Special Lecture GRK2240 WiSe 23/24

Differential forms and algebraic de Rham cohomology

The ring lectures will take place at HHU in room 25.21.HS 5E, 14:30 - 15:45 and at BUW in room G.13.18, 14:15 - 15:30 on Thursdays.
See also the schedule for more information on the lectures of the GRK.

Here is a rough syllabus for the course.

Talk	Day/Place	Speaker
1. De Rham complex on manifolds	12.10./HHU	Matthias Wendt
2. De Rham theorem on smooth manifolds, periods	19.10./BUW	Matthias Wendt
3. Sheaf cohomology, de Rham theorem via hypercohomology	26.10./HHU	Kay Rülling
4. Analytic de Rham and Dolbeault complex on complex manifolds	02.11./BUW	Kay Rülling
5. Harmonic forms and Hodge decomposition	09.11./HHU	Matthias Wendt
6. Spectral sequences (general and Hodge-to-de Rham)	16.11./BUW	Matthias Wendt
7. Mixed Hodge structures	23.11./HHU	Matthias Wendt
8. Algebraic de Rham cohomology	30.11./BUW 16:45 -18:00 in F.13.11!	Kay Rülling
9. Grothendieck's theorem ``algebraic de Rham = analytic de Rham''	14.12./BUW	Kay Rülling
10. Abel-Jacobi map and geometric applications	21.12./HHU	Matthias Wendt
11. de Rham cohomology in characteristic \(p>0\), Cartier operator	11.01./BUW	Kay Rülling
12. Deligne-Illusie	18.01./HHU	Kay Rülling
13. Proof of Deligne-Illusie	25.01./HHU	Kay Rülling

Literature

• G.E. Bredon, Topology and geometry, Springer, 1997.
• P. Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math.(1971), no. 40, 5–57.
• P. Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics Vol. 163, Springer, 1970.
• P. Deligne, L. Illusie Relèvements modulo \(p^2\) et décomposition du complexe de de Rham, Invent. Math. (1987), no. 2, 247-270.
• P. Griffiths, J. Harris, Principles of Algebraic Geometry. 2nd ed., John Wiley & Sons Ltd, 1994.
• A. Grothendieck, On the de Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math.(1966), no.29, 95-104.
• N. M. Katz, Nilpotent connections and the monodromy theorem: applications of a result of Turrittin, Publ. Math. IHES (1070), tome 39, p. 175-232.
• K. Kodaira, Complex manifolds and deformation of complex structures, Springer, 1986.
• D. Mumford, Pathologies of modular algebraic surfaces, Am. J. Math. (1961), vol. 83, no. 2, 339-342.
• J.-P. Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier (1955), vol 6, 1-42.
• L. W. Tu, An introduction to manifolds, Springer, 2010.
• C. Voisin, Hodge theory and complex algebraic geometry. I, Cambridge University Press, 2007.