Holomorphic shadows in the eyes of model theory

Details Holomorphic shadows in the eyes of model theory Holomorphic shadows in the eyes of model theory

2008-10-23

arxiv.org/abs/0810.4294v2

Transactions of the American Mathematical Society, 363 (2011), no. 6 3287-3307

Mathematics

Differential Geometry

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Scientific paper

10.1090/S0002-9947-2011-05219-0

We define a subset of an almost complex manifold (M,J) to be a holomorphic shadow if it is the image of a J-holomorphic map from a compact complex manifold. Notice that a J-holomorphic curve is a holomorphic shadow, and so is a complex subvariety of a compact complex manifold. We show that under some conditions on an almost complex structure J on a manifold M, the holomorphic shadows in the Cartesian products of (M,J) form a Zariski-type structure. Checking this leads to non-trivial geometric questions and results. We then apply the work of Hrushovski and Zilber on Zariski-type structures. We also restate results of Gromov and McDuff on J-holomorphic curves in symplectic geometry in the language of shadows structures.

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