Submission date: 23 February 2015.

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.

Mathematics Subject Classification: Primary: 03C60, Secondary: 03C45, 03C65, 32J99

Keywords and phrases: Model Theory, Compact Complex Manifold, Generic Automorphism, Zilber Dichotomy, Canonical Base Property

Full text arXiv 1502.06521: pdf, ps.