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Cohomological boundedness for flat bundles on surfaces and applications

Haoyu Hu, Jean-Baptiste Teyssier

TL;DR

The paper proves unconditional cohomological boundedness for flat bundles of bounded rank and irregularity on smooth projective surfaces by developing a D-module framework with irregularity $b$-divisors and a partial discrepancy operator. It introduces the moduli framework $ ext{MIC}_r(X,D,R)$ and establishes a Lefschetz recognition principle linking high-dimensional data to curve sections; it also derives a Grothendieck–Ogg–Shafarevich-type formula for the De Rham Euler characteristics via a formula for the characteristic cycle on surfaces. A key technical innovation is the partial discrepancy for nef Cartier $b$-divisors, enabling explicit control of turning points and cohomology. The results yield universal bounds on the turning locus and De Rham cohomology, with consequences for moduli of flat bundles and a Tannakian Lefschetz theorem for differential Galois groups, thereby connecting microlocal geometry, irregularity theory, and cohomological finiteness in a concrete, dimension-two setting.

Abstract

This paper explores the cohomological consequences of the existence of moduli spaces for flat bundles with bounded rank and irregularity at infinity and gives unconditional proofs. Namely, we prove the existence of a universal bound for the dimension of De Rham cohomology of flat bundles with bounded rank and irregularity on surfaces. In any dimension, we prove a Lefschetz recognition principle stating the existence of hyperplane sections distinguishing flat bundles with bounded rank and irregularity after restriction. We obtain in any dimension a universal bound for the degrees of the turning loci of flat bundles with bounded rank and irregularity. Along the way, we introduce a new operation on the group of b-divisors on a smooth surface (the partial discrepancy) and prove a closed formula for the characteristic cycles of flat bundles on surfaces in terms of the partial discrepancy of the irregularity b-divisor attached to any flat bundle by Kedlaya.

Cohomological boundedness for flat bundles on surfaces and applications

TL;DR

The paper proves unconditional cohomological boundedness for flat bundles of bounded rank and irregularity on smooth projective surfaces by developing a D-module framework with irregularity -divisors and a partial discrepancy operator. It introduces the moduli framework and establishes a Lefschetz recognition principle linking high-dimensional data to curve sections; it also derives a Grothendieck–Ogg–Shafarevich-type formula for the De Rham Euler characteristics via a formula for the characteristic cycle on surfaces. A key technical innovation is the partial discrepancy for nef Cartier -divisors, enabling explicit control of turning points and cohomology. The results yield universal bounds on the turning locus and De Rham cohomology, with consequences for moduli of flat bundles and a Tannakian Lefschetz theorem for differential Galois groups, thereby connecting microlocal geometry, irregularity theory, and cohomological finiteness in a concrete, dimension-two setting.

Abstract

This paper explores the cohomological consequences of the existence of moduli spaces for flat bundles with bounded rank and irregularity at infinity and gives unconditional proofs. Namely, we prove the existence of a universal bound for the dimension of De Rham cohomology of flat bundles with bounded rank and irregularity on surfaces. In any dimension, we prove a Lefschetz recognition principle stating the existence of hyperplane sections distinguishing flat bundles with bounded rank and irregularity after restriction. We obtain in any dimension a universal bound for the degrees of the turning loci of flat bundles with bounded rank and irregularity. Along the way, we introduce a new operation on the group of b-divisors on a smooth surface (the partial discrepancy) and prove a closed formula for the characteristic cycles of flat bundles on surfaces in terms of the partial discrepancy of the irregularity b-divisor attached to any flat bundle by Kedlaya.
Paper Structure (47 sections, 107 theorems, 163 equations)

This paper contains 47 sections, 107 theorems, 163 equations.

Table of Contents

  1. Geometric and $\mathcal{D}$-module preparations
  2. Base field
  3. Pair of varieties
  4. Transversality
  5. Universal hyperplane
  6. Characteristic cycle for coherent $\mathcal{D}$-modules
  7. De Rham cohomology
  8. The Solution and the Irregularity complexes
  9. Characteristic cycle and the De Rham and solution complexes
  10. Characteristic cycle and functorialities
  11. Meromorphic flat connections
  12. Good formal structure for connections
  13. Irregularity number
  14. $b$-divisors
  15. The Irregularity $b$-divisor
  16. ...and 32 more sections

Key Result

Theorem 1

Let $X$ be a smooth projective surface over $k$. Let $D$ be a normal crossing divisor of $X$. Then, there exists a quadratic polynomial $C : \mathop{\mathrm{Div}}\nolimits(X,D)\oplus \mathds{Z}\to\mathds{Z}$ affine in the last variable such that for every effective divisor $R$ of $X$ supported on $D

Theorems & Definitions (224)

  • Theorem 1
  • Definition 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Definition 1.1
  • Lemma 1.2
  • proof
  • ...and 214 more