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

Haoyu Hu, Jean-Baptiste Teyssier

TL;DR

This work develops a comprehensive, ℓ-independent framework to bound Betti numbers and cohomology of étale sheaves in positive characteristic. It introduces a unified approach combining Beilinson–Saito singular support theory with Abbes–Saito ramification filtrations, proving polynomial bounds in terms of rank and logarithmic conductors that remain uniform in families. The results extend Deligne’s finiteness phenomena beyond curves and smooth bases, provide bounds for inverse and higher direct images, and establish a relative Hermite–Minkowski-type finiteness for perverse sheaves, with refinements via a Langlands-based bypass of purity. The techniques leverage conductor-control by coherent sheaves, devissage along stratifications, and Lefschetz-pencil arguments, yielding explicit, optimal-degree polynomials in the affine case and broad, ℓ-robust bounds in general. Collectively, the paper advances a robust, geometric-microlocal method to control cohomology in families of schemes over finite fields and their singular degenerations, with broad implications for characteristic cycles and wild ramification theory.

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 $\ell$.

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

TL;DR

This work develops a comprehensive, ℓ-independent framework to bound Betti numbers and cohomology of étale sheaves in positive characteristic. It introduces a unified approach combining Beilinson–Saito singular support theory with Abbes–Saito ramification filtrations, proving polynomial bounds in terms of rank and logarithmic conductors that remain uniform in families. The results extend Deligne’s finiteness phenomena beyond curves and smooth bases, provide bounds for inverse and higher direct images, and establish a relative Hermite–Minkowski-type finiteness for perverse sheaves, with refinements via a Langlands-based bypass of purity. The techniques leverage conductor-control by coherent sheaves, devissage along stratifications, and Lefschetz-pencil arguments, yielding explicit, optimal-degree polynomials in the affine case and broad, ℓ-robust bounds in general. Collectively, the paper advances a robust, geometric-microlocal method to control cohomology in families of schemes over finite fields and their singular degenerations, with broad implications for characteristic cycles and wild ramification theory.

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 , there is a polynomial of degree such that the i-th Betti number of any rank r local system L on X-D is smaller than where 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 -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 .

Paper Structure

This paper contains 21 sections, 74 theorems, 216 equations.

Table of Contents

  1. Introduction
  2. Conductors of étale sheaves
  3. Ramification filtrations
  4. Conductor divisors
  5. Singular support and characteristic cycle of étale sheaves
  6. The singular support
  7. The characteristic cycle
  8. Characteristic cycle and cohomology
  9. Ramification of nearby cycles
  10. A conductor estimate
  11. Conductor estimate
  12. Estimates for Betti numbers on affine spaces
  13. Proof of \ref{['maintheoremAn']}
  14. Estimates for Betti numbers of étale sheaves
  15. Bounding the ramification with coherent sheaves
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $X$ be a proper smooth scheme of finite type of dimension $n$ over an algebraically closed field $k$ of characteristic $p>0$. Let $D$ be a reduced effective Cartier divisor of $X$ and put $U:=X-D$. Then, there exists $P\in \mathds{N}[x]^{n+1}$ with $P_i$ of degree $i$ for $i=0,\dots, n$ such tha

Theorems & Definitions (149)

  • Theorem 1.1: \ref{['general_boundedness_Betti_Xsmooth']}
  • Definition 1.2
  • Theorem 1.3: \ref{['maintheoremAn']}
  • Definition 1.4
  • Example 1.5
  • Theorem 1.6: \ref{['general_boundedness_Betti']}
  • Remark 1.7
  • Theorem 1.8: \ref{['Betti_inverse_image_lc']}
  • Theorem 1.9: \ref{['Coho_direct_image_lc']}
  • Remark 1.10
  • ...and 139 more