Hellenica World

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In differential geometry, a Kähler–Einstein metric on a complex manifold is a Riemannian metric that is both a Kähler metric and an Einstein metric. A manifold is said to be Kähler–Einstein if it admits a Kähler–Einstein metric. The most important special case of these are the Calabi–Yau manifolds, which are Kähler and Ricci-flat.

The most important problem for this area is the existence of Kähler–Einstein metrics for compact Kähler manifolds.

In the case in which there is a Kähler metric, the Ricci curvature is proportional to the Kahler metric. Therefore, the first Chern class is either negative, or zero, or positive.

When the first Chern class is negative, Aubin and Yau proved that there is always a Kähler–Einstein metric.

When the first Chern class is zero, Yau proved the Calabi conjecture that there is always a Kähler–Einstein metric. Shing-Tung Yau was awarded with his Fields medal because of this work. That leads to the name Calabi–Yau manifolds.

The third case, the positive or Fano case, is the hardest. In this case, there is a non-trivial obstruction to existence. In 2012, Chen, Donaldson, and Sun and independently by Gang Tian proved that in this case existence is equivalent to an algebro-geometric criterion called K-stability. Their proof appeared in a series of articles in the Journal of the American Mathematical Society.
References

Moroianu, Andrei (2007). Lectures on Kähler Geometry. London Mathematical Society Student Texts 69. Cambridge. ISBN 978-0-521-68897-0.