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The Seiberg-Witten and Gromov invariants. (English) Zbl 0854.57020
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.

57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
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