This project aims to study families of canonical Kähler metrics, their evolution under the Ricci flow, and their asymptotic behavior near singularities, in order to deduce structure theorems and address central classification problems. An important starting point is that the Ricci tensor of a Kähler metric can be expressed in terms of the complex Monge-Ampère operator. After Yau's celebrated solution to the Calabi conjecture, degenerate complex Monge-Ampèrere equations have been studied by Eyssidieux, Guedj, and Zeriahi to construct canonical Kähler metrics on singular varieties. While the construction is well understood, the asymptotic behavior of these metrics near the singularities remains mysterious.
An alternative and complementary approach is to use the Ricci flow from Riemannian geometry, a geometric flow that evolves a Riemannian metric by its Ricci tensor. If the initial metric is Kähler, the evolving metrics are also Kähler, and the flow is called the Kähler-Ricci flow. Following Perelman's resolution of the Poincaré conjecture, it is expected that the Kähler-Ricci flow could play an important role in classification problems of compact Kähler manifolds. Building on this expectation, Song and Tian proposed an analytic version of Mori's Minimal Model Program, using the Ricci flow. The expectation is that, starting from a smooth Kähler variety, the flow deforms it multiple times, restarts on new varieties, and eventually reaches a minimal model. The fact that these models can be singular motivates our team to define and study weak Kähler-Ricci flows.
The project is funded by the Charles Defforey foundation, Institut de France
Members
- Sébastien Boucksom, Directeur de Recherche CNRS, Sorbonne Université
- Alix Deruelle, Professeur, Université Paris-Saclay
- Eleonora Di Nezza, Professeure, Sorbonne Université
- Vincent Guedj, Principal Investigator, Professeur, Université de Toulouse
- Henri Guenancia, Directeur de Recherche CNRS, Université de Bordeaux
- Chinh H. Lu, Professeur, Université d'Angers
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