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More constraints on symplectic forms from Seiberg-Witten invariants. (English) Zbl 0854.57019
The author proves the following result for a closed oriented symplectic 4-manifold \((X,\omega)\) with \(b^+_2 \geq 2\):
If the Seiberg-Witten invariant for a class \(c \in H^2(X)\) is non-zero then \(|c \cdot [\omega]|\leq c_1(K) \cdot [\omega]\). Moreover, equality holds iff \(\pm c\) is equal to \(c_1(K)\).
Here \(K\) is the canonical line bundle for any almost complex structure on \(X\) which is compatible with \(\omega\). Along the same line of arguments, the author proves that the complex projective plane has no symplectic form \(\omega\) for which \(c_1(K) \cdot [\omega] > 0\). This result was recently improved by the same author who showed that indeed the complex projective plane has a unique symplectic structure [the author, ibid. No. 2, 221-238 (1995; Zbl 0854.57020); see the review below]. That paper is a very good survey on Seiberg-Witten invariants for symplectic 4-manifolds.

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