2022复几何青年论坛

时间：12月3日-5日。

地点：腾讯会议 429 686 832  无密码

学术委员会：

傅吉祥   复旦大学

李嘉禹   中国科学技术大学

刘克峰   美国加州大学洛衫矶分校

莫毅明   香港大学

张伟平   南开大学

郑方阳   重庆师范大学

周向宇   中国科学院

朱小华   北京大学

组织委员会：

李琼玲    南开大学   qiongling.li@nankai.edu.cn

饶胜       武汉大学    likeanyone@whu.edu.cn

万学远   重庆理工大学    xwan@cqut.edu.cn

肖建       清华大学   jianxiao@tsinghua.edu.cn

报告人：

楚健春（北京大学）

韩小利（清华大学）

李起峰（山东大学）

郦言（北京理工大学）

马梓铭（南方科技大学）

宁家福（中南大学）

孙俊（武汉大学）

汪志威（北京师范大学）

吴劲草（上海财经大学）

吴瑞聪（华东师范大学）

徐浩（浙江大学）

杨波（厦门大学）

杨晓奎（清华大学）

张德凯（上海大学）

张科伟（北京师范大学）

张振雷（首都师范大学）

赵晨（中国科学技术大学）

赵全庭（华中师范大学）

日程安排：

报告题目和摘要：

 题目: A Nakai-Moishezon type criterion for supercritical deformed Hermitian-Yang-Mills equation

报告人：楚健春（北京大学）

摘要: The deformed Hermitian-Yang-Mills equation is a fully nonlinear elliptic PDE on Kahler manifolds, which can be regarded as the complex analogue of the Lagrangian equation. In this talk, I will present a Nakai-Moishezon type criterion for supercritical deformed Hermitian-Yang-Mills equation, which in particular confirms Collins-Jacob-Yau's conjecture on projective manifolds. This is a joint work with Man-Chun Lee and Ryosuke Takahashi.

题目：Stability of Yang-Mills-Higgs Pairs on S^n

报告人：韩小利（清华大学）

摘要：In this talk,first we will introduce some notations about weekly stable Yang-Mills-Higgs pairs. Then we will talk the weakly stable Yang-Mills-Higgs on S^n (n>3). We prove that the connection is Yang-Mills and the norm of the Higgs field is constant while n=4 and the connection is flat and the norm of the Higgs field is constant while n>4 for weakly stable Yang-Mills-Higgs Pairs.

题目：球状簇的稳定性和退化

报告人：郦言  （北京理工大学）

摘要：球状簇是一类有高度对称性的几何空间。我们将回顾关于球状簇几何结构的基本理论，由此分类其等变退化，计算相关不变量并研究其稳定性、半稳定退化等问题。

题目: Unbendable rational curves of Goursat type and Cartan type

报告人： 李起峰(山东大学）

摘要: In this talk we discuss the relation between the geometry of unbendable rational curves on a complex manifold and natural differential systems on the corresponding Douady space parameterizing these curves. We give a correspondence between the germs of unbendable rational curves of Goursat type with ordinary differential equations of order three, and give a correspondence between the germs of unbendable rational curves of Cartan type with germs of contact manifolds. This is a joint work with Jun-Muk Hwang.

题目: Smoothing, scattering and a conjecture of Fukaya

报告人：马梓铭 （南方科技大学）

摘要：In this talk, we will discuss the reconstruction problem of the mirror manifold, starting with Fukaya's conjectural construction using counting gradient flow trees/counting holomorphic disks. We will prove an important part of this conjecture and relate it to the famous Gross-Siebert construction, using a newly constructed dgBV algebra of polyvector field. This is based on joint works with Kwokwai Chan and Naichung Conan Leung.

题目: The extension of (quasi-)plurisubharmonic functions on complex manifolds

报告人：宁家福  （中南大学）

摘要：In this talk, we will report some results about the extension of (quasi-)plurisubharmonic functions on complex manifolds.  In particular, we will introduce a partial solution of Coman-Guedj-Zeriahi problem under the assumption of the existence of the holomorphic retraction structure, which is the case of a joint work with Prof. Zhiwei Wang and Prof. Xiangyu Zhou.

题目：Holomorphic curves in Kahler surface

报告人: 孙俊（武汉大学）

摘要：In this talk, we will introduce two ideas approaching to the existence of holomorphic curves in Kahler surface from geometric analysis viewpoint. The first one is the symplectic mean curvature flow. We will talk about singularity analysis for the flow. The second one is variational method. We will introduce a new family of functionals and study the properties of the critical points of the new functionals as well as the gradient flows. This talk is based on joint work with Xiaoli Han and Jiayu Li.

题目： The Kawamata--Viehweg--Nadel-type vanishing theorem and the asymptotic multiplier ideal sheaf

报告人：吴劲草 （上海财经大学）

摘要: In this report, we will briefly talk about a recent work that establishes a Kawamata--Viehweg--Nadel-type vanishing theorem on a compact Kahler manifold. This vanishing theorem is suitable for an arbitrary pseudo-effective line bundle and the base manifold is not necessary to be projective. Moreover, we will also discuss a relative variant then.

