Complex Geometry Seminar

组织者 Organizer:二木昭人(组织者)
时间 Time: 周三15:20-16:55, 2019-9-11 ~ 12-11
地点 Venue:清华大学近春园西楼第三会议室

报告摘要 Abstract


Speaker: Chuyu Zhou (Peking University)

Title: K-stability under a view point of Birational Geometry

Abstract: In this talk, I will give an introduction of K-stability by language of Birational Geometry, and give some new criteria for uniformly K-stability. I will also introduce a local stability theory developed by Chi Li, Chenyang Xu, Xiaowe Wang, etc. and then complete local special test configuration theory.



2019-12-4, 15:20 -16:20 and 16:30 -17:30

Speaker 1: Yingying Zhang (Tsinghua University)

Title: Obstructions to the existence of coupled Kahler-Einstein metrics

Abstract: Coupled K\"ahler-Einstein metric was introduced by Hultgren and Witt-Nystrom. It is a new type canonical metric generalizing K\"ahler-Einstein metrics or K\"ahler Ricci solitons on a compact Kahler manifolds. In this talk, we will discuss two obstructions to the existence of the coupled K\"ahler-Einstein metrics. One is the Matsushima type obstruction, which is about the reductivity of the Lie algebra of automorphism. Another is an extension of original Futaki invariant. We will also discuss the localization formula of this generalized Futaki invariant and use it to verify the existence of coupled K\"ahler-Einstein metric on an example.  (This is the joint work with Professor Akito Futaki.)

Speaker 2: Hikaru Yamamoto (Tokyo University of Science)

Title: The moduli space of deformed Hermitian Yang-Mills connections

Abstract: The deformed Hermitian Yang-Mills connection was defined by Leung, Yau and Zaslow as the Fourier-Mukai transform of a special Lagrangian submanifold in the context of mirror symmetry. It is well-known as a work of McLean that the moduli space of special Lagrangian submanifolds is a smooth finite dimensional manifold and the dimension is the first Betti number of the ambient space. In this talk, I will show that this is also true for the moduli space of deformed Hermitian Yang-Mills connections. First, I will give some background of this study and if time permits I will introduce a similar result for G2 case. This is joint work with Kotaro Kawai.



Speaker: Ke Feng (Peking University)

Title: The Dirichlet Problem of Fully Nonlinear Equations on Hermitian Manifolds

Abstract: We consider the Dirichlet problems for a class of fully nonlinear equations on Hermitian manifolds and derive a priori C2 estimates which depend on the initial data on manifolds, the admissible subsolutions and the upper bound of the gradient of the solutions. In some special cases, we also obtain the gradient estimates, and hence we can solve, for example, the Dirichlet problems of the (strongly) Gauduchon (resp. the balanced) metrics on Hermitian (resp. Kahler) manifolds with admissible subsolutions. We also derive an alternative proof of the upper bound of the gradient of the solutions to the equations related to the (n-1)-plurisubharmonic functions and to the Gauduchon conjecture on the compact Hermitian manifolds without boundary. This is a joint work with Huabin Ge and Tao Zheng.



Speaker: Abdellah Lahdili (Peking University)

Title: Kaehler metrics with constant weighted scalar curvature and weighted K-stability

Abstract: We will introduce a notion of a Kaehler metric with constant weighted scalar curvature on a compact Kaehler manifold X, depending on a fixed real torus T in the reduced group of automorphisms of X, and two smooth (weight) functions defined on the momentum image of X. We will also define a notion of weighted Mabuchi energy adapted to our setting, and of a weighted Futaki invariant of a T-compatible smooth Kaehler test configuration associated to (X, T). After that, using the geometric quantization scheme of Donaldson, we will show that if a projective manifold admits in the corresponding Hodge Kaehler class a Kaehler metric with constant weighted scalar curvature, then this metric minimizes the weighted Mabuchi energy, which implies a suitable notion of weighted K-semistability. As an application, we describe the Kaehler classes on a geometrically ruled complex surface of genus greater than 2, which admits conformally Kaehler Einstein-Maxwell metrics.



