2025/02/16 by Haoyu Hu, Hu, Haoyu, Jean-Baptiste Teyssier +1
Computer Science · Mathematics · #11S15 #14F08 #14F20 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2502.11060
openalex publication_date 2025/02/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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≤ i ≤ n, there is a polynomial Pi 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 Pi(lcD(L))⋅ r where lcD(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 ℓ-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 ℓ.