Semi-continuity of conductors and ramification bound of nearby cycles
In this talk, we firstly discuss a lower semi-continuity property for conductors of étale sheaves on relative curves in the equal characteristic case, which supplement Deligne and Laumon's lower semi-continuity property of Swan conductors and is also an l-adic analogue of André's semi-continuity result of Poincaré-Katz ranks for meromorphic connections on complex relative curve. After that, we give a ramification bound for the nearby cycle complex of an étale sheaf ramifications along the special fiber of a regular scheme semi-stable over an equal characteristic henselian trait, which extends a main result in a joint work with Teyssier in 2019 and answers a conjecture of Leal in a geometric situation. If time permits, we discuss the common new ingredient behind the two aspects above, which is a decreasing property of the conductor divisor defined in terms of Abbes and Saito’s ramification theory after pull-backs.
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