Why certain Tannaka groups attached to abelian varieties are almost   connected
  We show that for a complex abelian variety X a certain Tannaka group G(X) attached to X is a pro-reductive group whose group of connected components is abelian, and hence isomorphic to the etale pro-finite abelian fundamental group of the dual abelian variety of X. 
Rainer Weissauer
math.AG