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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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