# Programme

Une réception vin et fromage aura lieu le mardi 18 juin après la dernière conférence à 16h30 dans le PK-5115, situé dans le Pavillon Président-Kennedy (201, ave du Président-Kennedy).

## Conférences

### Ronan Conlon

*A family of Kahler flying wing steady Ricci solitons*

Steady Kahler-Ricci solitons are eternal solutions of the Kahler-Ricci flow. I will present new examples of such solitons with strictly positive sectional curvature that live on \(C^n\) and provide an answer to an open question of H.-D. Cao in complex dimension \(n>2\). This is joint work with Pak-Yeung Chan and Yi Lai.

### Huai-Dong Cao

*Deformation of Fano Manifolds and Weil-Petersson metrics*

In this talk, I shall first discuss small deformations of Fano Kähler-Einstein (KE) manifolds with non-discrete automorphisms and the Weil-Petersson metric on the space of Fano KE manifolds. I then explain how to define a Weil–Petersson type metric on the space of Fano Kähler–Ricci solitons and provide a necessary and sufficient condition on when such a metric is independent of the choices of Kähler–Ricci soliton metrics. This is based on joint work with Xiaofeng Sun, S.-T. Yau, and Yingying Zhang.

### Charlie Cifarelli

*Explicit complete Calabi-Yau metrics and Kähler-Ricci solitons*

Since Calabi's original paper, the technique now known as the Calabi ansatz has been a key starting point for many geometric constructions in Kähler geometry. Among other things, Calabi showed us how to use this technique to construct a complete Ricci-flat metric on the canonical bundle \(K_B\) of a Kähler-Einsten Fano manifold \(B\), generalizing some well-known examples coming from physics. Starting with the work of Cao, it is now well-understood by the work of Cao, Feldman-Ilmanen-Knopf, Futaki-Wang, and Chi Li, that the same technique can be used to construct complete Kähler-Ricci solitons the total space of a line bundle \(L\) over \(B\) with \(L^k = K_B\) for \(k\) rational. These are shrinking if \(k < 1,\) steady if \(k = 1\), and expanding if \(k > 1\). Moreover, Chi Li generalized this same construction to the total space of the self-direct sum line bundle \(E = L\oplus ... \oplus L \to B,\) under a similar topological condition on \(E\).

I will present on some work in progress generalizing these results to the situation of more general direct sum line bundles \(E = L^{a_1} \oplus ... \oplus L^{a_k} \to B,\) using the theory of hamiltonian 2-forms, introduced by Apostolov-Calderbank-Gauduchon-Tønneson-Friedman. This generalizes the examples listed above, as well as recent work with Apostolov constructing complete Calabi-Yau metrics and steady solitons on \(C^n,\) as well as producing infinitely many new examples. We obtain complete Calabi-Yau metrics, as well as Kähler-Ricci solitons, coming in two distinct geometric flavors which we call Type 1 and Type 2. In the Type 1 cases, the Calabi-Yau metrics and expanding/steady solitons are known from the literature, whereas the shrinking solitons are new to our knowledge. In the Type 2 case, we only obtain Ricci-flat metrics and steady solitons, which are both new to our knowledge.

### Ruadhai Dervan

*The universal structure of moment maps in complex geometry*

Much of complex geometry is motivated by the question of linking the existence of solutions to geometric PDEs (producing “canonical metrics”) to stability conditions in algebraic geometry. I will address a basic question in this direction: what is the recipe to actually produce geometric PDEs in complex geometry which have links to stability conditions? The construction will be geometric, using a combination of universal families and tools from equivariant differential geometry. This is joint work with Michael Hallam.

### Paul Gauduchon

*Extremal Kähler metrics and gravitational instantons*

Kähler metric is called *extremal* if its scalar curvature is a *Killing potential*, i.e. the moment relative to the Kähler form of a Hamiltonian Killing vector field; it is called *toric extremal* if the latter belongs to a maximal, effective Hamiltonian toric action preserving the whole Kähler structure. The existence of such a Kähler structure in the conformal class of a class of four-dimensional *gravitational instantons* of ALF type, including the Euclidean version of well-known Lorentzian spaces, as well as the one-parameter family of instantons discovered in 2011 by Yu Chen and Edward Teo, plays a prominent rôle in its eventual complete classification, including a new descriptiont of the Chen–Teo instanton, presented in a recent joint paper with Olivier Biquard.

