Vol. 15, No. 6, 2021

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Wild ramification, nearby cycle complexes, and characteristic cycles of $\ell$-adic sheaves

Hiroki Kato

Vol. 15 (2021), No. 6, 1523–1564
Abstract

We prove a purely local form of a result of Saito and Yatagawa. They proved that the characteristic cycle of a constructible étale sheaf is determined by wild ramification of the sheaf along the boundary of a compactification. But they had to consider ramification at all the points of the compactification. We give a pointwise result, that is, we prove that the characteristic cycle of a constructible étale sheaf around a point is determined by wild ramification at that point. The key ingredient is to prove that wild ramification of the stalk of the nearby cycle complex of a constructible étale sheaf at a point is determined by wild ramification at that point.

Keywords
wild ramification, nearby cycle complexes, characteristic cycles
Mathematical Subject Classification 2010
Primary: 14F20
Secondary: 14C25, 14G20
Milestones
Received: 27 February 2020
Revised: 9 October 2020
Accepted: 12 December 2020
Published: 16 October 2021
Authors
Hiroki Kato
Laboratoire de Mathématiques d’Orsay
Université Paris-saclay
Orsay
France