Complex Differential Geometry (MAGIC044)
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Complex manifolds are central objects in many areas of mathematics: differential geometry, algebraic geometry, several complex variables, mathematical physics, topology, global analysis etc. Their geometry is much richer than that of real manifolds which leads to fascinating phenomena and the need for new techniques. The present course will give a brief introduction to basic notions and methods in complex differential geometry and complex algebraic geometry. The aim is to present beautiful and powerful classical results, such as the Hodge theorem, as well as to develop enough language and techniques to make the material of current interest accessible. Literature S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry", vol. II, John Wiley & Sons
A. Moroianu, "Lectures on Kähler Geometry", CUP
C. Voisin, "Hodge theory and complex algebraic geometry", vol. I, CUP
P. Griffiths and J. Harris, "Principles of Algebraic Geometry", John Wiley & Sons (chapter 0 only)
Additional reading material K. Fritzsche and H. Grauert, "From Holomorphic Functions to Complex Manifolds", Springer
D. Huybrechts, "Complex Geometry, Springer
W. Ballman, "Lectures on Kähler manifolds", EMS
R.O. Wells, "Differential analysis on complex manifolds", Springer
Autumn 2010 (Monday, October 4 to Friday, December 17)
Familiarity with basic notions of topological and differentiable manifolds, especially tensors and differential forms. Knowledge of such Riemannian concepts as the Levi-Civita connection and curvature will be helpful, but not essential.
1. Complex and almost complex manifolds
2. Holomorphic forms and vector fields
3. Complex and holomorphic vector bundles
4. Hermitian bundles, metric connections, curvature
5. Chern classes
6. Hermitian and Kähler metrics
7. Dolbeaut theory and the Hodge theorem
8. Curvature of Kähler manifolds; holomorphic sectional and Ricci curvature
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