Holomorphic shadows in the eyes of model theory

TL;DR

This paper introduces the concept of holomorphic shadows in almost complex manifolds, showing they form Zariski-type structures under certain conditions and connecting symplectic geometry with model theory.

Contribution

It defines holomorphic shadows and demonstrates their structure as Zariski-type, linking complex geometry with model-theoretic frameworks and extending results in symplectic geometry.

Findings

Holomorphic shadows form Zariski-type structures under specific conditions.

J-holomorphic curves are examples of holomorphic shadows.

Connections established between symplectic geometry and model theory.

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.