RECENT ADVANCES IN KÄHLER GEOMETRY IN CONJUNCTION WITH THE 30 th SHANKS LECTURE

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1 RECENT ADVANCES IN KÄHLER GEOMETRY IN CONJUNCTION WITH THE 30 th SHANKS LECTURE VANDERBILT UNIVERSITY NASHVILLE, MAY 18-22, 2015 Shanks Lecture: Shing-Tung Yau, Harvard University, USA Title: Perspectives in Kähler Geometry Abstract: In this talk, I shall give some historical account of Kähler geometry in the past century and the possible direction that it may be going. Vestislav Apostolov, Université du Québec à Montréal, Canada Title: Higher codimension CR structures, Levi Kähler reduction, and toric geometry Abstract: CR structures in co-dimension one play an increasingly important role in differential geometry, deeply intertwined with Kähler and Sasaki geometry. In this talk, based on a joint work in progress with D. Calderbank, P. Gauduchon and E. Legendre, I will discuss the relation between CR structures of higher co-dimension and Kähler geometry, through a process called Levi Kähler reduction. I will focus in particular on the toric case, where the Levi Kähler reduction provides a new way to construct distinguished Kähler metrics on toric varieties. For certain complex toric orbisurfaces, we show that their extremal Kähler metrics can be obtained as a Levi Kähler reduction of a product of CR 3-spheres, thus generalizing the Bryant Webster Bochner-flat Kähler metrics on the weighted projective spaces. Claudio Arezzo, ICTP, Italy Title: Extremal and Kcsc resolutions Abstract: In this talk I ll present two gluing results, both proved jointly with R. Lena and L. Mazzieri, about existence of extremal and Kähler constant scalar curvature metrics on resolutions of orbifolds with isolated singularities. 1

2 Hugues Auvray, Université Paris-Sud, France Title: Asymptotics of extremal Kähler metrics of Poincaré type Abstract: Given a simple normal crossing divisor D in a compact Kḧler manifold X, one defines Poincaré type Kähler metrics on X \ D as smooth Kähler metrics on X \ D roughly modelled on a product (Poincaré cusp metrics tranverse to the divisor) (smooth metrics along the divisor) near D. In this regard, the main purpose of this talk will be the following result: under some special curvature assumption such as constant scalar curvature or extremality in the sense of Calabi, Poincaré type Kähler metrics enjoy very sharp asymptotic description near the divisor. We will see in particular how the existence of such metrics with special curvature outside the divisor implies the existence of metrics with special curvature on the divisor. Charles P. Boyer, University of New Mexico, USA Title: Moduli Problems in Sasakian Geometry Abstract: I want to focus my talk on several foundational problems, mainly in the case of positive Sasakian structures. (1) Given a manifold determine how many inequivalent contact structures of Sasaki type there are. (2) Given an isotopy class of contact structures determine the moduli space of compatible positive Sasaki classes. (3) Determine the Sasakian structures within a fixed underlying pseudoconvex CR structure. (4) Determine those Sasakian structures with extremal representatives. (5) Given extremal Sasakian structures when do they have constant scalar curvature? (6) In the monotone case discuss the moduli space of Sasaki-Einstein structures. Of course we can give only partial answers to these problems and only in special cases. My talk is based in part on joint work with several colleagues: Leonardo Macarini, Justin Pati, Christina Tønnesen-Friedman, and Otto van Koert. Xiuxiong Chen, Stony Brook University, USA Title: On a new continuity path for constant scalar curvature Kähler metrics Abstract: In this talk, we will discuss a new continuity path which links the solution of csck metrics with a solution of certain 2nd fully nonlinear equation. This is largely an expository talk where we explain various aspects of geometric and analysis centered around this new path.

