Vanishing theorems for constructible sheaves on abelian varieties over finite fields

Let (kappa ) be a field, finitely generated over its prime field, and let k denote an algebraically closed field containing (kappa ). For a perverse (overline{mathbb {Q}}_ell )-adic sheaf (K_0) on an abelian variety (X_0) over (kappa ), let K and X denote the base field extensions of (K_0) and (X_0) to k. Then, the aim of this note is to show that the Euler–Poincare characteristic of the perverse sheaf K on X is a non-negative integer, i.e. (chi (X,K)=sum _nu (-1)^nu dim _{overline{mathbb {Q}}_ell }(H^nu (X,K))ge 0). This generalizes the result of Franecki and Kapranov [9] for fields of characteristic zero. Furthermore we show that (chi (X,K)=0) implies K to be translation invariant. This result allows to considerably simplify the proof of the generic vanishing theorems for constructible sheaves on complex abelian varieties of [11]. Furthermore it extends these vanishing theorems to constructible sheaves on abelian varieties over finite fields.