## Complex Geometry Seminar

## 报告人简介

**2019-12-11**

**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.

**-----------------------------------History------------------------------------**

**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.

**2019-11-27**

**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.

**2019-11-20**

**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.

** **

**2019-11-13**

**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.

**2019-11-6**

**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.

**2019-10-9**

**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.

** **

**2019-9-25**

**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.

** **

**2019-9-18**

**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.

** **

**2019-9-11**

**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 $k\geq2$.

** **

**2019-6-20**

**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.

** **

**2019-3-11**

**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.