turkmath.org

We will start by introducing basic definition and properties of complex abelian varieties. In particular we recall Poincare Reducibility Theorems for abelian varieties that state that an abelian variety is simple or isogenous to a product of abelian varieties of smaller dimensions. Then, we will improve the statement in the following way. Let Is^g_D be the locus in the moduli space of principally polarised abelian varieties of dimension g that contain an abelian subvariety of at most half dimension and restricted polarisation of type D. Then Is^g_D is an irreducible subvariety in the moduli space and the locus of non-simple abelian varieties is the countable union of all Is^g_D. Moreover, in the similar way to Humbert, we will produce equations for Is^g_D in the Siegel upper half-space.