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Summer School 2010 in Cologne

Scientific Programme

Morning Lectures

The academic programme consists of a series of morning lectures and afternoon working groups.
We will have two series of lectures introducing the main subjects. The goal of the lectures will be to communicate the fundamental motivating questions in each field, the tools used to address them, and the important results. We plan to have two lectures each morning.
Introduction to extremal Kähler Metrics. Series of four lectures, by Prof. Gauduchon
Extremal Kähler metrics were introduced by E. Calabi in the 80's to remedy the non-existence of Kähler-Einstein metrics, even of Kähler metrics of constant scalar curvature, on such a simple complex manifold as the blow-up of the projective plane at a point. In this series of lectures, we shall provide a self-contained presentation of these metrics and of their main properties, including a brief review of basic general facts in Kähler geometry. We then introduce some basic tools defined on the space of Kähler metrics of a compact complex manifold, which are currently used to explore uniqueness and existence issues for extremal metrics. In the last part of these lectures, we present a general construction of extremal Kähler metrics, which include the first constructions by Calabi of extremal Kähler metrics of non-constant scalar curvature and can be used to check current conjectures in the field.

E. Calabi, Extremal Kähler metrics, in Seminar of Differerential Geometry, ed. S. T. Yau, Annals of Mathematics Studies 102, Princeton University Press (1982), 259--290.
E. Calabi, Extremal Kähler metrics, II, in Differential Geometry and Complex Analysis, eds. I. Chavel and H. M. Farkas, Springer Verlag (1985), 95--114.
V. Apostolov, D. M. J. Calderbank, P. Gauduchon, Christina W. Tønnesen-Friedman, Hamiltonian 2-forms in Kähler geometry, III: Extremal metrics and stability, Invent. math. 173 (2008), 547-601.
P. Gauduchon, Calabi extremal metrics: An introduction, a book in progress...

Canonical metrics, stability, and non-linear partial differential equations.
Series of four lectures, by Prof. Phong.
The goal of this series of lectures is to provide a self-contained introduction to the problem of finding canonical metrics in Kähler geometry. Typically, canonical metrics are the solutions of some non-linear partial differential equation. Their existence has been conjectured by Yau to be equivalent to suitable notions of stability in geometric invariant theory, and several notions of stability have been proposed by Tian and Donaldson. We provide some of the motivation behind these conjectures. Our emphasis will be analytic, and we shall describe some of the recent progress in the study of the equations of constant scalar curvature, including on Monge-Ampere equations, on Kähler-Ricci flows, and on Calabi flows.

Main references

S.K. Donaldson, Scalar curvature and stability of toric varieties, J. Differential Geom. 59 (2002) 289-349.
G. Tian, Kähler-Einstein metrics of positive scalar curvature, Inventiones Math. 130 (1997) 1-37.
S.T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampere equation I, Comm. Pure Appl. Math. 31 (1978) 339-411.

Survey articles

D.H. Phong and J. Sturm, Lectures on stability and constant scalar curvature, arXiv math 0801.4179.
Y.T. Siu, Lectures on Hermitian-Einstein metrics for vector bundles and Kähler-Einstein metrics, Birkhauser (1987).

Afternoon Working Groups

In the working groups the mentors will answer questions and explain ideas which appeared in the morning lectures.
On Monday we will have an introductory lecture recalling basics of complex geometry.
On Tuesday afternoon there will be a lecture on Bergman kernel asymptotics, by G. Marinescu, explaining some ideas from Phong's lectures.
On Thursday afternoon Semyon Klevtsov (Universite Libre, Bruxelles) will give a lecture on Kähler metrics, 2d gravity and complex random matrices explaining relations between Kähler metrics, Mabuchi K-energy, Bergman approximations and physics.

Design © 2005-2006 Jiří Horák, Photos courtesy of Dierk Lürbke
last changed 8 September, 2010