Definability of restricted theta functions and families of abelian varieties

We consider some classical maps from the theory of abelian varieties and their moduli spaces and prove their definability, on restricted domains, in the o-minimal structure \Rae. In particular, we prove that the embedding of moduli space of principally polarized ableian varierty, Sp(2g,\Z)\backslash \CH_g, is definable in \Rae, when restricted to Siegel's fundamental set \fF_g. We also prove the definability, on appropriate domains, of embeddings of families of abelian varieties into projective space

Model theoretic stability and definability of types, after A. Grothendieck

We point out how the "Fundamental Theorem of Stability Theory", namely the equivalence between the "non order property" and definability of types, proved by Shelah in the 1970s, is in fact an immediate consequence of Grothendieck's "Crit{\`e}res de compacit{\'e}" from 1952. The familiar forms for the defining formulae then follow using Mazur's Lemma regarding weak convergence in Banach spaces

Definability of groups in ℵ0\aleph_0-stable metric structures

We prove that in a continuous $\aleph_0$-stable theory every type-definable group is definable. The two main ingredients in the proof are: \begin{enumerate} \item Results concerning Morley ranks (i.e., Cantor-Bendixson ranks) from \cite{BenYaacov:TopometricSpacesAndPerturbations}, allowing us to prove the theorem in case the metric is invariant under the group action; and \item Results concerning the existence of translation-invariant definable metrics on type-definable groups and the extension of partial definable metrics to total ones. \end{enumerate

Hyperbolic Ax-Lindemann theorem in the cocompact case

We prove an analogue of the classical Ax-Lindemann theorem in the context of compact Shimura varieties. Our work is motivated by J. Pila's strategy for proving the Andr\'e-Oort conjecture unconditionallyComment: To appear in Duke Mathematical Journa

Ax-Schanuel for Shimura varieties

We prove the Ax-Schanuel theorem for a general (pure) Shimura variety