Seminar on Algebraic Geometry and Ramification
- Speaker:Haoyu Hu (Nanjing University)
- Time:2021-03-15 15:00-16:00
- Title:Semi-continuity of conductors and ramification bound of nearby cycles
- Abstract: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.
- Speaker:Hiroki Kato (Universite Paris-Saclay)
- Time:2021-02-25 16:00-17:00
- Title:Etale cohomology of rigid analytic varieties via nearby cycles over general bases
- Abstract:One of the most fundamental results in the study of étale cohomology of rigid analytic varieties is the comparison with the nearby cycle cohomology, which gives a canonical isomorphism between the cohomology of an algebraizable rigid analytic variety and the cohomology of the nearby cycle. I will discuss a generalization of this comparison result to the relative case: For an algebraizable morphism, the compactly supported higher direct image sheaves are identified, up to replacing the target by a blowup, with a generalization of the nearby cycle cohomology, which is given by the theory of nearby cycles over general bases. This result can be used to show the existence of a tubular neighborhood that doesn’t change the cohomology for algebraizable families.
- Speaker:Will Sawin (Columbia university)
- Time:2021-02-25, 10:00-11:00
- Title:Stalks of perverse sheaves in characteristic p
- Abstract:Perverse sheaves are objects that efficiently encapsulate geometric information in multiple areas of algebraic geometry, number theory, representation theory, and topology. A key invariant of perverse sheaves is the characteristic cycle, which can be used to calculate the Euler characteristic or the rank of the vanishing cycles at a particular point. Massey showed that the characteristic cycle can be used to bound the stalk of the perverse sheaf at a particular point.
We generalize Massey's formula from characteristic 0 to characteristic p. This relies on the recent construction of the characteristic cycle in characteristic p. It has multiple applications to number theory over the ring of polynomials in one variable over finite fields, since many natural arithmetic functions in that setting arise from the stalks of perverse sheaves - most famously, automorphic forms.