Communications in Analysis and Geometry

Volume 13 (2005)

Number 3

Locally holomorphic maps yield symplectic structures

Pages: 511 – 525

DOI: https://dx.doi.org/10.4310/CAG.2005.v13.n3.a2

Author

Robert E. Gompf

Abstract

For a smooth map f:X⁴ → Σ² that is locally modeled by holomorphic maps, the domain is shown to admit a symplectic structure that is symplectic on some regular fiber, if and only if f^*[Σ] ≠ 0. If so, the space of symplectic forms on X that are symplectic on all fibers is non-empty and contractible. The cohomology classes of these forms vary with the maximum possible freedom on the reducible fibers, subject to the obvious constraints. The above results are derived via an analogous theorem for locally holomorphic maps f:X²ⁿ → Y^{2n −2} with Y symplectic.

Published 1 January 2005