Base number theorem for abelian varieties. An Infinitesima approach.

we gives a proof of the following : Every subvariety of the locus of g-dimensional complex Jacobians whose Néron-Severi group has rank bigger than one is of codimension ≥(g−1). The same statement holds with "Jacobians'' replaced by "hyperelliptic Jacobians'', "Prym varieties'' or "polarized abelian varieties''.