100, rue des maths 38610 Gières / GPS : 45.193055, 5.772076 / Directeur : Louis Funar

Zehao Sha

The Calabi conjecture and Kähler-Einstein metrics
Jeudi, 18 Janvier, 2024 - 17:00 à 18:00
Résumé : 

In Kähler geometry, the Calabi conjecture presented by E. Calabi in 1954 is a prescribed Ricci curvature problem within the cohomology class of a given Kähler metrics on closed complex manifolds. More precisely, for any real differential (1,1)-form \theta which represents the first Chern class of the Kähler manifold, can we find another Kähler metric in the same Kähler class of a given Kähler metric whose Ricci form is \theta?

Calabi transformed this conjecture into a complex Monge-Ampère equation. Later, in 1978, S.-T. Yau proved the Calabi conjecture by constructing a solution of this equation using the continuity method. The solution of this equation is a Kähler-Einstein metric(Ricci-flat) and such Kähler manifolds are called Calabi-Yau manifolds.
In this talk, we will see all the concepts mentioned above and give a brief strategy on how to prove the Calabi conjecture. If time permits, we will discuss further the existence of Kähler-Einstein metrics on closed Kähler manifolds which depends on the sign of the first Chern class.


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