov invariants exist for symplectic 4-manifolds. As corollaries, one obtains for example the following results: 1. Let \(K\) be the canonical line bundle for any almost complex structure on \(X\) which is compatible with the symplectic form \(\omega\). Then the Poincaré dual to \(c_1(K)\) is represented by an embedded symplectic curve.2. If \(c_1 (K) \cdot c_1(K) < 0\) the \(X\) can be symplectically blown down along a symplectic curve of self-interaction \(-1\).As for 4-manifolds with \(b^+_2 =1\), the methods of the paper imply that the complex projective plane has a unique symplectic structure. Reviewer: P.Teichner (Berkeley) Cited in 17 ReviewsCited in 54 Documents MSC: 57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) Keywords:Seiberg-Witten invariants; pseudo-holomorphic curves; Gromov invariants; symplectic 4-manifolds; 4-manifold Citations:Zbl 0854.57019 PDFBibTeX XMLCite \textit{C. H. Taubes}, Math. Res. Lett. 2, No. 2, 221--238 (1995; Zbl 0854.57020) Full Text: DOI