Dipartimento di Matematica "Giuseppe Peano"

Lezione lagrangiana

The Calabi Problem and Birational Geometry

Relatore:
Carolina Araujo
IMPA Brasil

Data e luogo:
29 Maggio 2025 dalle 14.30 alle 15.30
in Aula A, Palazzo Campana, Via Carlo Alberto 10, Torino

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.

Short Bio