B\'ezout's theorem for abelian varieties

Abstract

Let $X$, $Y$ be closed irreducible subvarieties of an absolutely simple abelian variety of dimension $g$ over a field. If $\dim(X) + \dim(Y) \le g$, we prove that the addition morphism $X \times Y \to X + Y$ is semismall. As a consequence, we deduce that if $\dim(X) + \dim(Y) \ge g$, the subvarieties $X$ and $Y$ must meet (B\'ezout's theorem). If we drop the assumption that the abelian variety is absolutely simple, we prove that B\'ezout's theorem still holds if $X$ satisfies a nondegeneracy condition. These results were previously known only in characteristic zero. Our proof of the semismallness statement is based on the theory of perverse sheaves: using results of Kr\"amer and Weissauer, we prove that for perverse sheaves $K$ supported on $X$, and $L$ supported on $Y$, the convolution product $K * L$ is again perverse.