BASIC NOTIONS SEMINAR Friday 16 January at 11:00 - Carolina Araujo (IMPA)

[ICTP Math Section]

_*BASIC NOTIONS SEMINAR*_

Friday 16 January,  at 11:00


*Venue:* Leonardo Building - Luigi Stasi Seminar Room (In-Person)

*Speaker:* Carolina Araujo (IMPA)

*Title:* The Calabi Problem

*Abstract: *In his 1954 ICM lecture, Eugenio Calabi popularized a 
formidable problem at the confluence of differential and algebraic 
geometry, which became known as the Calabi problem. The problem asks 
which compact complex manifolds admit a special type of 
constant-curvature metric, known as a Kähler-Einstein metric.

For Kähler manifolds with either flat or negative curvature, the Calabi 
problem was solved in the 1970s by Yau and by Aubin and Yau. They 
confirmed Calabi’s prediction, showing that such manifolds always admit 
a Kähler-Einstein metric. However, in the case of positively curved 
projective manifolds - known as Fano manifolds - the existence of a 
Kähler-Einstein metric is not guaranteed.

In the past decade, surprising and deep connections have emerged between 
the existence of Kähler-Einstein metrics and birational geometry, 
leading to significant advances in the Calabi problem. In this talk, I 
will discuss the problem itself, its links with birational geometry, and 
the current state of the art in dimension three.