MAT 683 

Advanced Topics in
Differential Geometry 

Spring 2006 

MF 12:50-2:10
Physics P-123 

Prof. Claude LeBrun

Office hours:
MF 11-12, Math Tower 3-108.

Extremal Kähler metrics

A polarized complex manifold is by definition a compact complex manifold (M, J) equipped with a Kähler class [ω]. To the algebraic geometer, such triples (M, J, [ω]) are objects of central interest, because any non-singular variety in a complex projective space carries a natural polarization, given by the hyper-plane class.

In search of a natural bridge between algebraic and differential geometry, Calabi therefore proposed that one should try to represent each Kähler class on any compact complex manifold by a constant-scalar-curvature Kähler metric. Later, he modified his proposal to allow for extremal Kähler metrics, meaning Kähler metrics for which the gradient of the scalar curvature is a holomorphic vector field.

While it turns out that not every Kähler class need contain such a metric, Calabi's idea was close enough to the mark so as to generate a huge body of deep mathematics, including the the existence theory for Kähler-Einstein metrics. Unfortunately, the general case is much harder than the Kähler-Einstein case, as the relevant PDE can no longer be reduced to the second-order Monge-Ampère equation. Nonetheless, considerable progress has recently been made concerning existence, non-existence, and uniqueness of solutions. This course will offer an introduction to some of the key ideas in the subject. In particular, we will explore some of the recent key papers of Donaldson, Mabuchi, Arezzo, Pacard, Ross, Thomas, and others.

Prerequisite: MAT 569, or permission of the professor.

Grades will be based upon class participation.

The Professor may be reach by e-mail at .

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DSS advisory. If you have a physical, psychiatric, medical, or learning disability that may affect your ability to carry out the assigned course work, please contact the office of Disabled Student Services (DSS), Humanities Building, room 133, telephone 632-6748/TDD. DSS will review your concerns and determine what accommodations may be necessary and appropriate. All information and documentation of disability is confidential.