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

TL;DR

This paper establishes bounds on Betti numbers of constructible sheaves in characteristic p>0, depending only on rank, stratification, and wild ramification, with applications to local systems and perverse sheaves.

Contribution

It provides new polynomial bounds for Betti numbers based on logarithmic conductors, extending Deligne's finiteness to singular schemes and algebraic families.

Findings

Betti numbers are bounded by polynomials in the logarithmic conductor and rank.

Betti numbers of direct images are controlled by rank and conductor.

Extended Deligne's finiteness to singular schemes and families.

Abstract

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 $0\leq i \leq n$, 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 $P_i(lc_D(L))\cdot r$ where $lc_D(L)$ 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 $\ell$-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 $\ell$.