Semi-continuity of conductors, and ramification bound of nearby cycles

Abstract

Abstract For a constructible étale sheaf on a smooth variety of positive characteristic ramified along an effective divisor, the largest slope in Abbes and Saito’s ramification theory of the sheaf gives a divisor with rational coefficients called the conductor divisor. In this article, we prove decreasing properties of the conductor divisor after pull-backs. The main ingredient behind is the construction of étale sheaves with pure ramifications. As applications, we first prove a lower semi-continuity property for conductors of étale sheaves on relative curves in the equal characteristic case, which supplement Deligne and Laumon’s lower semi-continuity property of Swan conductors ([33]) and is also an ℓ {\ell} -adic analogue of André’s semi-continuity result of Poincaré–Katz ranks for meromorphic connections on complex relative curves. ([6]). Secondly, we give a ramification bound for the nearby cycle complex of an étale sheaf ramified along the special fiber of a regular scheme semi-stable over an equal characteristic henselian trait, which extends a main result in a joint work with Teyssier ([20]) and answers a conjecture of Leal ([35]) in a geometric situation.