Selected topics in Algebraic geometry: An introduction to mixed Hodge modules

About the course:

This is a course I taught in Bonn in the Winter semester of 2022. 

The aim of this course is to give a motivated introduction to the theory of mixed Hodge modules. We will start with the classical Hodge theory,  then a quick tour to Griffiths’ theory of variation of Hodge structures and finally land into M. Saito’s theory of mixed Hodge modules (with a focus on the pure case). Along the way, we will show how D-modules naturally arise and give a quick introduction. At the end of the course, some applications of Hodge modules to algebraic geometry will be given.

Papers discussed:

Lecture 3: Katz and Oda, On the differentiation of De Rham cohomology classes with respect to parameters,  J. Math. Kyoto Univ. 8(2): 199-213 (1968).

Lecture 4: Steven Zucker, Hodge Theory with Degenerating Coefficients: L2 Cohomology in the Poincare Metric, Annals (1979);

Wilfried Schmid, Variation of Hodge Structures: The Singularities of the Period Mapping, Inventiones (1973);

Carlos Simpson, Higgs bundles and local systems, Publication IHES (1992). [For Deligne's theorem, see section 1].

Lecture 5: Zucker's paper and section 6 of the MHM project.

Lecture 6: Zucker's paper and section 6 of the MHM project.

Lecture 7: Christian Schnell, section 8-12 of An overview of Morihiko Saito’s theory of mixed Hodge modules. (Note that we use left D modules in class and Schnell uses right D modules, which results some notation changes).

Lecture 8: Christian Schnell, section 16-17 of An overview of Morihiko Saito’s theory of mixed Hodge modules.

Prerequisites:

I will try to give a self-contained exposition, but it will be better if you have some backgrounds on

- De Rham cohomology of differential manifolds.

- Sheaf theory and cohomology on complex manifolds or algebraic varieties (Chapter 0-1 of Griffiths-Harris or Chapter 2-3 of Hartshorne).

References:

- Pierre Deligne, Théorie de Hodge: II, Publications mathématiques de l’I.H.É.S., tome 40 (1971), p. 5-57.

- Morihiko Saito, Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988), no. 6, 849–995 (1989).

- Morihiko Saito, Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26 (1990), no. 2, 221–333.

- Morihiko Saito, A young person's guide to mixed Hodge modules.

- Claude Sabbah and Christian Schnell, the Complex Mixed Hodge Module project.

- Christian Schnell, notes on D-modules.

- Christian Schnell, An overview of Morihiko Saito’s theory of mixed Hodge modules.

- Claire Voisin, Hodge theory and complex algebraic geometry, first volume.

Applications of Hodge modules

-Stefan Kebekus and Christian Schnell, Extending holomorphic forms from the regular locus of a complex space to a resolution of singularities, J. Amer. Math. Soc. 34 (2021), no. 2, 315–368.

- Mihnea Popa and Christian Schnell, Kodaira dimension and zeros of holomorphic one-forms, Ann. of Math. 179 (2014), no. 3, 1109–1120.

- Mihnea Popa and Christian Schnell, Viehweg's hyperbolicity conjecture for families with maximal variation, Invent. Math. 208 (2017), no. 3, 677–713.

- Mihnea Popa: D-modules in birational geometry, ICM 2018.