Remarks on the nonvanishing of cohomology groups for perverse sheaves on   abelian varieties
  It is shown that for a perverse sheaf $K$ on an abelian variety $X$ the integers $i$ for which the cohomology $H^i(X,K)$ does not vanish define an interval in the number line (under certain conditions on the field of definition of $K$) 
Rainer Weissauer
math.AG