Seminar on Hodge Theory in Positive Characteristic, Fall 2016

This is the webpage of the student learning seminar on algebraic geometry at Columbia. The predecessor of this seminar is the Remynar, from which the template for this page is partially copied.

For the first half of the semester, we will study Deligne-Illusie's [DI] algebraic proof of degeneration of the Hodge-de Rham spectral sequence in characteristic 0 that proceeds by degeneration to positive characteristic. While everyone is invited to study the original paper, we will instead follow [I], which is a re-exposition (in English) of the original covering the background needed to understand the proofs. We expect [I1] to be readable with knowledge of algebraic geometry at the level of Hartshorne, along with fundamental notions from homological algebra, e.g. abelian categories, (co-)limits, spectral sequences, etc.

In the second half of the semester, we will study Deligne's "Weil II" paper [D], beginning with the lecture notes of Katz [K]. We will review étale cohomology only briefly.

References:

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Organizers: Shizhang Li, Carl Lian.

Time and location: Tuesday 16:30-18:00 in Math 507. **except Tuesday 25 October, Math 622.

Date Speaker Title Ref.
6 Sep Shizhang Li Introduction; Differentials and the de Rham Complex [I] Sec. 0-1
13 Sep Carl Lian Smoothness and Liftings [I] Sec. 2
20 Sep Dmitrii Pirozhkov Frobenius and Cartier Isomorphism [I] Sec. 3
27 Sep Monica Marinescu Derived Categories and Spectral Sequences [I] Sec. 4
4 Oct Dan Gulotta Decomposition, Degeneration, and Vanishing Theorems [I] Sec. 5
11 Oct Yogesh More From Characteristic p>0 to Characteristic 0 [I] Sec. 6
18 Oct Remy van Dobben de Bruyn Counterexamples in Characteristic p [L], [S]
25 Oct** Qixiao Ma Review of Étale Cohomology Remynar
1 Nov Qixiao Ma Introduction to Weil II; Reductions [K] Sec. 1
8 Nov No talk: Univ. holiday
15 Nov Dmitrii Pirozhkov Reduction to the Purity Theorem [K] Sec. 2
22 Nov Dmitrii Pirozhkov Reduction to the Monodromy Theorem [K] Sec. 3
29 Nov Raymond Cheng Proof of the Monodromy Theorem [K] Sec. 4
6 Dec Dan Gulotta Proof of the Mondromy Theorem, cont.; Applications [K] Sec. 4
13 Dec Dingxin Zhang Decomposition Theorems [BBD]