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 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.     

Symplectic varieties with invariant Lagrangian subvarieties

Doklady Mathematics -     

Continuous edge chromatic numbers of abelian group actions

We prove that for any generating set S of Z~n,the continuous edge chromatic number χ'_c（G） of the Schreier graph of the Bernoulli shift action G=F(S,2^Z~n) is χ'（G）+1=|S|+1.In particular,for the standard generating set,the continuous edge chromatic number of F(2^Z~n) is 2n+1.     

Abelian group actions on algebraic varieties with one fixed point

Dedicated to the memory of Ed Floyd with admiration and friendship