ICCM Lectures on Geometry

主讲人:Xiaokui Yang
时间: 周五 15:00 -16:00
地点:线上

报告人简介

ICCM Lectures on Geometry

Organizers: 
Shing-Tung Yau (Harvard University)
Nai Chung Leung (Chinese University of Hong Kong)
Si Li (YMSC, Tsinghua University)
Kefeng Liu (University of California, Los Angeles)
Chin-Lung Wang (Taiwan University)
Xiaokui Yang (YMSC, Tsinghua University)

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Date: 2020-07-10

Title: On the Ohsawa-Takegoshi extension theorem

Speaker: Prof. Junyan Cao (Universite Paris 6)

Abstract: Since it was established, the Ohsawa-Takegoshi extension theorem turned out to be a fundamental tool in complex geometry.

We establish a new extension result for twisted canonical forms defined on a hypersurface with simple normal crossings of a projective manifold with a control on its L^2 norm. It is a joint work with Mihai Păun.

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Date: 2020-07-03
Title: A characterization of non-compact ball quotient
Speaker:Ya DENG (IHES)
Abstract: In 1988 Simpson extended the Donaldson-Uhlenbeck-Yau theorem to the context of Higgs bundles, and as an application he proved a uniformization theorem which characterizes complex projective manifolds and quasi-projective curves whose universal coverings are complex unit balls. In this talk I will give a characterization for quasi-projective manifolds to be uniformized by complex unit balls, which generalizes the uniformization theorem by Simpson. 


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Date:2020-6-26
Title: Localization of eta invariant
Speaker: Prof. Bo Liu (East China Normal University)
Abstract: The famous Atiyah-Singer index theorem announced in 1963 computed the index of the elliptic operator, which is defined analytically, in a topological way. In 1968, Atiyah and Segal established a localization formula for the equivariant index which computes the equivariant index via the contribution of the fixed point sets of the group action. It is natural to ask if the localization property holds for the more complex spectral invariants, e.g., eta-invariant.
The eta-invariant was introduced in the 1970's as the boundary contribution of index theorem for compact manifolds with boundary. It is formally equal to the number of positive eigenvalues of the Dirac operator minus the number of its negative eigenvalues and has many applications in geometry, topology, number theory and theoretical physics. It is not computable in a local way and not a topological invariant.
In this talk, we will establish a version of localization formula for equivariant eta invariants by using differential K-theory, a new research field in this century. This is a joint work with Xiaonan Ma.

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Date:2020-06-19

Title: On a canonical bundle formula with $\R$-coefficients

Speaker: Zhengyu Hu (Chongqing University of Technology)
Abstract:  In this talk, I will discuss a canonical bundle formula for a proper surjective morphism 
(not necessarily with connected fibers) with  $\R$-coefficients and its applications. Moreover, I will discuss the inductive property of the moduli divisor.

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Date:2020-06-13

Title: Projective manifolds whose tangent bundle contains a strictly nef subsheaf

Speaker: Wenhao Ou (AMSS)
Abstract: In this talk we will discuss the structure of projective manifold $X$ whose tangent bundle contains a locally free strictly nef subsheaf. 

We establish that $X$ is isomorphic to a projective bundle over a hyperbolic manifold. 

Moreover, if the fundamental group $\pi_1(X)$ is virtually abelian, then $X$ is isomorphic to a projective space. 
This is joint work with Jie Liu (MCM) and Xiaokui Yang (YMSC).


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Date:2020-06-05

Title: Complex structures on Einstein four-manifolds of positive scalar curvature

Speaker: Peng Wu (Fudan University)
Abstract: In this talk we will discuss the relationship between complex structures and Einstein metrics of positive scalar curvature 
on four-dimensional Riemannian manifolds. One direction, that is, when a four-manifold with a complex structure admits a compatible Einstein metric of positive scalar curvature has been answered by Tian, LeBrun, respectively. We will consider the other direction, that is, when a four-manifold with an Einstein metric of positive scalar curvature admits a compatible complex structure. We will show that if the determinant of the self-dual Weyl curvature is positive then the manifold admits a compatible complex structure.
Our method relies on Derdzinski's proof of the Weitzenbock formula for the self-dual Weyl curvature.


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