Model Theory of Compact Complex Manifolds with an Automorphism

TL;DR

This paper develops the model theory of compact complex manifolds with a generic automorphism, revealing properties like geometric elimination of imaginaries, the canonical base property, and the Zilber dichotomy, with applications to complex dynamics.

Contribution

It introduces the theory CCMA, extending difference-closed fields to complex manifolds with automorphisms, and establishes key model-theoretic properties and classifications.

Findings

CCMA admits geometric elimination of imaginaries but not full elimination.

The canonical base property holds for finite-dimensional types in CCMA.

Finite-dimensional types of SU-rank one are either one-based or almost internal to the fixed field.

Abstract

Motivated by possible applications to meromorphic dynamics, and generalising known properties of difference-closed fields, this paper studies the theory CCMA of compact complex manifolds with a generic automorphism. It is shown that while CCMA does admit geometric elimination of imaginaries, it cannot eliminate imaginaries outright: a counterexample to 3-uniqueness in CCM is exhibited. Finite-dimensional types are investigated and it is shown, following the approach of Pillay and Ziegler, that the canonical base property holds in CCMA. As a consequence the Zilber dichotomy is deduced: finite-dimensional types of SU-rank one are either one-based or almost internal to the fixed field. In addition, a general criterion for stable embeddedness in TA (when it exists) is established, and used to determine the full induced structure of CCMA on projective varieties, simple nonalgebraic complex tori, and simply connected nonalgebraic strongly minimal manifolds.