Characteristic cycle and wild ramification for nearby cycles of \'etale sheaves

Abstract

In this article, we give a bound for the wild ramification of the monodromy action on the nearby cycles complex of a locally constant \'etale sheaf on the generic fiber of a smooth scheme over an equal characteristic trait in terms of Abbes and Saito's logarithmic ramification filtration. This provides a positive answer to the main conjecture in Isabel Leal's article "On the ramification of \'etale cohomology groups" for smooth morphisms in equal characteristic. We also study the ramification along vertical divisors of \'etale sheaves on relative curves and abelian schemes over a trait.