Model theory of compact complex manifolds with an automorphism

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: finitedimensional minimal types are either one-based or almost internal to the fixed field. In addition, a general criterion for stable embeddedness in T A (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.

Model theoretic dynamics in Galois fashion

“…(1) One could ask about the weak elimination of imaginaries in T mc G . The theory CCMA from [2] is an example of T mc G which fits into our assumptions, but does not eliminate imaginaries (Corollary 3.6 in [2]), hence does not allow the weak elimination of imaginaries -otherwise by [7, Proposition 1.7] and Lemma 4.37, we would obtain a contradiction.…”

“…Proof. We start with proving implication from (1) to (2). Assume that q = tp (1) Type q is the non-forking extension of p.…”

“…The following proof of the geometric elimination of imaginaries (i.e. every imaginary element is interalgebraic with a finite real tuple) is based on similar proofs from [8, Paragraph 1.10], [9, Paragraph 2.9] and an observation from [2]. However, there are some tricky steps in our adaptation, therefore we include a whole proof.…”

“…It is quite common to use results from [9] during studying a new theory equipped with an automorphism. We give an example about this: in [2], the authors introduce a new theory which describes compact complex manifolds with an automorphism. Its model companion, denoted by CCMA, exists and automatically inherits simplicity and other tameness properties due to the main results of [9].…”

“…Proof. We start with proving implication from (1) to (2). Assume that q = tp D L (a/B) for some small a ⊂ D. Let M N D be such that A ⊆ M and B ⊆ N .…”