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The Seiberg-Witten invariants and symplectic forms. (English) Zbl 0853.57019
Recently, Seiberg and Witten introduced a remarkable new equation which gives differential-topological invariants for a compact oriented 4-manifold with a distinguished integral cohomology class [see for example E. Witten, Lectures at MIT and Harvard, Fall 1994; and ‘Monopoles and four-manifolds’, Math. Res. Lett. 1, 769-796 (1994)].
The author obtains the following interesting result on the above-mentioned invariants: Let \(X\) be a compact oriented 4-manifold with \(b^{2+} \geq 2\). Let \(\omega\) be a symplectic form on \(X\) with \(\omega \wedge \omega\) giving the orientation. Then the first Chern class of the associated almost complex structure on \(X\) has Seiberg-Witten invariant equal to \(\pm 1\). As a consequence, it can be proved that connected sums of 4-manifolds with non-negative definite intersection forms do not admit symplectic forms which are compatible with the given orientation. For example, when \(n>1\) and \(m\geq 0\), then \((\#_n \mathbb{C} P^2) \# (\#_m \overline {\mathbb{C} P}^2)\) has no symplectic form which defines the given orientation.

57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
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