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Kähler Geometry

Jacopo Stoppa
Course Type: 
PhD Course
Academic Year: 
Oct - Feb
60 h

Kaehler geometry studies complex manifolds with a hermitian metric adapted to the complex structure. I plan to start from the basics and then concentrate on some aspects of the theory of canonical metrics on compact Kaehler manifolds (including the Calabi-Yau theorem). Some other topics (e.g. the hyperkaehler condition; variational approach to Kaehler-Einstein metrics; deformation theory of Kaehler-Einstein metrics) may be covered depending on the interests of the audience.

Some references:

  • Daniel Huybrechts: Complex Geometry, an Introduction, Springer Universitext, 2005.
  • Gabor Szekelyhidi: An Introduction to Extremal Kaehler Metrics, Graduate Studies in Mathematics
    vol. 152, American Mathematical Society, 2014.

Some main topics:
Complex manifolds. Almost complex structures: algebraic aspects, integrability. Decomposition of the exterior differential on a complex manifold. Hermitian metrics. The Kaehler condition. Covariance constancy of almost complex structure. Holomorphic normal coordinates. The Fubini-Study metric. Christoffel symbols and curvature in the Kaehler case. The curvature tensor of CP^n. Ricci curvature and Ricci form. The first Chern class. The case of hypersurfaces. The Kaehler-Einstein condition. Introduction to the Aubin-Yau and Yau theorems. Introduction to the Fano case. Examples of complete Ricci flat metrics (Gibbons-Hawking ansatz). The Laplace operator of a Kaehler matric. Solving the Poisson equation on a compact Kaehler manifold. Recollections on the Hodge decomposition in the Riemannian case. The ddbar-Lemma. The Lefschetz operator and its
formal adjoint. Algebraic aspects: the representation of sl(2) on cohomology, other commutators, primitive forms and the Leftschetz decomposition. The Kaehler identities. The Hodge decomposition for a Kaehler metric. The Hard Lefschetz theorem. The Aubin-Yau theorem: reduction to a complex Monge-Ampere equation. Uniqueness, continuity method. Yau's estimates: C^0, trace inequality, C^2. The Calabi-Yau theorem: reduction to C^0 estimate and Moser iteration.

Possible additional topics (in no specific order):
First results on constant scalar curvature Kaehler (cscK) and extremal metrics. Deformation theory of cscK metrics. Natural functionals in Kaehler geometry and the variational approach. Kaehler-Einstein metrics on del Pezzos and Fanos. The hyperkaehler condition.


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