Math 269Z: Topics in Hodge theory
Science Center 238.
Advancement of the class:
- Week 25-27 January: Complex manifolds, holomorphic vector bundles, Dolbeault cohomology, Hermitian and Kähler metrics, Chern and Levi-Civita connections. Chapters 2-3 of Voisin's book.
- Week 1-3 February: Curvature of connections, L2 metrics, elliptic operators, Kähler identities, harmonic forms and Hodge decomposition. Chapters 5-6 of Voisin's book.
- Week 8-10 February: Lefschetz decomposition, hard Lefschetz, Hodge index theorem, Lefschetz theorem on (1,1) classes. Chapters 4-6-7 of Voisin's book.
- Week 15-17 February: Holomorphic de Rham complex, Hodge-to-de Rham spectral sequence, hypercohomology, GAGA principle. Chapters 7-8 of Voisin's book.
- Week 22-25 February: Mixed Hodge structures, Hodge structure on open varieties, families of complex manifolds, Kodaria-Spencer map. Chapters 8-9 of Voisin's book.
- Week 01-3 March: Local systems, flat vector bundles, local period map, variation of Hodge structures. Chapters 9-10 of Voisin's book.
- Week 8-10 March: Period domains, global period map, cycle class map, Hodge and absolute classes, Hodge loci.
- Week 15-17 March: Spring break.
- Week 22-24 March: Mumford-Tate groups and domains, Hodge loci.
- Week 29-31 March: Definability of arithmetic quotients I.
- Week 5-7 April: Definability of arithmetic quotients II.
- Week 12-15 April: Degenerations of Hodge structures.
- Week 19-21 April: Definability of period maps, end of the proof.
- Week 26 April: Conjectures.
- Meeting times: Tuesday-Thursday 09:00 AM-10:15 AM.
- Room: Science Center 112.
- First meeting: Tuesday, January 25, 2022.
- Office hours & problem review sessions: Monday 1 to 2PM and Wednesday 5 to 6PM, Science Center 238.
This course is an introduction to Hodge theory, which is a powerful framework for studying algebraic varieties and a very active area of research. We will start by introducing Hodge structures (pure and mixed), variations of Hodge structures, their period domains and degenerations of Hodge structures. Our next focus will be the study of Hodge loci. We will prove their algebraicity properties using tools from o-minimal geometry. Other topics may include: fields of definition of Hodge loci, Shimura varieties, and typical and atypical intersections.
For the second part of the class, we will be following the proof of algebraicity of Hodge loci following this article:
- Benjamin Bakker, Bruno Klingler, Jacob Tsimerman, Tame topology of arithmetic quotients and algebraicity of Hodge loci.
Further references include:
- Phillip Griffiths, Topics in transcendental algebraic geometry.
- Mark Green, Philip Griffiths, Matt Kerr, Mumford-Tate groups and domains.
- Claire Voisin, Hodge theory and complex algebraic geometry I & II.
- Philip Griffiths and Joseph Harris, Principles of algebraic geometry.
- Chris Peters and Joseph Steenbrink, Mixed Hodge structures.
More references will be added for the second part of the class.
Good background in complex analysis (Math 113), algebraic geometry (Math 232), differential geometry (Math 136), and algebraic topology (Math 231).
Exams and grading:
There will be weekly homework which will count for 80% of the final grade. No late homework will be accepted (except under special circumstances) and the lowest score will be dropped. Collaborative work on homework is accepted but you must write your own solution as well as the names of the collaborators. There will be a final project which consists of writing a short report on a topic tangentially treated in class and giving a 50 minute presentation about it. Together with participation, the final project will count for the remaining 20% of the grade.