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Mixed Hodge structures

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Cours doctoral

Title:

Mixed Hodge structures

Topic: Algebraic and analytic complex geometry

Key words: Hodge Theory, deformation and degeneration of complex manifolds, coho- mological tools.

Lecturer: Christophe Mourougane, Université Rennes 1.

Schedule: Thursday 11/02/2021: 10h15-12h15 and 15h-17h Friday : 12/02/2021: 10h15-12h15.

Summary

I will first present the definition and some examples of mixed Hodge structures. I will then dis- play, on the example of degenerations of projective varieties, the existence of weight structures and their compatibility with geometric morphisms. This provides a great deal of information on the singularities of the special fiber. On the way, I will recall cohomological methods like spectral sequences associated to filtration of sheaves.

Some knowledge on analytic or algebraic geometry, with basics on Hodge structures is required.

References

[1] Fouad El Zein and Lê D˜ung Tráng. Mixed Hodge structures. In Hodge theory, volume 49 ofMath. Notes, pages 123–216. Princeton Univ. Press, Princeton, NJ, 2014.

[2] David R. Morrison. The Clemens-Schmid exact sequence and applications. In Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982) Ann. of Math. Stud., vol- mue 106, pages 101–119. Princeton Univ. Press, Princeton, NJ, 1984.

[3] Chris A. M. Peters and Joseph H. M. Steenbrink. Mixed Hodge structures, volume 52 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics[Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Sur- veys in Mathematics]. Springer-Verlag, Berlin, 2008.

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