3 Akito Futaki, University of Tokyo, Japan Title: Weighted Laplacian on real and complex complete metric measure spaces Abstract: Weighted Laplacian appears in real Riemannian manifolds and complex Kähler manifolds with weighted measures in natural manners. While there is not much difference on compact manifolds, we see, on noncompact complete manifolds, there is a significant difference between real and complex manifolds. Henri Guenancia, Stony Brook University, USA Title: Kähler-Einstein metrics: from cones to cusps Abstract: In this talk, I will explain the following result and outline its proof: Let X be a compact Kähler manifold and D a smooth divisor such that K X + D is ample. Then the negatively curved Kähler-Einstein metric with cone angle β along D converges to the cuspidal Kähler-Einstein metric of Tian-Yau when β tends to zero. Hans-Joachim Hein, University of Maryland, USA Title: Asymptotically conical Calabi-Yau manifolds Abstract: A Riemannian cone is a warped product of the form dr 2 + r 2 g L for some closed Riemannian manifold (L, g L ) called the link of the cone. We say that such a cone is Calabi-Yau if the cone metric is Ricci-flat Kähler with a parallel holomorphic volume form; Sasaki-Einstein manifolds are by definition the links of Calabi-Yau cones. In this talk, I will explain a method to determine all complete noncompact Calabi-Yau manifolds asymptotic to some given Calabi-Yau cone at infinity. Given the cone, this reduces the classification of these manifolds to a purely algebraic problem, which can be often be solved explicitly. Joint work with Ronan Conlon. Claude LeBrun, Stony Brook University, USA Title: The Einstein-Maxwell Equations and Conformally Kähler Geometry Abstract: Any constant-scalar-curvature Kähler (csck) metric on a complex surface may be viewed as a solution of the Einstein-Maxwell equations, and this allows one to produce solutions of these equations on any 4-manifold that arises as a compact complex surface with b 1 even. However, not all solutions of the Einstein-Maxwell equations on such manifolds arise in this way; new examples can be constructed by means of conformally Kähler geometry.

4 Chi Li, Stony Brook University, USA Title: Moduli space of smoothable Kähler-Einstein Q-Fano varieties Abstract: I will talk about the construction of moduli spaces of smoothable Kähler-Einstein Q-Fano varieties. I will then discuss the projectivity of moduli spaces, and prove in particular the quasi-pojectivity of moduli spaces of Kähler-Einstein Fano manifolds. This is joint work with Xiaowei Wang and Chenyang Xu. Toshiki Mabuchi, Department of Mathematics, Osaka University, Japan Title: Strong K-stability for general polarizations Abstract: In this talk, we discuss the Yau-Tian-Donaldson Conjecture for general polarizations from the viewpoints of strong K-stability introduced in [1]. The natural questions one can ask are the following: (1) How strong is this stability notion? (2) Is this notion closely related to the existence of constant scalar curvature Kähler metrics in the polarization class? The purpose of this talk is to explain our approach (and some related results) in answering these questions. References [1] T. Mabuchi: The Donaldson-Futaki invariant for sequences of test configurations, in Geometry and Analysis on Manifolds, Progr. Math. 308, Birkhäuser Boston, 2015, Nefton Pali, Université Paris-Sud, France Title: Variational stability of Kähler Ricci Solitons Abstract: I will explain the solution of the variational stability problem for Kähler Ricci Solitons Julius Ross, University of Cambridge, UK Title: The Complex Homogeneous Monge-Ampère Equation and the Hele-Shaw Flow Abstract: (This is joint work with David Witt-Nyström) As is well known, the Dirichlet problem for the Complex Homogeneous Monge-Ampère Equation corresponds to finding geodesics in the space of Kähler potentials which has motivated much of the study of this problem. In this talk we will discuss a certain duality between a particular form of this problem and a certain flow in the plane called the Hele-Shaw flow. In fact this duality holds between what should be considered the simplest (non-trivial) version of this Dirichlet problem and the simplest (non-trivial) planar flow. As such we can get a rather concrete understanding of the associated geodesic, in particular with regard to its regularity.

