Subvarieties of Abelian Varieties

Let (A; ) be a principally polarized abelian variety (ppav) of dimension g over the eld C of complex numbers. This means that is an ample divisor on A, well-determined up to translation, with h(A; ) := dimH(A; ) = 1. Let [ ] 2 H(A;Z) be the cohomology class of the theta divisor . Then the cohomology class [ ] e is divisible by (g e)!. The class [ ] g e (g e)! is not divisible and it is called the dimension e minimal cohomology class in (A; ). This class is positive in the sense that some multiple of it can be represented by an algebraic subvariety (for instance [ ] e is the class of a complete intersection of g e general translates of ) and, furthermore, any subvariety whose class is a multiple of [ ] g e (g e)! is nondegenerate, i.e., generates A as a group. We are interested in the representability of multiples of the minimal classes by algebraic subvarieties of A. We begin by discussing two special cases.