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The goal of this article is to prove the comparison theorem between algebraic and topological nearby cycles of a morphism without slopes. We prove in particular that for a family of holomorphic functions without slopes, if we iterate comparison isomorphisms for nearby cycles of each function the result is independent of the order of iteration.
We introduce a novel approach to Bertini irreducibility theorems over an arbitrary field, based on random hyperplane slicing over a finite field. Extending a result of Benoist, we prove that for a morphism $phi colon X to mathbb{P}^n$ such that $X$ i
We prove a monomialization theorem for mappings in general classes of infinitely differentiable functions that are called quasianalytic. Examples include Denjoy-Carleman classes (of interest in real analysis), the class of infinitely differentiable f
For a proper semistable curve $X$ over a DVR of mixed characteristics we reprove the invariant cycles theorem with trivial coefficients (see Chiarellotto, 1999) i.e. that the group of elements annihilated by the monodromy operator on the first de Rha
We study restriction of logarithmic Higgs bundles to the boundary divisor and we construct the corresponding nearby-cycles functor in positive characteristic. As applications we prove some strong semipositivity theorems for analogs of complex polariz
For a constructible etale sheaf on a smooth variety of positive characteristic ramified along an effective divisor, the largest slope in Abbes and Saitos ramification theory of the sheaf gives a divisor with rational coefficients called the conductor