Holomorphic shadows in the eyes of model theory

Kessler, Liat

doi:10.1090/S0002-9947-2011-05219-0

Holomorphic shadows in the eyes of model theory
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Trans. Amer. Math. Soc. 363 (2011), 3287-3307 Request permission

Abstract:

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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