The classification of ruled symplectic 4-manifolds. (English) Zbl 0874.57019

A 4-manifold \((M^4,\omega)\) is said to be ruled if it is the total space of an \(S^2\)-fibration \(\pi: M\to\Sigma\) over a compact Riemann surface \(\Sigma\). The symplectic form \(\omega\) is compatible with the ruling \(\pi\) if it is nondegenerate on the fibers. The authors prove that there is, up to isomorphism, at most one symplectic form on \(M\) in each cohomology class. More precisely, their main theorem is the following: Let \(\omega_0\) and \(\omega_1\) be two cohomologous symplectic forms on the ruled 4-manifold \(M\). Then there is a diffeomorphism \(f:M\to M\) such that \(f^*(\omega_0)=\omega_1\). Moreover, if \(\omega_0\) and \(\omega_1\) are both compatible with the ruling, then they are isotopic. Since the possible cohomology classes of symplectic forms are known [see for example the second author, J. Am. Math. Soc. 3, 679-712 (1990; Zbl 0723.53019); Erratum: ibid. 5, 987-988 (1992; Zbl 0799.53039)], the above result completes the classification of symplectic forms on the ruled 4-manifolds.


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