Topics in Algebraic geometry: Hodge Theory
This is the course page for the Fall 2022 Hodge theory course. We will loosely follow Voisin’s books, but with lots of supplemental material to be posted here.
Voisin, Hodge Theory and Complex Algebraic Geometry I
Voisin, Hodge Theory and Complex Algebraic Geometry II
Office Hours: T1-2, BA6172
Tentative Schedule
Week 1: What is Hodge theory? Intro to complex manifolds, Kähler structures, etc.
Week 2: de Rham cohomology, Laplacians, elliptic operators
Week 3: The Hodge and Lefschetz decompositions, and the Hodge index theorem
Week 4: Hodge structures, polarizations, the Hodge-to-de Rham spectral sequence
Week 5: Algebraic de Rham cohomology, comparison, Deligne-Illusie and reduction mod p
Week 6: Some applications — generic vanishing
Week 7: Variations of Hodge structure, period maps
Week 8: Some applications — IVHS, generic Torelli theorems, monodromy
Week 9: Cycle classes
assessment
Students will be asked to write a short note and give a short presentation on a topic of Hodge theory, individually or in pairs. Some possible topics include:
The \(T^1\)-lifting theorem and the Tian-Todorov theorem (unobstructedness of Calabi-Yau manifolds) (lots of sources, e.g. here)
Katz-Oda’s algebraic description of the Gauss-Manin connection
The Ceresa cycle is not always algebraically equivalent to zero (there are now many proofs of this — we could even do more than one)
Bloch’s conjecture (Voisin II, Chapter 11 and Mumford’s Rational equivalence of zero-cycles on surfaces)