Characteristic cycle and wild ramification for nearby cycles of etale sheaves

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 [24] 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.