5 Cristiano Spotti, University of Cambridge, UK Title: Kähler-Einstein metrics on smoothable K-polystable Fano varieties Abstract: I will discuss the existence of Kähler-Einstein metrics on Q-smoothable K-polystable Q-Fano varieties and show its relevance in the study of compactified moduli spaces of Kähler-Einstein manifolds with positive curvature scalar. Joint work with Song Sun and Chengjian Yao. Song Sun, Stony Brook University, USA Title: Gromov-Hausdorff limits of Kähler-Einstein manifolds Abstract: In this talk I will discuss algebraic structure on the Gromov-Hausdorff limits of Kähler-Einstein manifolds, presenting some old and new results. This is based on joint work with Simon Donaldson. Gábor Székelyhidi, University of Notre Dame, USA Title: Kähler-Einstein metrics along the smooth continuity method Abstract: I will discuss an equivariant version of the Yau-Tian-Donaldson conjecture on the existence of Kähler-Einstein metrics, strengthening the result of Chen-Donaldson-Sun. This potentially gives new examples of Kähler-Einstein manifolds, and it can also be applied to the existence problem for Kähler-Ricci solitons. It is joint work with Ved Datar. Craig van Coevering, U.S.T.C., China Title: Some results related to Stability in Sasakian Geometry Abstract: We consider some new results in Sasakian geometry involving K-stability and K- energy, which have been so profitable in Kähler geometry. Adapting analytic arguments of R. Berman and B. Berndtsson to the transversally Kähler geometry of a Sasakian manifolds we are able prove the convexity of the K-energy along weak geodesics. From this it follows that constant scalar curvature Sasakian metrics (cscs) and more generally Sasaki-extremal metrics are unique, in a given polarization. Also, by considering a slice argument we obtain some results, known for Kähler manifolds, on the space of Sasakian structures associated to a fixed contact structure (η, ξ), namely that K-polystability is sufficient for deforming a cscs metric. The same technique also shows that small deformations of cscs structures are K-semistable, and analogously have the K-energy bounded below. There are still gaps, but a goal of this project is to construct a good moduli space of cscs structures associated to a fixed contact structure (η, ξ).

6 Valentino Tosatti, Northwestern University, USA Title: The Kähler-Ricci flow: finite-time singularities and long-time behavior Abstract: The behavior of the Ricci flow on compact Kähler manifolds is intimately related to the complex structure of the manifold. It is known that the maximal existence time of the flow can be computed from simple cohomological data. In the case when this is finite, I will describe a result (joint with T. Collins) which gives a geometric description of the set where finite-time singularities occur, answering a conjecture of Feldman-Ilmanen-Knopf and Campana. When the maximal existence time is infinite, assuming that the canonical bundle is semiample, we have a rather good understanding of the metric behavior, even when the normalized flow is collapsing (joint with B. Weinkove, X. Yang and Y. Zhang). Jeff Viaclovsky, University of Wisconsin, Madison, USA Title: Scalar-flat Kähler ALE metrics on minimal resolutions metrics Abstract: Scalar-flat Kähler ALE surfaces have been studied in a variety of settings since the late 1970s. All previously known examples have group at infinity either cyclic or contained in SU(2). I will describe an existence result for scalar-flat Kähler ALE metrics with group at infinity G, where the underlying space is the minimal resolution of C 2 /G, for all finite subgroups G of U(2) which act freely on S 3. I will also discuss a non-existence result for Ricci-flat metrics on certain spaces, which is related to a conjecture of Bando-Kasue-Nakajima. If time permits, I will also present some new examples self-dual metrics on connected sums of complex projective planes. This is joint work with Michael Lock. Bing Wang, University of Wisconsin, Madison, USA Title: The Kähler Ricci flow on Fano manifolds Abstract: Based on the compactness of the moduli of non-collapsed Calabi-Yau spaces with mild singularities, we set up a structure theory for polarized Kähler Ricci flows with proper geometric bounds. Our theory is a generalization of the structure theory of non-collapsed Kähler Einstein manifolds. As applications, we prove the Hamilton-Tian conjecture and the partial C 0 -conjecture of Tian. In this talk, we will describe the framework of the solution, explain where are the technical difficulties and show how to overcome them. This is a joint work with X.X. Chen. Ben Weinkove, Northwestern University, USA Title: Gauduchon metrics with prescribed volume form Abstract: We prove that on any compact complex manifold one can find Gauduchon metrics with prescribed volume form. This is equivalent to prescribing the Chern-Ricci curvature of the metrics, and thus solves a conjecture of Gauduchon from This is a joint work with Gábor Székelyhidi and Valentino Tosatti.

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