Estimates for Betti numbers and relative Hermite-Minkowski theorem for perverse sheaves

We prove estimates for the Betti numbers of constructible sheaves in characteristic p>0 depending only on their rank, stratification and wild ramification. In particular, given a smooth proper variety of dimension n over an algebraically closed field and a divisor D of X, for every $0\le i\le n$, there is a polynomial $P_i$ of degree $\max \{i,2n-i\}$ such that the i-th Betti number of any rank r local system L on X-D is smaller than $P_i(l{c}_D(L))\cdot r$ where $l{c}_D(L)$ is the highest logarithmic conductor of L at the generic points of D. As application, we show that the Betti numbers of the inverse and higher direct images of a local system are controlled by the rank and the highest logarithmic conductor. We also reprove Deligne’s finiteness for simple $\mathrm{\ell }$-adic local systems with bounded rank and ramification on a smooth variety over a finite field and extend it in two different directions. In particular, perverse sheaves over arbitrary singular schemes are allowed and the bounds we obtain are uniform in algebraic families and do not depend on $\mathrm{\ell }$.