# K-stability

# Monday September 28th-Friday October 2nd, 2020

**Update:** The videos and the slides of the conference can be found **here**.

## Organisers:

Paolo Cascini, Ivan Cheltsov, James McKernan and Chenyang Xu

## Speakers

Jarod Alper

Sebastien Boucksom

Harold Blum

Ruadhai Dervan

Kristin De Vleming

Simon Donaldson

Anne-Sophie Kaloghiros

Chi Li

Yuchen Liu

Zsolt Patakfalvi

Hendrik Suess

Chenyang Xu

Ziquan Zhuang

Please contact one of the organisers if you need the link and password to connect with Zoom.

## Schedule

In California time (add 3 hours for the East Coast, 8 hours for UK and 9 hours for Continental Europe):

Monday Sep. 28

8:30-9:20 Simon Donaldson

9:30-9:45 Coffee break

9:45-10:35 Chenyang Xu

10:45-11:35 Yuchen Liu

Tuesday Sep. 29

8:30-9:20 Ruadhai Dervan

9:30-9:45 Coffee break

9:45-10:00 Jesus Martinez Garcia

10:00 - 10:15 Carlo Scarpa

10:15 - 10:30 Lu Qi

10:45-11:35 Ziquan Zhuang

Wednesday Sep. 30

8:30-9:20 Zsolt Patakfalvi

9:30-9:45 Salvatore Tambasco

9:45-10:00 Quentin Posva

10:00-10:15 Enrico Fatighenti

10:15-10:30 Jiyuan Han

10:45-11:35 Chi Li

Thursday Oct. 1

8:30-9:20 Sebastien Boucksom

9:30-9:45 Coffee break

9:45-10:35 Hendrik Suess

10:45-11:35 Jarod Alper

Friday Oct. 2

8:30-9:20 Anne-Sophie Kaloghiros

9:30-9:45 Coffee break

9:45-10:35 Kristin De Vleming

10:45-11:35 Harold Blum

## Titles and Abstracts

**Jarod Alper **- Existence of separated good moduli spaces of K-polystable Fanos

Abstract: We will survey how recent advances in moduli theory allow for the construction of a separated algebraic space parameterizing K-polystable Fano varieties. The argument depends on establishing certain valuative criteria, called S-completeness and \Theta-reductivity, for the moduli stack of K-semistable Fanos. This is joint work with Harold Blum, Daniel Halpern-Leistner and Chenyang Xu.

**Harold Blum **- Uniqueness of K-polystable degenerations of Fano varieties

Abstract: In this talk, I will explain that given a family of Fano varieties over a punctured curve any K-polystable degeneration is unique. The result and its proof have a couple of interesting applications: (1) it is an input in constructing the separated good moduli space parametrizing K-polystable Fano varieties and (2) it can be used to show that K-polystable Fano varieties have reductive automorphism group. This talk is based on joint work with Jarod Alper, Daniel Halpern-Leistner, and Chenyang Xu.

**Sebastien Boucksom** - Non-Archimedean pluripotential theory and K-stability

Abstract: I will survey joint work with Mattias Jonsson in which we develop a non-Archimedean analogue of complex pluripotential theory. This provides a convenient framework for the study of K-stability, in which the valuation and filtration approaches fit in naturally.

**Ruadhai Dervan** - Stability of fibrations

Abstract: I will describe an analogue of K-stability for fibrations. The notion generalises K-stability of polarised varieties when the base of the fibration is a point, and slope stability of a vector bundle when the fibration is the projectivisation of a vector bundle. Conjecturally, this notion of stability should allow one to form moduli spaces of stable fibrations over a fixed base, and should be equivalent to the existence of certain canonical metrics. These conjectures are analogues of central conjectures in the theory of polarised varieties and vector bundles (essentially all proven for bundles and Fano varieties). The main result will prove one direction of our conjecture linking stability of fibration with the existence of appropriate canonical metrics, which we call "optimal symplectic connections". This is joint work with Lars Sektnan.

