Bézout's theorem for abelian varieties (Q1312857)

The aim of this short article is to give an elementary algebraic proof of the following result, originally proved by \textit{Barth} for simple complex tori: if \(X\) and \(Y\) are irreducible closed subvarieties of a simple abelian variety \(A\) defined over an algebraically closed field of characteristic 0, such that \(\dim (X) + \dim(Y) \geq \dim (A)\), then \(X \cap Y \neq \emptyset\). This is obtained by considering the difference morphism \(n:X \times Y \to A\): if it is not surjective, the general fiber \(F\) is positive-dimensional and the tangent spaces along its smooth part are all contained (after translation) into a proper subspace of \(T_ 0A\). But then \(F\) cannot generate \(A\), contradiction. Hence \(n\) is surjective, and 0 is in its image. The case where the base field has positive characteristic remains open.