# Hodge Theory

## Fall Semester 2011

 Instructor: Matt Kerr Office: Cupples I, Room 114 e-mail: matkerr [at] math.wustl.edu Office Hours: Tuesday and Thursday 4-5:30, and by appointment Course Outline: I. Kaehler geometry and sheaf cohomology II. The Hodge decomposition and residue theory III. Variations of Hodge structure and the period mapping IV. Mixed Hodge structures V. Algebraic cycles The course will begin with a review of complex manifolds, holomorphic vector bundles, differential forms, Hermitian and Kaehler metrics, and sheaf cohomology. I'll follow this with a thorough treatment of the Hodge theorem, starting with the general analytic result for elliptic operators, and its consequences in the Kaehler case. For computing Hodge structures of hypersurfaces and how they vary, we'll study the connection to commutative algebra given by Griffiths's residue theorem. This leads naturally on to transversality, period maps, and period domains, as well as applications to algebraic cycles and the Hodge conjecture. We will touch on topics such as mirror symmetry, normal functions, and the conjectures of Bloch and Beilinson, and conclude with a discussion of current research and a wide array of open problems. Prerequisites: Math 5022, 5032, 5042, or permission of the instructor. Class Schedule: Lectures are on Monday, Wednesday and Friday from 11-12 in Cupples I Rm. 113. First class is Wednesday Aug. 31 and last class is Friday Dec. 9 (with holidays Sept. 5, Oct. 14, Nov. 23, and Nov. 25). Midterm Exam: (take-home) due Tuesday Nov. 22 by 5 PM (solutions) Final Exam: no exam; project will substitute for final (see below) Assignments: For the first month or so, I'm going to actually collect weekly homework, since I think having some initial feedback is important. After that, I may collect one or two more but mainly just post problems and some solutions. Problem Set 1 (solutions) Problem Set 2 (solutions) Problem Set 3 (solutions) Problem Set 4 (solutions) Problem Set 5 (not to hand in) Problem Set 6 (not to hand in) More important will be your mini-project. The options will be unveiled in early October and we'll discuss these one-on-one. The idea will be to explore topics and/or do computations not covered (or maybe only mentioned) in the lectures. You'll prepare a short paper and give a 10-minute talk at the end. Lecture Notes: Will be scanned and posted here as I write them. The hope is that this makes taking notes optional. I. Kaehler geometry and sheaf cohomology A. Linear algebra B. Calculus on manifolds C. Complex manifolds D. Almost complex structures E. Kaehler manifolds F. Sheaves and cohomology G. Riemann surfaces and complex tori II. The Hodge decomposition and residue theory A. Sobolev spaces B. Elliptic differential operators C. Harmonic representatives D. Hodge decomposition E. Lefschetz theorems F. Residue theory G. Hodge structures III. Variations of Hodge structure and the period mapping A. Leray spectral sequence B. Gauss-Manin connection C. Polarized variations of HS D. Curvature of period domains IV. Mixed Hodge structures A. Examples and Definitions B. Open and singular varieties C. Extensions of MHS D. Limit MHS and boundary components V. Algebraic cycles A. Abel's theorem B. Normal functions and Lefschetz (1,1) C. The Hodge Conjecture D. Algebraic vs. homological equivalence E. A theorem of Green and Voisin Appendix A: Results from complex algebraic geometry Books: [Note - I have placed a copy of each of the books set apart from the text below on reserve.] Claire Voisin, Hodge theory and complex algebraic geometry (tr. Schneps), v. 1 & 2; Cambridge is the recommended textbook, because it is modern and clearly written and contains many of the topics I'll discuss. Another good book which covers the subject generally but in a very different style is James D. Lewis, A survey of the Hodge conjecture; CRM. For the first part of the course, where we prove the Hodge theorem, classic references are R. O. Wells, Differential analysis on complex manifolds; Springer, Frank Warner, Foundations of differentiable manifolds and Lie groups; Springer [or Scott, Foresman and Co.], and Chapter 0 of Phillip Griffiths and Joseph Harris, Principles of algebraic geometry; Wiley. There is also the book on complex geometry by Huybrechts. Later on, when we study variations of Hodge structure and period mappings, J. Carlson, S. Muller-Stach, and C. Peters, Period mappings and period domains; Cambridge will be useful; and for the parts of the course after that, Chris Peters and Joseph Steenbrink, Mixed Hodge structures; Springer and Lewis's book are both good. There are many other good (but more advanced) books by Demailly, Kato and Usui, Kulikov, Griffiths, Green, Voisin, etc. all from the point of view of transcendental algebraic geometry. Grading Policy: The project, at 60%, is the main component of your grade; midterm and HW both count 20%. Grades for individual items will appear on telesis.