### Weiyong He

*Geometric flows related to isotropy problems in symplectic four manifolds*

We discuss some recent developments of two geometric flows, nonlinear Hodge flow and hypersymplectic flow on a symplectic four manifold, to address the isotropy problems of a symplectic form in a given class.

### Yoshinori Hashimoto

*Non-Archimedean aspects of the modified Mabuchi energy*

Variational aspects of the Mabuchi energy and constant scalar curvature Kähler metrics have been much clarified in recent years, thanks to the deep understanding of the completion of the space of Kähler metrics in both Archimedean and non-Archimedean settings. The modified Mabuchi energy is an energy functional whose critical point is the extremal metric. In this talk, we present some results that extend some known properties of the Mabuchi energy to the modified Mabuchi energy.

### Simon Jubert

*Coercivity of the weighted Mabuchi functional*

In this talk, we will discuss the existence of a large class of Kahler metrics with a special curvature condition, called weighted cscK metric. These metrics were introduced by Lahdili with the goal of unifying several natural problems in Kahler geometry such as finding cscK metrics, Kahler-Ricci solitons (and their weighted extensions), Sasaki-Einstein metrics, and others. I will present recent work (not yet on arXiv) in collaboration with Lahdili and Di Nezza, where we demonstrate that the coercivity of the weighted Mabuchi functional implies the existence of a weighted cscK metric (the converse was obtained in previous work with Apostolov and Lahdili). First, I will introduce the notion of weighted cscK metric. Next, I will discuss applications of the result. Finally, I will explain the proof strategy, which is a generalization of that used by Chen and Cheng in the cscK case.

### Abdellah Lahdili

*The weighted Hermite-Einstein equations*

In this talk, I will report on my joint work with Michael Hallam. We introduce a weighted generalization of the Hermite-Einstein equation for torus equivariant vector bundles over a compact Kahler manifold. As a main result, we prove the weighted Kobayashi–Hitchin correspondence, namely that a T-equivariant vector bundle admits a weighted Hermite–Einstein metric if and only if the vector bundle is weighted slope polystable. We illustrate our theory by showing that the weighted slope stability of some natural extensions of the holomorphic tangent bundle of a Fano manifold can be used to bound the greatest positive lower bound of the Bakry-Emery-Ricci curvature on Kahler metrics in the anticanonical polarization.

### Eveline Legendre

*The Einstein-Hilbert functional and the Donaldson-Futaki invariant*

I will present briefly a correspondence between the constant scalar curvature Kähler problem on a polarized manifold and a family of CR-Yamabe problems on the associated circle bundle. I will focus on the algebraic part of the story, where we establish a direct link with the Collins-Szekelyhidi approach to the Donaldson-Futaki invariant of polarized cones. This is a joint work with Abdellah Lahdili and Carlo Scarpa.

### Chi Li

*Valuations and Kahler-Ricci flows*

I will first survey application of algebraic minimization problem for valuations to the study of normalized Kahler-Ricci flow on Fano varieties. We then propose a conjectural algebraic description of shrinking Kahler-Ricci solitons along Kahler-Ricci flow at finite time singularities by using certain minimizing valuations.

### Nick McCleerey

*Lelong Numbers for m-Subharmonic Functions*

The Lelong number plays an important role in pluripotential theory as a measurement of the singularities of a psh function at a point. We discuss analogous measurements for m-subharmonic functions, where the situation is greatly complicated and extremely little is known; a key difficulty is that one can no longer consider only isolated singularities. We address the case of singularities located along a (local) complex submanifold, and arrive at some satisfactory answers when the codimension is sufficiently small.

### Yasufumi Nitta

*Extremal Kähler metrics and Mabuchi solitons on Fano manifolds*

A purpose of this talk is to clarify the relation between two kinds of canonical Kähler metrics on Fano manifolds, the Calabi’s extremal Kähler metrics and the Mabuchi solitons. These are both generalizations of the concept of Kähler-Einstein metrics. Mabuchi showed that the existence of Mabuchi solitons implies that of extremal Kähler metrics representing the first Chern class. It is also known that the converse is true for Fano manifolds of dimension up to two. Based on the above, we present examples of Fano manifolds in all dimensions greater than two which admit extremal Kähler metrics in every Kähler class, but do not admit Mabuchi solitons. A key ingredient is the Mabuchi constant, a holomorphic invariant of a Fano manifold, and its product formula. Furthermore, we show that for Fano manifolds whose Mabuchi constants are less than 1, the existence of extremal Kähler metrics representing the first Chern class implies that of Mabuchi solitons.

