We consider some classical maps from the theory of abelian varieties and their moduli spaces, and we prove their definability on restricted domains in the o-minimal structure . In particular, we prove that the projective embedding of the moduli space of the principally polarized abelian variety is definable in when restricted to Siegel’s fundamental set . We also prove the definability on appropriate domains of embeddings of families of abelian varieties into projective spaces.
Ya’acov Peterzil. Sergei Starchenko. "Definability of restricted theta functions and families of abelian varieties." Duke Math. J. 162 (4) 731 - 765, 15 March 2013. https://doi.org/10.1215/00127094-2080018