Misha Verbitsky

Instituto Nacional de Matemática Pura e Aplicada

Hodge theory

Slides:

• Lecture 1: Complex manifolds, 17.03.2025
• Lecture 2: Cauchy formula (for home perusal)
• Lecture 3: Frobenius theorem, 19.03.2025
• Lecture 4: Real analytic manifolds 24.03.2025
• Lecture 5: Newlander-Nirenberg theorem 26.03.2025
• Lecture 6: Kähler structures on homogeneous spaces, 31.03.2025
• Lecture 7: Levi-Civita connection, 02.04.2025
• Lecture 8: Intrinsic torsion, 07.04.2025
• Lecture 9: Bismut connection, 09.04.2025
• Lecture 9-bis, a supplement: Harmonic forms and cohomology (for home reading).
• Lecture 10: Lefschetz identities and supersymmetry on Kähler manifolds, 14.04.2025
• Lecture 11: Sheaves and cohomology, 28.04.2025
• Lecture 12: Torus action and Dolbeault cohomology of a torus, 30.04.2025
• Lecture 13: Poincaré-Dolbeault-Grothendieck lemma, 05.05.2025
• Lecture 14: Chern connection, 07.05.2025
• Lecture 15: Hodge theory for forms with coefficients in holomorphic bundle, 12.05.2025
• Lecture 16: Kodaira-Nakano vanishing theorem, 14.05.2025
• Lecture 16: Kodaira-Nakano vanishing theorem, 14.05.2025
• Lecture 16-bis, a supplement: Generalized functions and currents (for home reading).
• Lecture 17: Non-bounded operators on Hilbert spaces, 19.05.2025
• Lecture 18: L²-cohomology, 21.05.2025
• Lecture 19: Holomorphically convex manifolds, 26.05.2025
• Lecture 20: Stein manifolds, 28.05.2025

Home assignments

• Assignment 1: complex structures, 17.03.2025. Results.
• Assignment 2: Frobenius form, 24.03.2025. Results.
• Assignment 3: Integrability of almost complex structures, 31.03.2025. Results.
• Assignment 4: Complex structures on homogeneous spaces, 07.03.2025. Results.