### Sean T. Paul

*Constant Scalar Curvature Metrics on Algebraic Manifolds*

According to the Yau-Tian-Donaldson conjecture, the existence of a constant scalar curvature Kähler (cscK) metric in the cohomology class of an ample line bundle L on a compact complex manifold X should be equivalent to an algebro-geometric “stability condition” satisfied by the pair (X,L). The cscK metrics are the critical points of Mabuchi’s K-energy functional M, defined on the space of Kähler potentials, and an important result of Chen-Cheng shows that cscK metrics exist iff M satisfies a standard growth condition (coercivity/properness). Recently the speaker has shown that the K-energy is indeed proper if and only if the polarized manifold is stable. Our stability condition is closely related to the classical notion of Hilbert-Mumford stability as well as “K-Stability”. The speaker will give a non-technical account of the many areas of mathematics that are involved in the proof. In particular, he hopes to discuss the surprising role played by arithmetic geometry in the spirit of Arakelov, Faltings, and Bismut-Gillet-Soule.

### Rosa Sena-Dias

*Einstein metrics from the Calabi ansatz via Derdzinski duality*

One of the main sources of examples of Einstein metrics has been Kähler geometry. Yet we know that several Kähler manifolds do not carry Kähler-Einstein metrics. It is therefore natural to relax the Kähler condition in looking for Einstein metrics. In the 80's, Derdzinski obtained a local construction for 4d conformally Kähler, Einstein metrics using extremal Kähler metrics. Together with G. Oliveira, we set out to use Derdzinski's methods on a class of extremal Kähler metrics arising from an ansatz due to Calabi. In this talk, I shall report on our findings. I will start by reviewing Derdzinski's results and the Calabi anstaz. Then I shall explain how our construction includes Einstein fillings of \(S^3/\mathbb{Z}_m.\) I also discuss a cone-angle Einstein deformation of the Page metric on \(\mathbb{CP}^2\sharp \overline{\mathbb{CP}^2}.\) If time permits, I shall describe a limiting procedure yielding an asymptotically hyperbolic Einstein metric not arising from the Derdzinski construction. This is joint work with G. Oliveira.

### Christina Tønnesen-Friedman

*Weighted Extremal Kähler Twins*

This talk will describe some ongoing joint work with C. P. Boyer, E. Legendre, and H. Huang. I will discuss the existence of what we call *weighted extremal twins* in which a Kähler metric is weighted extremal (for now, in the sense of V. Apostolov and D. M. J. Calderbank, rather than the more general definition due to A. Lahdili) with respect to two different (even up to rescale) Killing potentials. This generalizes the twinning phenomenon appearing among certain strongly Hermitian solutions (found by C. LeBrun) to the Einstein-Maxwell equations on the first Hirzebruch surface. Under appropriate conditions, the weighted extremal twins will correspond to two extremal Sasaki structures both compatible with a fixed pseudo-convex CR structure, (*D,J*), of Sasaki type on a compact smooth manifold *M*. With some very special (Bochner-flat) exceptions, the existence of twins is probably a discrete phenomenon. In the Sasaki setting, this can be contrasted with the well known fact (proven by C. P. Boyer, K. Galicki, and S. Simanca) that the set of extremal Sasakian structures is open when one can deform within the isotopy classes.

### Xiaowei Wang

*Moment map and convex functions*

In this talk, we discuss a setup of moment maps coupled with an \(Ad_K\)-invariant convex function \(f\) on \(\mathfrak{k}^*\), the dual of Lie algebra of \(K\), and study the structure of the stabilizer of the critical point of \(f\) composing with the moment map.

As an outcome, we are able to obtain a general Calabi-Matsushima decomposition based only on the convexity of \(f\) so that all existing Calabi-Matsushima type of decomposition theorems fall into this new framework. This work is motivated by the work of Donaldson (2017) together with the goal of finding a natural interpretation of Tian-Zhu’s Calabi-decomposition for Käher-Ricci solitons (2002), which are examples of infinite dimensional version of our setting. (Joint with King-Leung Lee and Jacob Sturm)