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The Kaehler-Ricci flow and Singular Calabi-Yau manifolds

Periodic Reporting for period 1 - KRF-CY (The Kaehler-Ricci flow and Singular Calabi-Yau manifolds)

Reporting period: 2015-07-01 to 2017-06-30

"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.
I started working on the regularity properties of the Kähler-Ricci flow. On a compact Kähler manifold the Kähler-Ricci flow detects special metrics: in the case of non-negative curvature for example, the flow, starting from any Kähler form as initial data, converges to the unique Kähler-Einstein metric. The question is then to understand what happens when the initial data is not anymore a Kähler form but a singular object, i.e. a positive current. With a collaborator of mine, Chinh Lu, we prove that one can run the flow with any positive current as initial data. Moreover we prove that the flow has regularizing properties: after some time the metric that is evolving along the flow becomes smooth.
We also have some partial results regarding the uniqueness of the flow starting from a degenerate initial data. The problem of uniqueness in full generality is still open.

These results obtained a lot of interest: I was invited to communicate them in the weekly seminar of several University, such as Cambridge and in international conferences (for example in Banff and at MSRI).

I then exploited the singular behaviour of singular solutions of degenerate elliptic complex Monge-Ampère equations on compact Kähler manifolds. With Chinh Lu and Tamas Darvas we studied the singularity type of full mass positive currents in big cohomology classes. The motivation to work with such objects comes from the need to study singular special metrics (Kähler-Einstein metrics for example) on a compact normal Kähler space. The existence of such metrics reduces indeed to a degenerate complex Monge-Ampère equation (DCMAE for short). The first step is then to look for a weak solution (a priori singular) of a DCMAE. Such weak solutions belong to energy classes, called Monge-Ampère energy classes. In the work with Lu and Darvas we studied the first energy class in this singular setting: the class E. In particular we characterised in a precise way the singularity type of potentials in the class E. As a corollary of our result we have give a positive answer to an open question in pluripotential theory.

The collaboration with Lu and Darvas then led also to the study of complex Monge-Ampère equations with prescribed singularities. We obtained existence of singular Kähler-Einstein metrics with prescribed singularity type on Kähler manifolds of general type and Calabi-Yau.

In the last 20 years there have been a lot of work in order to study the space of Kähler metrics on compact Kähler manifold. The motivation for this was give by Donaldson and was related to the uniqueness of constant scalar curvature Kähler metrics. As one can imagine, the properties of such a space are strictly related to geodesics. The study of geodesics joining two Kähler potential is again related a DCMAE on a compact Kähler manifold with boundary. In a paper in collaboration with Vincent Guedj we studied the geometry and topology of the space of Kähler metrics, on singular variety and we generalised most of the results that are valid in the smooth case.

I communicate the result of this work in several seminar and in international institutions such as the Simon's Center.
"As I already discussed above, during the project my collaborators and I succeeded in having a better understanding of the singular behaviour of solutions of degenerate (elliptic and parabolic) complex Monge-Ampère equations on compact Kähler manifolds.
These kind of results are interesting also in relation to the Minimal Model Program (MMP). The MMP is a part of the birational classification of algebraic varieties. Its goal is to construct a birational model of any complex projective variety which is as simple as possible, the so-called ""minimal model"". The Italian school investigated the case of surfaces. The situation in higher dimension is much more complicated since a smooth minimal model does not always exist: this is why one has to allow singularities.
Searching for a Kähler-Einstein metric on a singular variety is then equivalent to study a degenerate complex Monge-Ampère equations on a smooth resolution.

With my collaborators, Chinh Lu and Tamas Darvas we are also currently studying the geometry of the space of singularity types. A singularity type is roughly speaking an equivalence class of quasi-plurisubharmonic functions on a compact Kähler manifold: two potentials are in the same class if and only if the have the same asymptotic behaviour at minus infinity. This is a project in progress that it will not be finished by the end of the project but whose developments are going to clarify the type of singularities that may occur.

This is why we think that there is no doubt that our results in the singular setting will play a leading role in subsequent applications and will impact the community working in differential/complex geometry.

At this point, there is no obvious potential for commercial exploitation."