Complex Geometry Seminar

时间: 周三15:20-16:55, 2019-9-11 ~ 12-4



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: 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.



Speaker: Dan Xie (Tsinghua University)

Title: TBA

Abstract: TBA



Speaker: Zhenlei Zhang (Capital Normal University)

Title: TBA

Abstract: TBA



Speaker: Abdellah Lahdili (Peking University)

Title: TBA

Abstract: TBA



Speaker: Ke Feng (Peking University)

Title: TBA

Abstract: TBA


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

Speaker 1: Yingying Zhang (Tsinghua University)

Title: TBA

Abstract: TBA


Speaker 2: Hikaru Yamamoto (Tokyo University of Science)

Title: TBA

Abstract: TBA




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.

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