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ICCM Lectures on Geometry

## 报告摘要 Abstract

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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ICCM Lectures on Geometry

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Title: Finite ball quotients and algebraicity of the Bergman kernel

Speaker: Prof. Hang Xu (UC San Diego)

Time: 10:00 -11:00 (Friday, 2020-09-25)

Abstract: The Bergman kernel is an important biholomorphic invariant of domains in $\mathbb{C}^n$ and, more generally, of complex analytic spaces. It is a classical problem to characterize simple “model” domains by properties of their Bergman kernels or Bergman metrics. In this talk, we shall discuss a characterization of two dimensional finite ball quotients by algebraicity of their Bergman kernels, and some properties of the Bergman metrics on finite ball quotients. This is a joint work with P. Ebenfelt and M. Xiao.

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Title: Automorphism groups of smooth hypersurfaces

Speaker: Prof. Xun Yu (Tianjin University)

Time: 10:00 -11:00 (Friday, 2020-09-11)

Abstract: I will discuss automorphism groups of smooth hypersurfaces in the projective space and explain an approach to classify automorphism groups of smooth quintic threefolds and smooth cubic threefolds. This talk is based on my joint works with Professor Keiji Oguiso and Li Wei.

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Title: Almost complex Hodge theory

Speaker: Prof. Weiyi Zhang (The Univeristy of Warrick, UK)

Time: 16:30 -17:30 (Saturday, 2020-09-05)

Abstract: In this talk, I will introduce an effective method to solve the $\bar\partial$-harmonic forms on the Kodaira-Thurston manifold endowed with an almost complex structure and an Hermitian metric. Using the Weil-Brezin transform, we reduce the elliptic PDE system to countably many linear ODE systems. By studying the Stokes phenomenon on linear ODE systems, we reduce the problem of finding $\bar\partial$-harmonic forms to a generalised Gauss circle problem.

We show how this is applied to almost complex Hodge theory. In particular, we answer a question of Kodaira and Spencer in Hirzebruch's 1954 problem list that Hodge numbers can vary with different choices of Hermitian metric. This is a joint work with Tom Holt.

Time: 10am -11am (Friday, 2020-08-28)

Title: Positivity in hyperkaehler manifolds via Rozansky—Witten theory

Speaker: Prof. Chen Jiang (Shanghai Center for Mathematical Sciences, Fudan University)

Abstract: For a hyperk\"{a}hler manifold $X$ of dimension $2n$, Huybrechts showed that there are constants $a_0, a_2, \dots, a_{2n}$ such that $$\chi(L) =\sum_{i=0}^n\frac{a_{2i}}{(2i)!}q_X(c_1(L))^{i}$$for any line bundle $L$ on $X$, where $q_X$ is the Beauville--Bogomolov--Fujiki quadratic form of $X$. Here the polynomial $\sum_{i=0}^n\frac{a_{2i}}{(2i)!}q^{i}$ is called the Riemann--Roch polynomial of $X$.

In this talk, I will discuss recent progress on the positivity of coefficients of the Riemann--Roch polynomial and also positivity of Todd classes. Such positivity results follows from a Lefschetz-type decomposition of the root of Todd genus via the Rozansky—Witten theory.

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Time: 10am -11am (Friday, 2020-08-14)

Title: Counterexamples to Fujita's conjecture in positive characteristic

Speaker: Prof. Yi Gu (Suzhou University)

Abstract: In this talk, we shall present counterexamples to Fujita’s conjec ture in positive characteristic. More precisely, given any fifield k of positive characteristic and any integer n ∈ N+, we construct a smooth projective surface S over k along with an ample line bundle L on it so that the adjoint line bundle KS + nL is not free of base points. This is a joint work with Lei Zhang and Yongming Zhang.

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Date: 10am -11am (Friday, 2020-08-07)

Title: An
eigenvalue estimate for the $\bar{\partial}$-Laplacian associated to a nef line
bundle

Speaker: Prof. Jingcao Wu (Fudan University)

Abstract: The asymptotic estimate for the order of the cohomology group $H^{p,q}(X,L^k)$ is a complicated problem in complex geometry.

In this lecture, we will follow B. Berndtsson’ s idea to make an approach on the estimate when $L$ is nef.

First we develop the harmonic theory associated with a nef line bundle.

Then we give an estimate of the number of the eigenforms.

In particular, when eigenvalue equals zero, it will lead to the asymptotic estimate for the order of the corresponding cohomology group.

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Time: 10am -11am (Friday, 2020-07-31)

Title: Generic
scarring for minimal hypersurfaces along stable hypersurfaces

Speaker: Professor Antoine Song (UC Berkeley)

Abstract: Minimal hypersurfaces are natural geometric analogues of eigenfunctions of the Laplacian and a problem of interest is the study of their spatial distribution in the ambient manifold.

I will discuss a joint work with Xin Zhou, where we prove that in a generic closed Riemannian manifold of low dimension, any 2-sided stable minimal hypersurface is the "scarring limit" of a sequence of minimal hypersurfaces whose index and area diverge to infinity.

This phenomenon contrasts with the previously existing generic equidistribution result.

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Date: 10am -11am (Thursday, 2020-07-23)

Title: Reflexive sheaves, Hermitian-Yang-Mills connections, and tangent cones

Speaker: Professor Song Sun (UC Berkeley)

Abstract: The Donaldson-Uhlenbeck-Yau theorem relates the existence of Hermitian-Yang-Mills connections over a compact Kahler manifold with algebraic stability of a holomorphic vector bundle. This has been extended by Bando-Siu in 1994 to a class of singular Hermitian-Yang-Mills connections on reflexive sheaves. We study tangent cones of these singular connections in the geometric analytic sense, and show that they can be characterized in terms of new algebro-geometric invariants of reflexive sheaves. Based on joint work with Xuemiao Chen (University of Maryland).

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Date: 2020-07-17

Title: M-theory is time-reversal invariant

Speaker: Dan Freed (University of Texas, Austin)

Abstract: In joint work with Mike Hopkins we prove that there is no parity anomaly in M-theory in the low-energy field theory approximation.

There are two sources of anomalies: the Rarita-Schwinger field and the cubic form for the C-field. I will explain the general principles behind these anomalies, since they apply in many problems. Then I'll turn to the specific computations we did to verify this anomaly cancellation. They include topologial and geometric methods for computing eta-invariants as well as homotopy-theoretic techniques for computing bordism groups.

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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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