Research Seminar on Algebraic Topology, Summer Term 2024

Julian Holstein, Birgit Richter

In the summer term 2024 we will meet roughly every second Thursday from 14:15h to 17:45 in H2 (Geomatikum) or online. The precise dates are: 4.4., 18.4., 25.4., 16.5., 6.6., 20.6., and 4.7.2024.

The topic for our working seminar is Hodge Theory and the plan for that is as follows:

1. Define Hochschild and cyclic homology of associative algebras and the module of Kähler differentials and differential forms of a commutative algebra. What are these differentials for a polynomial algebra? Describe the antisymmetrization map [Loday 1.1,2.1,1.3 with emphasis on 1.3.10-1.3.12].
2. For smooth algebras the antisymmetrization map induces an isomorphism. That's the famous Hochschild-Kostant-Rosenberg (HKR) Theorem. Tell us what smooth algebras are and prove the theorem. This also gives a description of cyclic homology for smooth algebras in characteristic zero [Loday, 3.4].
3. For any commutative algebra in characteristic zero whose underlying module is flat, the Hochschild homology groups split into a direct sum. This decomposition is the so-called Hodge decomposition of Hochschild homology. One proof uses the fundamental spectral sequence [Loday 3.5].
4. Define Hodge structure on smooth projective varieties. Consider the smooth de Rham complex and the holomorphic de Rham complex as a resolutions of the constant sheaf. Recall de Rham cohomology [Voisin 4.37]. Truncate the holomorphic de Rham complex to obtain a filtration, and the Fröhlicher (aka Hodge-to-de Rham) spectral sequence [Voisin 8.2.1, 8.3.3]. Take collapse of the spectral sequence as a black box and define abstract Hodge structure [Voisin 7.1.1] and consider Examples [Voisin, 7.2.1, 7.2.2]. Talk about polarization [7.1.2] in the unlikely case there’s more time.
5. Prove degeneration of the Fröhlicher spectrial sequence, depending on taste. Either: Introduce the different Laplacians and harmonic forms on complex manifolds [Voisin 5.1], represent cohomology by harmonic [5.3.1], talk about the Kähler identities and Hodge decomposition [6.1] Or: Sketch the algebraic proof of the degeneration of the Fröhlicher spectral sequence using reduction to characteristic p, following [Illusie]. (This is only recommended with solid prior knowledge of algebraic geometry in characteristic p.)
6. Compare the Hodge filtrations on Hochschild homology and de Rham cohomology of a smooth variety [Weibel, §1]. Talk briefly about the cyclic case and examples of smooth projective varieties [§2-4].

• Luc Illusie, Frobenius and Hodge Degeneration, in: Bertin, José, Demailly, Jean-Pierre, Illusie, Luc, Peters, Chris: Introduction to Hodge theory. Translated from the 1996 French original by James Lewis and Peters, SMF/AMS Texts Monogr., 8 American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2002.
• Jean-Louis Loday, Cyclic homology, Grundlehren Math. Wiss., 301, Springer-Verlag, Berlin, 1998.
• Daniel Quillen, Gordon Blower, Topics in Cyclic Theory, London Mathematical Society Student Texts 97, Cambridge University Press, 2020.
• Claire Voisin, Hodge theory and complex algebraic geometry. Translated from the French by Leila Schneps. Reprint of the 2002 English edition. Cambridge Stud. Adv. Math., 76 and 77, 2007, Cambridge University Press, Cambridge.
• Charles Weibel, The Hodge filtration and cyclic homology. K-Theory 12, 1997, no.2, 145--164.

If you want to give a talk in the working seminar, send an email to us. If you are interested in participating, then please register for the mailing list of the seminar by sending "subscribe" to topologieseminar.math-request@lists.uni-hamburg.de