2020-06-30 来源：数学科学研究中心

第七周论文讨论班

时间：北京时间69日上午9:00-10:00

时间：北京时间62日上午9:00-10:00

ICCM:

14.

The following is the information of the lecture on geometry:

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18430443981
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ICCM Lectures on Geometry
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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.

13.

The following is the information of the lecture on geometry:

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ICCM Lectures on Geometry

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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.

12.

The following is the information of the lecture on geometry:

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ICCM Lectures on Geometry

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Title:  Positivity in hyperkaehler manifolds via RozanskyWitten theory

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

Time: 16:00 -17:00 (Friday, 2020-08-28)

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 RozanskyWitten theory.

11.

10.

Thefollowingistheinformationofthelectureongeometry:Zoom:ID：18430443981Password：ICCMDetailsareasbelow:ICCMLecturesonGeometry----------------------------------------------------------------------Title:Aneigenvalueestimateforthe$\bar{\partial}$-LaplacianassociatedtoaneflinebundleSpeaker:Prof.JingcaoWu(FudanUniversity)Time:10am-11am(Friday,2020-08-07)Abstract:Theasymptoticestimatefortheorderofthecohomologygroup$H^{p,q}(X,L^k)$isacomplicatedproblemincomplexgeometry.Inthislecture,wewillfollowB.Berndtsson’sideatomakeanapproachontheestimatewhen$L$isnef.Firstwedeveloptheharmonictheoryassociatedwithaneflinebundle.Thenwegiveanestimateofthenumberoftheeigenforms.Inparticular,when eigenvalueequalszero,itwillleadtotheasymptoticestimatefortheorderofthecorrespondingcohomologygroup.

9.

8.

The following is the information of the lecture on geometry:

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ICCM Lectures on Geometry

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

Speaker:  Professor Song Sun (UC Berkeley)

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

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).

ICCM:

7.

The following is the information of the lecture on geometry:

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ICCM Lectures on Geometry

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Title:  M-theory is time-reversal invariant

Speaker:  Dan Freed (University of Texas, Austin)

Time: 9:30am -10:30am (Friday, 2020-07-17)

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.

ICCM:

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5.

The following is the information of the lecture on geometry:

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ICCM Lectures on Geometry

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Title: A characterization of non-compact ball quotient

Speaker: Prof. Ya Deng (IHES)

Time: 3pm -4pm (Friday, 2020-07-03)

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.

4.

The following is the information of the lecture on geometry:

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ICCM Lectures on Geometry

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Title: Localization of eta invariant

Speaker: Prof. Bo Liu (East China Normal University)

Time: 10:00 am -11:00 am (Friday, 2020-06-26)

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.

3.

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ICCM Lectures on Geometry

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

Speaker: Zhengyu Hu (Chongqing University of Technology)

Time: 10:00 am -11:00 am (Friday, 2020-06-19)

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.

2.

The ICCM lecture on Geometry is rescheduled this week. It is at 10:00 am -11:00 am (Saturday, 2020-06-13).

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ICCM Lectures on Geometry

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

Speaker: Wenhao Ou (AMSS)

Time: 10:00 am -11:00 am (Saturday, 2020-06-13)

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).

1.

ICCM Lectures on Geometry

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

Speaker: Peng Wu (Fudan University)

Time: 10:00 am -11:00 am (Friday, 2020-06-05)

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.