Model theory of compact complex manifolds with an automorphism

Bays, Martin; Hils, Martin; Moosa, Rahim

doi:10.1090/tran/6941

Model theory of compact complex manifolds with an automorphism
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by Martin Bays, Martin Hils and Rahim Moosa PDF

Trans. Amer. Math. Soc. 369 (2017), 4485-4516 Request permission

Abstract:

Motivated by possible applications to meromorphic dynamics, and generalising known properties of difference-closed fields, this paper studies the theory $\operatorname {CCMA}$ of compact complex manifolds with a generic automorphism. It is shown that while $\operatorname {CCMA}$ does admit geometric elimination of imaginaries, it cannot eliminate imaginaries outright: a counterexample to $3$-uniqueness in $\operatorname {CCM}$ 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 $\operatorname {CCMA}$. As a consequence the Zilber dichotomy is deduced: finite-dimensional minimal types 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 $\operatorname {CCMA}$ on projective varieties, simple nonalgebraic complex tori, and simply connected nonalgebraic strongly minimal manifolds.

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