Title: On the sum map for subvarieties of simple abelian varieties

Abstract: Let $X$ and $Y$ be subvarieties of a simple abelian variety $A$ defined over an algebraically closed field and let $Z=X+Y\subseteq A$. If $\dim(X)+\dim(Y)\le \dim( A)$, we prove that the addition map $X\times Y \to Z$ is semismall. In particular, $Z$ has the expected dimension $\dim(X)+\dim(Y)$. Over the field of complex numbers, the latter statement was proved in 1982 by Barth, and Prasad gave in 1993 a very simple proof in characteristic zero. Surprisingly enough, it did not seem to be known in general in positive characteristics. In the above situation, we prove the semismallness of the addition map in all characteristics using perverse sheaves. This is joint work with Ben Moonen.