Extremal metrics and relative K-stability – EMARKS

Kähler geometry is at the intersection of various fields of research in pure mathematics and is a very active world for the last 40 years. Without being exhaustive, it is intrinsically related to symplectic geometry, complex analysis, algebraic geometry, Riemannian geometry, PDE analysis, deformation theory, quantization and has applications in all these fields and also in others like mathematical physics via String Theory. The origin of this extraordinary relationship lies in the very definition of a Kähler manifold that allows one to define the metric tensor using simply one potential function, implying a long list of “miracles”. We refer to J-P. Bourguignon's enlightening paper “The unabated vitality of Kählerian geometry” (Mathematical works of E. Kähler, de Gruyter 2003) where the importance of the quest of Kähler metrics with special curvature properties and the impact of Kähler geometry on different fields are stressed.

Coming back to the definition of a Kähler manifold, one can ask if there are natural/canonical metric in a given Kähler class. In the early 80's, Calabi stated precisely this question and suggested Kähler extremal metrics as candidates. These metrics arise as critical points of a functional involving the scalar curvature and turn out to be solutions of a 4-th order non-linear PDE. Kähler—Einstein and constant scalar curvature Kähler metrics are the most commonly known examples of extremal Kähler metrics. Since the breakthrough of S. T. Yau in the 70's about Einstein's equations of general relativity, most efforts in this field are related to the so-called Yau-Tian-Donaldson conjecture.

Given a projective manifold X equipped with a polarization L, this conjecture predicts that there is an “extremal” Kähler metric in the class c1(L) if and only if (X,L) is K-polystable relative to a maximal torus of automorphisms. We will give more explanations in the sequel, but let us mention for the moment that this formulation is due to G. Székelyhidi and is based on the ideas of G. Tian and S.K. Donaldson for the last 20 years. Together with J. Stoppa, G. Székelyhidi confirmed that the existence of an extremal metric implies relative K-polystability. In the context of constant scalar curvature Kähler metrics, we will explain later the very recent progress on the Yau-Tian-Donaldson conjecture that exhibits the great vitality of the field.

The Yau-Tian-Donaldson conjecture is still wide open for extremal metrics and many experts think that the notion of relative K-stability needs to be strengthened.

We wish to study this key conjecture from different aspects.

Firstly it is crucial to obtain new examples of extremal Kähler manifolds (that are not constant scalar curvature Kähler metrics, so there is a non-trivial group of isometries) and analyze the stability of their underlying structures. This is one of the main goals of our project. In order to do so, we shall investigate certain geometries with natural symmetries, including toric and spherical geometry, the symplectic reduction process and its relation with Kähler metrics, compact orbifolds and Hamiltonian 2-forms. By extension, it is natural to address these issues in Sasakian geometry.
Another main objective of our project is to understand the precise behavior of extremal metrics in various natural geometric problems, as will be described in detail later in our proposal. Among other things, we have in mind to set up a new theory relating extremal Kähler metrics and stationary Lagrangian submanifolds that would give some insights into the geometry of the moduli of such objects.
Finally, we would like to tackle some wide open questions about extremal Kähler metrics, in particular the possibility of a stabilization process for unstable manifolds (or manifolds that do not carry extremal Kähler metrics), which is certainly related to an extension of the Arezzo-Pacard theory of blow-ups.