Hodge II Reading Group


Due to scheduling issues, before each session we will meet on the 2nd floor of Eckhart and enter the first available room (if there are none available, we will move up to the 3rd floor). The meetings will usually take place Thursday afternoons, scheduled around the REU Number Theory course.

The goal of the reading group is, roughly, to understand Hodge II, and perhaps more importantly to get a handle on mixed Hodge Theory in some applications and examples.

Schedule:
Week 1 (Friday June 21st, 2pm, Eckhart 206) - Dan, Classical Hodge Theory
Week 2 (Thursday June 27th, 1pm, Eckhart 2nd floor east) - Sean, Introduction to Mixed Hodge Theory and Hodge II
Week 3 (Friday July 5, 3:00-4:30pm, Eckhart 2nd floor east) - Sean, Introduction to Mixed Hodge Theory and Hodge II, continued. Notes
Week 4 (Thursday July 11th, 1:00-2:30pm, Eckhart 2nd floor east) - Jingren, Section I-II (Filtrations, Mixed Hodge Structures)
Week 5-(1) (Tuesday July 16th, 1:00-2:30pm, Eckhart 2nd floor east) - Jingren, Existence of mixed Hodge structure
Week 5-(2) (Thursday July 18th, 1:00-2:30pm, Eckhart 2nd floor east) - Jonathan, Section III (Log complex)
Week 6 (Tuesday July 22nd, 3:15pm-4:45pm) - Simion, The Decomposition Theorem
Week 7 - No talk
Week 8 (Tuesday August 6th, 1:30-3:30pm) - Fan, Katz - Nilpotent connections and the monodromy theorem
Week 9 - No talk
Week 10 (Friday August 23rd, 1:00-2:30pm) - Sean, Theorem of the Fixed Part


Here are some ideas for future talks (please e-mail me if you want to volunteer for one):
*Hodge theory of singular varieties (Examples, results of Hodge III)
*Differentials on hypersurface complements (see note in Resources after the Griffiths paper)
*Hodge I (the Hodge - l-adic dictionary)
*Harmonic analysis and extensions of mixed hodge structures (e.g. reproducing Greens functions, polylogarithms)
*Griffiths transversality, VHS
*More applications!

Resources:
Deligne -
* Hodge I, II, and III
* Théorème de Lefschetz et critères de dégénérescence de suites spectrales
Extremely roughly, Hodge I is Deligne's ICM talk and is mostly motivational. Hodge II treats the case of open varieties, and Hodge III adds in singular varieties. We will be focusing on Hodge II in the seminar. The last paper on degeneration of spectral sequences is a precursor to Hodge II and is helpful in understanding the first section of Hodge II, as well as some of the applications.

Griffiths -
On the Periods of Certain Rational Integrals: I
Section 8 of this paper contains a nice application which we might work towards understanding in relation to Hodge II (c.f. Section 9.2 of Hodge III).

For classical Hodge theory in the compact Kahler manifold case, Griffiths and Harris (Chapter 0, plus Chapter 3 for the Frohlicher spectral sequence) and Voisin are reasonable. Voisin also has nice coverage of variations of Hodge structures.

De Cataldo - The Decomposition Theorem, Perverse Sheaves, and the Topology of Algebraic Maps