**Kristin De Vleming** - K-moduli of curves on a quadric surface and K3 surfaces

Abstract: By regarding a (d,d) curve C on a quadric surface as a log Fano pair (P1xP1, aC), where a is a rational number, one can use K moduli to study a family of compactifications of the moduli spaces of these curves as a varies. Of particular interest is the case d = 4: a general hyperellipic degree 4 K3 surface is a double cover branched over such a curve. In this case, we show that K stability provides a natural way to interpolate between the GIT moduli space of (4,4) curves and the Baily-Borel compactification of the K3 surfaces. Furthermore, Laza and O’Grady have shown how to interpolate between these moduli spaces via a series of explicit VGIT wall crossings, and we show that these VGIT walls coincide exactly with the K moduli wall crossings. This is joint work with Kenneth Ascher and Yuchen Liu.

**Simon Donaldson **- Stability and differential geometry.

Abstract: The talk will be of survey type, with no new results. We will review some of the history, starting with bundles over algebraic curves and the work of Narasimhan and Seshadri, and the role of moment maps in forming a bridge between differential geometry and algebro-geometric stability. We will discuss some similarities and differences between the theories for varieties and bundles and the specific example of deformations of the Mukai-Umemura Fano threefold. In the last part of the talk we will discuss the constant scalar curvature (non-Fano) case and in particular toric varieties.

**Enrico Fatighenti**** **- Fano varieties from representation theory.

Abstract: The idea of classifying Fano varieties using homogeneous vector bundles was behind Mukai's classification of prime Fano 3-folds. In this short talk, we give a survey of some recent progress along the same lines, including a biregular rework of the non-prime Mori--Mukai 3-folds classification and some examples of higher-dimensional Fano varieties with special Hodge-theoretical properties.

**Jiyuan Han**** **- A prelude to the optimal degeneration of Kahler-Ricci flows on Fano manifolds.

Abstract: This short talk serves as a “more analytic natured” prelude to the following talk given by Chi Li. We will discuss some preliminary topics in regard of the proof of the algebraic-uniqueness for Kahler-Ricci flow limits on Fano manifolds. These topics include the Hamilton-Tian conjecture, the R-test configurations induced by Kahler-Ricci flows (introduced by Chen-Sun-Wang), the H-functional, some results of optimal degeneration (of Kahler-Ricci flow) by Dervan-Szekelyhidi and Hisamoto.

**Anne-Sophie Kaloghiros **- On Calabi’s conjecture for smooth Fano 3-folds

I will report on joint work with Araujo, Castravet, Cheltsov, Fujita, Martinez-Garcia, Shramov, Süss and Viswanathan in which we study K poly/semistability of smooth Fano 3-folds.

**Chi Li **- ** **Nonlinear optimal degeneration problem for Fano varieties

Abstract: For any Q-Fano variety, we first introduce a nonlinear $\tilde{\beta}$-functional on the space of real valuations. We use the tools from birational algebraic geometry to show the existence of minimizers. We then prove that the minimizing valuation that induces a special R-test configuration is unique, and has a K-semistable Fano central fibre $(W, \xi)$. Moreoverthere is a unique K-polystable degeneration of $(W, \xi)$. As an application, we confirm the conjecture of Chen-Sun-Wang about the algebraic-uniqueness for normalized K\"{a}hler-Ricci flow limits on Fano manifolds. The results/techniques are global analogues of local counterparts in the study normalized volume functional of valuations (centered at a closed point). This is a joint work with Jiyuan Han.

**Yuchen Liu **- Openness of K-semistability for Fano varieties

Abstract: In this talk, I will explain the proof of openness of K-semistability for Fano varieties. This is one major step in the algebraic construction of Fano K-moduli spaces. We also establish a correspondence between weakly special test configurations and log canonical places of bounded complements. Our proof is a combination of valuative criteria for K-semistability due to Fujita and Li, boundedness of complements due to Birkar, and approximation techniques. This talk is based on joint work with Harold Blum and Chenyang Xu.

