Math 445 Winter

Winter: Hodge theory

Textbooks: Warner and Wells

- Covariant derivatives on vector bundles and curvature
- Review of de Rham complex
- Laplace-Beltrami operator
— Harmonic Forms
— Hodge Decomposition Theorem
— Bochner's vanishing theorem for harmonic one-forms
— Definition of Kahler manifolds, and proof of the main Kahler identities (Weil's proof)
- Chern-Weil theory: Chern classes, integrality of periods, functoriality, Chern character
— Statement of Hirzebuch's signature and Riemann-Roch theorems

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