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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.
References
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).
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