Locally holomorphic maps yield symplectic structures (Q2583311)

In the past several years the relationship between the differential topology of closed 4-manifolds and their symplectic structures has been shown. A map \(f : X^ {2n}\to Y^ {2m}\) between smooth, oriented manifolds is locally holomorphic if for each \(x\in X\), there are smooth, orientation-preserving coordinate charts about \(x\) and \(f(x)\) in which \(f\) is given by a holomorphic map. In this paper, the author proves that if \(f: X\to\Sigma\) is a locally holomorphic map from a closed, oriented, connected 4-manifold \(X\) to a closed, oriented surface \(\Sigma\), then \(X\) admits a symplectic structure that is symplectic on some regular fiber, if and only if \(f^ *[\Sigma]\neq 0\in H^ 2_{dR}(X)\). If \({\mathcal S}\) is the space of symplectic forms on \(X\) that are symplectic on all fibers, then \({\mathcal S}\) is non-empty and contractible. The cohomology classes of these forms vary with the maximum possible freedom on the reducible fibers. The above results are generalized by showing that if \(X^{2n}\), \(Y^{2n-2}\) are closed, oriented manifolds with a symplectic form \(\omega_ Y\) on \(Y\) and an \(\omega_ Y\)-compatibly locally holomorphic map \(f : X \to Y\), then \(X\) admits a symplectic structure.