## 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.

### MSC:

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

### Citations:

Zbl 0723.53019; Zbl 0799.53039
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