Quantitative sheaf theory

Sawin, Will; Forey, A.; Fresán, J.; Kowalski, E.

doi:10.1090/jams/1008

Quantitative sheaf theory
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by Will Sawin, A. Forey, J. Fresán and E. Kowalski

J. Amer. Math. Soc. 36 (2023), 653-726

DOI: https://doi.org/10.1090/jams/1008

Published electronically: August 17, 2022

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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.

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