Abstracts for Moduli of K-stable varieties

In Kahler geometry, an important and intriguing circle of ideas has been the connections between the analytic aspect which usually involves geometric PDEs, and the algebraic aspect which is related to degenerations and stability of algebraic varieties. An example is the recent result of Chen-Donaldson-Sun on the criterion for the existence of Kahler-Einstein metrics on Fano manifolds in terms of K-stability.

In this series of lectures I will explain several results along this direction. The main focus is to describe the bridges that allow one to reduce a hard problem in geometric PDE to a problem in algebraic geometry, and also allow one to draw algebro-geometric conclusions using the geometric PDEs. This also generates new interesting questions in algebraic geometry, some of which have seen recent progress by many people.

In this mini-course, I will give an introduction to the theory of GIT-, K- and KSBA stability of polarized varieities. We will then focus on the the application of K-stability to the study of moduli space Fano varieities.

Kawamata log terminal (klt) singularities appear ubiquitously in higher dimensional geometry, and have been intensively studied by birational geometers for more than three decades. Guided by the recent work of tangent cone in differential geometry, we explore a new prospective. Indeed we expect for any klt singularity, conjecturally there is a canonical (semi)stable degeneration induced by the valuation which minimises the normalised volume function (introduced by Chi Li). This study combines many different topics developed in higher dimensional geometry e.g. minimal model program, K-stability, asymptotic invariants etc. In particular, we also aim to answer the conjecture by Donaldson-Sun that the tangent cone is indeed an algebraic construction for singularities (appeared on the GH limit of KE Fano manifolds).

In Lecture 1, we will make the definitions, propose the questions, and compute some examples.

In Lecture 2, we will give a brief discussion of the cone cases, which relates to Fujita's work on Ding stability.

In Lecture 3, we will discuss how we can use minimal model program to investigate the property, and give a rather complete picture in the case when the minimiser is a divisorial valuation (so called quasi-regular).

In Lecture 4, we will briefly discuss the case when the minimiser is a quasi-monomial valuation of higher rank. If time permits, we will propose more questions for the future study.

This is quickly moving forward topic. Many works are established recently by various people, notably Harold Blum, Chi Li, Yuchen Liu and others.

We show relationships between uniform K-stability and plt blowups of log Fano pairs.

The Duistermaat-Heckman localization formula gives an interesting relation between the integral of a periodic Hamiltonian and its value on critical points. We give a slight generalisation of this formula and apply this on the Einstein-Hilbert functional on the cone of Reeb vector fields.

Kähler-Einstein metrics with conic singularities are playing a key role in the proof of Yau-Tian-Donaldson conjecture over Fano manifolds. The generalization of these type of metrics to the constant scalar curvature equation are not so well understood (and there are even several possible definitions).  We provide a Fredholm alternative result for the Lichnerowicz operator associated to a K\"ahler metric with conic singularities along a divisor.  With this result in hands, we can deduce several existence results of constant scalar curvature K\"ahler metrics with conic singularities: under small deformations of Kähler classes or over certain ruled manifolds. In this last case, we consider the projectivisation of a parabolic stable holomorphic bundle. This requires to prove that the existing Hermitian-Einstein metric on this bundle enjoys a regularity property along the divisor on the base.  Eventually, the study leads us naturally to a log Yau-Tian-Donaldson conjecture for projective bundles when the base is a curve.

In this talk we introduce a variational/pluripotential approach to the study of K-stability of Kähler manifolds with transcendental cohomology class, extending a classical picture for polarised manifolds. Our approach is based on establishing a formula for the asymptotic slope of the K-energy along certain geodesic rays, from which we deduce that cscK manifolds are K-semistable. Combined with a recent properness result of R. Berman, T. Darvas and C. Lu we further deduce uniform K-stability of cscK manifolds with discrete automorphism group, thus confirming one direction of the YTD conjecture in this setting.

If time permits we also discuss possible extensions of these results to the case of compact Kähler manifolds admitting non-trivial holomorphic vector fields.

In this talk, based on joint work with Song Sun, I will describe how to study concrete examples of degenerations of Kähler-Einstein Fano manifolds in dimension three and higher, discussing the relevance of such investigations in the study of existence of KE metrics in given families and in finding explicit examples of Fano K-moduli compactifications.

For smooth del Pezzo surface it is known that they either admit a Kaehler-Einstein metric or they are toric and therefore admit a non-trivial Kaehler-Ricci soliton. Moreover, by results of Odaka-Spotti-Sun we know which Gorenstein del Pezzo surfaces admit a Kaehler-Einstein metric. Hence, it is natural to ask which of the remaining ones admit a (non-trivial) Kahler-Ricci soliton. We are approaching this question by determining K-stability of pairs (X,v) consisting of a del Pezzo surfaces X and a holomorphic vector field v on X. Moreover, we classify all Gorenstein del Pezzo surfaces, which can be complemented to a K-stable pair.

Let us fix the complex numbers as our base field. We call a variety canonically polarized if it satisfies the mildest possible assumptions to have a reasonable divisor theory and some positive multiple of the canonical class is an ample Cartier divisor. Then by the union of results of Berman, Guenancia, Odaka and Xu for canonically polarized varieties it is equivalent (i) to admit a Kähler-Einstein metric (ii) (X,K_X) to be K-stable and (iii) X to be KSBA stable. Then the general theory of K-stability yields a CM line bundle on the moduli space of KSBA stable varieties. I present a joint work with Chenyang Xu showing that this line bundle is ample.