The authors give a complete classification of rational and ruled symplectic 4-manifolds. A symplectic manifold \((M,\omega)\) is said to be ruled if \(M\) is a \(S^2\)-bundle over a Riemann surface \(B\) such that the symplectic form \(\omega\) is nondegenerate on each fiber. \((M,\omega)\) is said to be rational if it can be obtained from the standard \(\mathbb{C} P^2\) by a sequence of blow-ups and blow-downs. The first results concerning this classification problem have been given by Gromov and Taubes. Now, the authors have completed the classification. The main result is the following:
Theorem. Let \((M,\omega)\) be a ruled symplectic manifold over a closed orientable surface. Then \(\omega\) is isotopic to a standard Kähler form on \(M\).
(Here, isotopic means that both symplectic forms can be joined by a path of cohomologous symplectic forms.) This result was obtained by combining techniques developed independently by the two authors.
The paper is self-contained and gives an excellent survey on the classification problem as well as other related problems.
For the entire collection see [Zbl 0852.00028].