Reconstructing Abelian Varieties via Model Theory

In 2012, Zilber used model-theoretic techniques to show that a curve of high genus over an algebraically closed field is determined by its Jacobian (viewed only as an abstract group with a distinguished subset for an image of the curve). In this paper, we consider an analogous problem for arbitrary (semi)abelian varieties $A$ over algebraically closed fields $K$ with a distinguished subvariety $V$. Our main result characterizes when the data $(A(K),+,V(K))$ (as a group with distinguished subset) determines the pair $(A,V)$ in the strongest reasonable sense. As it turns out, the situation is best understood by developing a theory of factorizations for such pairs $(A,V)$. In the final sections of the paper, we develop such a theory and prove unique factorization theorems (one for abelian varieties and a weaker one for semi-abelian varieties). In this language, the main theorem mentioned above (in the abelian case) says that the pair $(A,V)$ is determined by the data $(A(K),+,V(K))$ precisely when $(A,V)$ is simple and $0<\dim(V)<\dim(A)$.