Ramification and nearby cycles for \ell-adic sheaves on relative curves
Amphitéâtre Léon Motchane (IHES)
Amphitéâtre Léon Motchane
Le Bois Marie
35, route de Chartres
I will present a new approach for a formula of Deligne and Kato that computes the dimension of the nearby cycle complex of an $\ell$-adic sheaf on a smooth relative curve over a strictly henselian trait of characteristic $p \ne \ell$. Deligne considered the case where the sheaf has no vertical ramification and Kato extended the formula to the general case. My approach is based on ramification theory of Abbes and Saito. It computes the nearby cycle complex in terms of the refined Swan conductor. In fact, I compare Abbes-Saito's refined Swan conductor with Kato's Swan conductor with differential values, which is the key ingredient in Kato's formula; the case of rank one sheaves is due to Abbes and Saito. My approach provides also a new independent proof of Delinge-Kato's formula.