Introduction to Hodge theory

Lecturers: Tiago J. Fonseca, Nils Matthes

Time and location: Trinity Term 2021, Monday 2-3, Thursday 10-11, online.

Course description: In a nutshell, Hodge theory is the study of cohomology groups of complex manifolds using harmonic differential forms. With its many extensions and ramifications, it has become a very important tool in algebraic/complex geometry, and has interrelations with other subjects such as number theory and string theory.

The goal of this course is to give a first introduction to Hodge theory. The first half of this lecture will be devoted to presenting some basic foundational results of the theory, such as the Hodge decomposition and the hard Lefschetz theorem, which are very important in algebraic geometry. In the second half, several further topics will be discussed. Possible topics include: (mixed) Hodge structures; Hodge theory and algebraic cycles; families of Hodge structures and period maps

Prerequisites: Knowledge of differentiable manifolds and complex analysis in one variable will be assumed. Some knowledge of algebraic geometry will be helpful especially in the second half of the lecture but is not a strict requirement.

Broadening: For those Oxford students who would like to do a broadening: please do get in touch with Tiago and myself via email.

Literature: The main reference for this course will be:

• C. Voisin: Hodge theory and complex algebraic geometry. I,II. Cambridge Studies in Advanced Mathematics, 76. Cambridge University Press, Cambridge, 2002. x+322 pp.
Other references include:
• J. Bertin, J.-P. Demailly, L. Illusie, C. Peters: Introduction à la théorie de Hodge. Panoramas et Synthèses [Panoramas and Syntheses], 3. Société Mathématique de France, Paris, 1996. vi+273 pp.

• P. Griffiths, J. Harris: Principles of algebraic geometry. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York, 1978. xii+813 pp.

• C.A.M. Peters, J.H.M. Steenbrink: Mixed Hodge structures. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 52. Springer-Verlag, Berlin, 2008. xiv+470 pp.