GV‐subschemes and their embeddings in principally polarized abelian varieties

We prove that any embedding of a ‐subscheme in a principally polarized abelian variety does not factor through any nontrivial isogeny. As an application, we present a new proof of a theorem of Clemens–Griffiths identifying the intermediate Jacobian of a smooth cubic threefold to the Albanese variety of its Fano surface of lines.     

Families of irreducible principally polarized abelian varieties isomorphic to a product of elliptic curves

For each greater than or equal to two, we give a family of
-dimensional, irreducible principally polarized abelian varieties isomorphic to a product of elliptic curves. This family corresponds to the modular curve .

     

Abelian varieties with finite abelian group action

An automorphism of an abelian variety induces a decomposition of the variety up to isogeny. There are two such results, namely the isotypical decomposition and Roan’s decomposition theorem. We show that they are essentially the same. Moreover, we generalize in a sense this result to abelian varieties with action of an arbitrary finite abelian group. An early version of this article was inadvertently published before all the revisions had been completed and then retracted [https://doi.org/10.1007/s00013-018-1244-3]. This article is the final peer reviewed version.

     

A Fourier‐Mukai approach to the enumerative geometry of principally polarized abelian surfaces

We study twisted ideal sheaves of small length on an irreducible principally polarized abelian surface $({\mathbb T},\ell )We study twisted ideal sheaves of small length on an irreducible principally polarized abelian surface $({\mathbb T},\ell )$. Using Fourier‐Mukai techniques we associate certain jumping schemes to such sheaves and completely classify such loci. We give examples of applications to the enumerative geometry of ${\mathbb T}$ and show that no smooth genus 5 curve on such a surface can contain a $g^1_3$. We also describe explicitly the singular divisors in the linear system |2?|.     

A homological criterion for reducibility of analytic spaces,with application to characterizing the theta divisor of a product of two general principally polarized abelian varieties

A closed subset of pure codimension one in an analytic space, consisting entirely of local normal crossings double points, is called an ordinary rank two double locus. We give a topologically computable upper bound on the number of connected components of an ordinary rank two double locus in a given space. This leads to criteria for global reducibility of spaces. The first is that a simply connected space with a non empty ordinary rank two double locus is always reducible. A finer criterion implies that a principally polarized abelian variety A is isomorphic to a product of two positive dimensional principally polarized abelian varieties, each with smooth theta divisor, if and only if the theta divisor of A contains a non empty ordinary rank two double locus. Analogous reducibility results apply to certain complete intersection varieties, and to divisors on such varieties.