UC Berkeley

We develop notions of "almost complex analytic subsets" of almost complex manifolds, modelled after complex analytic subsets of complex manifolds. Basic analytic-geometric results are presented, including an identity principle for almost complex maps, and a proof that the singular locus of an almost complex analytic set is itself an "equational" almost complex analytic set, under certain conditions.

This work is partly motivated by geometric model theory. B. Zilber observed that a compact complex manifold, equipped with the logico-topological structure given by its complex analytic subsets, satisfies the axioms for a so-called "Zariski geometry", kick-starting a fruitful model-theoretic study of complex manifolds. Our results point towards a natural generalization of Zilber’s theorem to almost complex manifolds, using our notions of almost complex analytic subset. We include a discussion of progress towards this goal.

Our development draws inspiration from Y. Peterzil and S. Starchenko’s theory of nonstandard complex analytic geometry. We work primarily in the real analytic setting.