题目：局部正交映射与实二次超曲面间的映射问题

报告人：吴瑞聪 （华东师范大学）

摘要：复射影空间里的实二次超曲面为多复变领域里中一个经典的研究对象。这次报告中我们会看到这些超曲面之间的全纯映射会自然地给出复射影空间之间一种特殊映射，我们称之为局部正交映射。透过研究局部正交映射的一些基本性质，我们可以获得不少关于实二次曲面之间的全纯映射的刚性结果。这次报告内容是基于讲者与高云最近的一些合作成果。

题目: Title: Linear isometric invariants of bounded domains

报告人: 汪志威 （北京师范大学）

摘要: Given a bounded domain D and a positive number p, we denote by A^p(D) the space of L^p holomorphic functions on D. In this talk, we will show that, under certain conditions, the complex structure of the domain is complete determined by the linear space A^p(D), equipped with a norm, where p is not even.  This talk is based on joint work with Professors Fusheng Deng, Liyou Zhang, Jiafu Ning and Xiangyu Zhou.

题目：Spectral rigidity of Kahler manifolds

报告人：徐浩 （浙江大学）

摘要：For each fixed nonnegative integers p, if a compact Kahler manifold M of complex dimension $n$ has the same p-spectra as CP^n equipped with the Fubini-Study metric, we give explicit range of n such that this Kahler manifold is holomorphically isometric to CP^n. This extends previous works of Tanno, Chen-Vanhecke, Goldberg for p<= 2 and Li for even p. Our result is the best available at least in two cases: when p is odd or when the isospectral condition is relaxed to that the first three heat coefficients of M and CP^n are equal. As application, we give a simple proof that if two compact Kahler manifold has the same p-spectra for all p>=0, and one of them has constant holomorphic sectional curature, then the other also has constant holomorphic sectional curvature. This is joint work with K. Liu, X. Huang and Y. Zhi.

题目: Some results on complete Kahler manifolds with nonnegative holomorphic sectional curvature

报告人：杨波 （厦门大学）

摘要：Holomorphic sectional curvature (H for short) is just Riemannian sectional curvature restricted on a $J$-invariant plane in the tangent space. It is natural to study Kahler manifolds with a suitable condition on $H$. Recently Wu-Yau studied complete noncompact Kahler manifolds with negative pinched H. In this talk we are interested in complete noncompact Kahler manifolds with $H \geq 0$. We will discuss some preliminary results on existence, non-existence, and function theory on such manifolds.

题目：Strictly nef vector bundles and characterization of projective spaces

报告人：杨晓奎（清华大学）

摘要：In this talk, we will present some recent progress on the geometry of strictly nef vector bundles,  with focus on the characterization of projective spaces.

题目：A deformed Hermitian Yang-Mills flow

报告人：张德凯 （上海大学）

摘要：In this talk, we consider a new deformed Hermitian Yang-Mills flow on a compact Kähler manifold in the supercritical case. We first show the longtime existence of the solution of the flow under the supercritical condition on the initial data. Under Collins-Jacob-Yau's condition on the subsolution, the longtime solution converges to the solution of the deformed Hermitian Yang-Mills equation, which has been solved by Collins-Jacob-Yau by the continuity method. As an application of the flow, we show that on a compact Kähler surface, if there exists a semi-subsolution, the flow converges smoothly to a singular solution to the deformed Hermitian Yang-Mills equation away from a finite number of curves of negative self-intersection. This is a joint work with Prof. Jixiang Fu and Prof. Shing-Tung Yau.

题目：The algebraic and transcendental aspects of Kahler-Einstein metrics

报告人：张科伟 （北京师范大学）

摘要：In the framework of the Yau-Tian-Donaldson conjecture, the study of Kahler-Einstein metrics has revealed numerous deep connections between geometric analysis and algebraic geometry. We will briefly survey the history of this area and present some recent progress, mainly focusing on how the transcendental methods are intertwined with the algebraic ones.

题目: Limiting behavior of Kahler-Ricci flow

报告人：张振雷（首都师范大学）       

摘要: It is a survey talk on the long time behavior of Kahler-Ricci flow on smooth manifolds with semi-ample canonical bundles.

题目： L2-extension of adjoint bundles and Kollar's conjecture

报告人：赵晨 （中国科学技术大学）

摘要：I will talk about Kollar's package on the derived pushforward of the L2 extension of the dualizing sheaf twisted by a certain Hermitian vector bundle. In particular we give an L2-theoretic proof to Kollar's conjecture, which has been proved by M. Saito via the theory of mixed Hodge module. This is a work joint with Junhao Shentu.

题目：Bismut Kahler-like manifolds and Bismut torsion parallel manifolds

报告人：赵全庭 （华中师范大学）

摘要：The Kahler-like condition is introduced by Yang-Zheng and

Angella-Otal-Ugarte-Villacampa, which says the curvature of a metric connectionhas the same symmetry as the one of a Kahler metric. In this talk, we mainly discuss whenthe Bismut-Strominger connection is Kahler-like, BKL for short. We can show that a Hermitian manifold is BKL if and only if the metric is pluri-closed and the torsion is Bismut parallel. Furthermore, we give a sufficient and necessary condition for Bismut torsion parallelleness, BTP for short, in terms of Bismut curvature. Then we will prove structural theorems for BKL threefolds and BTP threefolds.