**Jesus Martinez Garcia**** **- Strong calabi dream rational surfaces

Abstract: Chen and Cheng introduced the notion of Calabi dream manifold as a compact manifold which admits an extremal metric for each Kähler class. They also gave a sufficient condition for a surface of general type to be Calabi dream. In this talk, we introduce strong Calabi dream manifolds as those which admit a Kähler metric with constant scalar curvature (and hence extremal) in each Kähler class. We further classify strong Calabi dream rational surfaces.

**Zsolt Patakfalvi ** - A simple proof of semi-positivity and the nefness threshold of the CM line bundle via the stability threshold

Abstract: This is a joint work with Giulio Codogni. I will present a simplified proof using the stability threshold of the following earlier results of ours: 1.) the semi-positivity of the CM line bundle for families of K-semistable Q-Fano varieties, 2.) the nefness threshold for the relative anti-canonical divisor with respect to the CM-line bundle, for families of Q-Fano varieties over a curve with uniformly K-stable general fiber. Our new constant for point 2.) is better than our earlier one. Another simple proof of 1.) was given by Xu and Zhuang, via a study of filtrations.

**Quentin Posva **- Positivity of the CM line bundle for K-stable log Fanos

Abstract: In this talk, we consider the Chow—Mumford line bundle on families of log Fano pairs. This line bundle is expected to give a polarisation on the good moduli space of K-stable log Fanos pairs. I will sketch a proof that it is indeed ample for families of uniformly K-stable log Fano pairs with maximal variation.

**Lu Qi**** **- On local volumes and boundedness of singularities.

Abstract: A folklore conjecture predicts that the set of local volumes of klt singularities is discrete away from zero (resp. satisfies ACC) if the coefficients of the boundary divisors belong to a finite (resp. DCC) set. In this talk, we will prove this conjecture when the ambient spaces are analytically bounded. We will also explore the relation between local volumes, log canonical thresholds and certain boundedness conditions related to \epsilon-plt blow-ups. This is based on ongoing joint work with Jingjun Han and Yuchen Liu.

**Carlo Scarpa** - The Hitchin-cscK system

A classic result in the study of Kähler metrics with special curvature properties is that the cscK equation can be realized as the moment map equation for an infinite-dimensional Kähler reduction. Building on a result by Biquard and Gauduchon we present a natural hyperkähler extension of this moment map picture, obtaining a new system of equations reminiscent of Hitchin's equations for Higgs bundles. This generalizes a previous result on complex curves by Donaldson to complex manifolds of arbitrary dimension.

**Hendrik Suess **- On the K-stability of G-varieties of complexity one

Abstract: The complexity of an action of a reductive group on a variety is the codimension of a generic orbit of the corresponding Borel subgroup. Normal G-varieties of complexity 0 are called spherical. If G=T is a torus, then the variety is called toric. In my talk I am trying to present a melange of Thibaut Delcroix' approach to the K-stability of spherical varieties and my earlier work on T-varieties of complexity one. If time permits, I will also talk about the application of these ideas to some SL2-threefolds. This is joint work with my PhD student Jack Rogers and part of a larger joint project with plenty of collaborators.

**Salvatore Tambasco **- On the volume of some Fano K-moduli spaces

Abstract: We compute the CM volume (that is the degree of the descended CM line bundle, which could be also understood differential geometric as the Weil-Petersson volume) of the Fano K-moduli space of quartic del Pezzo varieties in any dimension and of the K-moduli space of the log Fano hyperplane arrangements.

**Chenyang Xu **-** **K-stability of Fano varieties: an algebraic theory

Abstract: While K-stability was first defined to characterize whether a complex variety has a canonical metric, a purely algebraic research for Fano varieties has blossomed in higher dimensional geometry in recent years. In this talk, I will survey the progress on people’s understanding of the topic, especially on different characterizations of K-stability and using it to construct the K-moduli space of Fano varieties.

**Ziquan Zhuang** - Compatible divisors

Abstract: In this talk, I'll introduce the notion of compatible divisors. This is a special class of basis type divisors that contains many interesting information of a Fano variety. I'll then discuss some applications of compatible divisors, focusing on some recent progress on the K-stability of explicit Fano varieties and on the algebraic theory of K-stability. Partly based on joint work with Hamid Ahmadinezhad