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:

- [BBD] Beilinson, Bernstein, Deligne, "Faisceaux pervers"
- [D] Deligne, "La conjecture de Weil: II"
- [DI] Deligne-Illusie, "Relèvements modulo p^2 et decomposition du complexe de de Rham"
- [I1] Illusie, "Frobenius and Hodge degeneration" (available as pp. 108-157 in this document.)
- [K] Katz, "L-functions and monodromy: four lectures on Weil II"
- [L] Lang, "Examples of liftings of surfaces and a problem in de Rham cohomology"
- [S] Serre, "Sur la topologie des variétés algébriques sur caractéristique p" ("Mexico paper," pp. 36-65 here.)

Please e-mail the organizers if you'd like to receive notifications.

**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 |
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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] |