Speaker: Zhenlei Zhang (Capital Normal University)

Title: Relative volume comparison of Ricci flow

Abstract: In this talk I will present a relative volume comparison of Ricci flow. It is a refinement of Perelman pseudolocality theorem. It is a joint work with Professor Tian.



Speaker: Dan Xie (Tsinghua University)

Title: Sasaki-Einstein manifolds and AdS/CFT correspondence

Abstract: Given a Sasaki-Einstein five or seven manifolds, one can construct an AdS/CFT pair which relates a string theory on  and a dual superconformal field theory. I will review some basics of Sasaki-Einstein geometry and then discuss what geometric structures are important for AdS/CFT correspondence.



Speaker: Kewei Zhang (Peking University)

Title: Delta invariant and K-stability of Fano type manifolds

Abstract: In this talk I will mainly discuss the delta invariant, which was recently introduced by Fujita-Odaka. This invariant plays important roles in the study of Kahler-Einstein problems on Fano type manifolds and it is closely related to the notion of K-stability. I will discuss various applications of this invariant, including some recent results in my joint work with Ivan Cheltsov and Yanir Rubinstein.  For instance I will give some new examples of log Fano surfaces admitting conical Kahler-Einstein metrics. Moreover, I’ll show that the delta invariant coincides with greatest Ricci lower bound of Fano manifolds.



Speaker: Yalong Shi (Nanjing University)

Title: Examples of Kahler manifolds with proper K-energy

Abstract: I shall discuss some examples of Kahler manifolds with proper K-energy, by studying the J-equation. This implies existence of cscK metrics by the work of Chen-Cheng. These are joint works with H. Li-Y. Yao, W. Jian-J. Song and C. Arezzo-A. Della Vedova.



Speaker: Laurant La Fuente-Gravy (University of Luxembourg)

Title: Moment map and closed Fedosov star products

Abstract: I will describe a moment map on the space of symplecic connections on a given closed symplectic manifold. The value of this moment map at a symplectic connection is contained in the trace density of the Fedosov star product attached to this connection. Moreover, this Fedosov star product can only be closed when the symplectic connection lies in the vanishing set of the moment map. Considering closed Kaehler manifolds, I will show that the problem of finding zeroes of the moment map is an elliptic partial differential equation. I will also discuss obstructions to the existence of zeroes of the moment map, which means obstructions to the closedness of the Fedosov star product attached to the considered Kaehler data.



Speaker: Yan Li (Peking University)

Title: Tian's $\alpha_{m,k}^{\hat K}$-invariants on group compactifications

Abstract: In this talk, we will first review Tian's $\alpha_{m,k}^{\hat K}$-invariant on a polarized manifold and a related conjecture. Then we give computation of $\alpha_{m,k}^{K\times K}$-invariants on $G$-group compactification, where $K$ denotes a connected maximal compact subgroup of $G$. Finally we prove that Tian's conjecture is true for $\alpha_{m,k}^{K\times K}$-invariant on such manifolds when $k=1$, but it fails in general by showing counter-examples when $k\geq2$.



Speaker: Satoshi Nakamura (Fukuoka University)

Title: Deformation for coupled K\”ahler Einstein metrics

Abstract: The notion of coupled K\"ahler-Einstein metrics was introduced recently by Hultgren-Nystr\"om. In this talk we discuss the deformation of coupled K\"ahler-Einstein metrics on Fano manifolds. In particular we obtain a necessary and sufficient condition for a coupled K\"ahler-Einstein metric to be deformed to a coupled K\"ahler-Einstein metric for another close decomposition for Fano manifolds admitting non-trivial holomorphic vector fields. This generalizes Hultgren-Nystr\”om's result.



Speaker: Laurent La Fuente-Gravy (University of Liege, Belgium)

Title: Deformation quantization of Kähler manifolds

Abstract: I will start by a brief introduction to deformation quantization. Then, following the work of Karabegov, I will show how star products with the separation of variable property on Kähler manifolds are parametrized by formal deformations of the Kähler form. After that, I will describe one (or two) geometric ways to obtain star products with the separation of variable property on Kähler manifolds. If time permits, I will explain the role of the scalar curvature of the Kähler manifold in the notion of trace for star products.