Locally holomorphic maps yield symplectic structures

Details Locally holomorphic maps yield symplectic structures Locally holomorphic maps yield symplectic structures

2005-11-15

arxiv.org/abs/math/0511385v1

Comm. Anal. and Geom. 13 (2005), 511-525

Mathematics

Symplectic Geometry

10 pages, no figures

Scientific paper

For a smooth map $f:X^4\to\Sigma^2$ 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 $f^*[\Sigma]\ne0$. If so, the space of symplectic forms on $X$ that are symplectic on all fibers is nonempty 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 $f:X^{2n}\to Y^{2n-2}$ with $Y$ symplectic.

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