Cohomological boundedness for flat bundles on surfaces and applications

TL;DR

This paper establishes universal bounds for the cohomology and degrees of flat bundles with bounded rank and irregularity on surfaces, introduces a new operation on b-divisors, and provides applications in algebraic geometry.

Contribution

It proves the existence of bounds for De Rham cohomology and turning loci of flat bundles, and introduces the partial discrepancy operation on b-divisors.

Findings

Universal bound for De Rham cohomology dimensions on surfaces.

Lefschetz recognition principle for flat bundles in any dimension.

Closed formula for characteristic cycles using partial discrepancy.

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.