Characteristic cycle and wild ramification for nearby cycles of étale sheaves
Let \(S\) be a henselian trait with a perfect residue field of characteristic \(p>0\). Let \(\eta\) and \(\tilde{\eta}\) be respectively the generic and geometric generic point of \(S\), \(G\) the Galois group of \(\tilde{\eta}\) over \(\eta\), \(X\) a scheme and \(f:X\rightarrow S\) a morphism of finite type. Let \(\ell \neq p\) be a prime number, \(\Lambda\) a finite field of characteristic \(\ell\) and \(\mathcal{K}\) a bounded below complex of étale sheaves of \(\Lambda\)-modules on \(X\). Introduced by Grothendieck in the 1960s, the nearby cycles complex \(R\Psi (\mathcal{K},f)\) and the vanishing cycles complex \(R\Phi (\mathcal{K},f)\) of \(\mathcal{K}\) relative to \(f\) are complexes on the geometric special fiber of \(f\) endowed with a \(G\)-action, called the monodromy action. The authors give a bound for the wild ramification of the monodromy action on the nearby cycles complex of a locally constant étale 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 a conjecture of Leal for smooth morphisms in equal characteristic. The ramification along vertical divisors of étale sheaves on relative curves and abelian schemes over a trait is also studied.
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