Quantitative sheaf theory

TL;DR

This paper introduces a new measure of complexity for complexes of sheaves on algebraic varieties and proves that the fundamental six operations preserve this complexity in a controlled manner, with applications to exponential sums.

Contribution

It defines a notion of complexity for sheaves and demonstrates that the six operations are continuous with respect to this complexity, providing uniform bounds across characteristics.

Findings

Complexity bounds for sheaf operations are established.

The complexity measure controls Betti number sums uniformly.

Applications include horizontal equidistribution results for exponential sums.

Abstract

We introduce a notion of complexity of a complex of ell-adic sheaves on a quasi-projective variety and prove that the six operations are "continuous", in the sense that the complexity of the output sheaves is bounded solely in terms of the complexity of the input sheaves. A key feature of complexity is that it provides bounds for the sum of Betti numbers that, in many interesting cases, can be made uniform in the characteristic of the base field. As an illustration, we discuss a few simple applications to horizontal equidistribution results for exponential sums over finite fields.