"One of the most important problems in Kähler geometry is to search for canonical metrics on a given Kähler manifold X, where the word ""canonical"" stands for extremal/costant scalar curvature (cscK for short)/twisted cscK or Kähler-Einstein (KE).
It was conjectured by Calabi and then proved by Yau and Aubin in the late 70's that when the first Chern class of the manifold is identically zero or negative there always exists a unique Kähler-Einstein metric. The Fano case (i.e. positive first Chern class) turned out to be much more difficult because in this case KE metrics do not always exist. The obstruction to their existence is encoded in the notion of K-stability. Only recently was it proved that on a Fano manifold X there exists a KE metric if and only if X is K-stable.
The Minimal Model Program is part of the birational classification of algebraic varieties, leads to work with singular varieties. In the last few years, Eyssidieux, Guedj and Zeriahi have established the existence of a unique singular Kähler-Ricci flat metric on a very wide class of Calabi-Yau varieties. Their work reduces to study a degenerate complex Monge-Ampère equation and it establishes the existence of such singular Kähler-Ricci flat metric. Nevertheless, it does not establish the asymptotic behavior near the singular points. The main goal of this proposal is to study the asymptotic behavior and the regularity properties of these metrics/potentials near singularities. More generally, given a Kähler-Einstein metric on a singular variety, it is interesting to understand how we can relate the asymptotic behavior of such a metric to the singularities of the variety.
The above problem is of great interest in theoretical physics. Indeed, since the seminal paper of Candelas and de la Ossa in the 90's, physicists have guessed that Calabi-Yau 3-folds with the simplest isolated singularities should admit incomplete Kähler-Ricci flat metrics which near each singularity look like the conifold metric. If this were denied by some devolopments in this area, physicists should change their vision to ""see things"".
The analytic approach to the Minimal Model Program, proposed by Song and Tian in 2007, consists in reaching the ""minimal model"" of a given variety via the Kähler-Ricci flow. In order to do so one needs to start the flow from a degenerate initial data. One of the objectives of the proposal is to start the flow from a singular data and to investigate the regularity properties of the Kähler-Ricci flow running from an arbitrary positive closed current.
"
