Semi-continuity of conductors, and ramification bound of nearby cycles (Q6060808)

For a constructible étale sheaf on a smooth variety, its largest slope in Abbes-Saito's ramification theory gives a divisor with rational coefficients called the conductor divisor. This divisor can sharply bound the conductor of the sheaf after restricting to curves on the variety. In the under-review paper, the author proves that the conductor divisor decreases after pull-backs. One of the main ingredients is the construction of étale sheaves with pure ramifications. As applications, the author proves a lower semi-continuity property for conductors of étale sheaf on relative curves in equi-characteristic cases, and gives a ramification bound for the nearby cycle of a semistable regular scheme over an equi-characteristic Henselian trait, which confirms a conjecture of \textit{I. Leal} [J. Reine Angew. Math. 749, 295--304 (2019; Zbl 1412.14013)] in a geometric situation.