UMD 2023 Hodge Theory RIT

The Hodge Theory RIT is a reading group for Fall 2023 open to anyone who wants to learn the basics of Hodge Theory as it applies to algebraic geometry. Apart from an introductory lecture given by the RIT organizer Patrick Brosnan, the lectures will (at least mostly) be given by students. The first meeting is scheduled for Friday, September 8 from 3-4 in MTH 1311.

Proposed Schedule

We will meet weekly, but, unfortunately, not all meetings can be at Friday at 3pm (due to occasional conflicts with the UMD Math Colloquium). Here is a proposed list of topics.
  1. Introduction and statement of basic theorems (9/8, 3-4, MTH 1311)
  2. De Rham cohomology and the Hodge theorem I
  3. Hodge theorem II (some indications of proof)
  4. Kahler manifolds and the Hodge decomposition
  5. Category of Hodge structures and mixed Hodge structures
  6. Statement of Deligne's theorems on existence of Mixed Hodge stuctures of complex algebraic varieties
  7. Period domains and classifying spaces of Hodge structures
  8. Smooth, proper families of algebraic varieties and monodromy
  9. Gauss-Manin connection
  10. Variations of Hodge structure
  11. Schmid's sl2 and nilpotent orbit theorems

Notes

References

For items 2-4, the books by Warner and Wells seem like pretty good references. For 5 and 6, Deligne's papers Hodge 2 and Hodge 3 are the most authoritative. For all of the topics, the books of Peters and Steenbrink and Voisin are good general references. And, of course, Griffiths and Harris is an important reference as well.