Geometry Seminar

Speaker:Li Ziyu, Feng Jing, Miao Minghao
Organizer:Akito Futaki
Time:Saturday, 13:00 - 17:15 (PM), Mar. 16, 2023
Venue:Jing Zhai 304

Upcoming Talks: 

Time:Saturday, 13:00 - 17:15 (PM), March 16, 2024
Venue:Jing Zhai 304
13:00 - 14:15 Li, Ziyu (Tsinghua Unv.)
14:30 - 15:45 Feng, Jing (Tsinghua Unv.)
16:00 - 17:15 Miao, Minghao (Nanjing Univ.)


13:00 - 14:15 Li, Ziyu (Tsinghua Unv.)
Title: Kobayashi Conjecture and Siu's Strategy
Abstract: The famous conjecture of Shoshichi Kobayshi is that a generic algebraic hypersurface of dimension n and of sufficiently large degree d ≥ d_n in the complex project space \mathbb{P}^{n+1} is hyperbolic. A fundamental vanishing theorem states that any entire curve must satisfy some differential equations. If there are enough independent differential equations, then all entire curves must be constant. Based on this observation, Yum-Tong Siu presented his proof in 2015.

14:30 - 15:45 Feng, Jing (Tsinghua Unv.)
Title: 4d N=1 AdS/CFT correspondence for three dimensional quotient singularity
Abstract: We study the AdS/CFT correspondence induced by D3 branes probing three dimensional Gorenstein quotient singularity C^3/G. The field theory is given by the McKay quiver, which has a vanishing NSVZ beta function assuming that all the chiral fields have the U(1)_R charge 2/3. Various physical quantities such as quiver Hilbert series, superconformal index, central charges, etc. are computed from both the field theory and the gravity side. Our computation gives confirmation that the McKay quiver indeed gives one description of the SCFT on D3 branes probing the quotient singularity.

16:00 - 17:15 Miao, Minghao (Nanjing Univ.)
Title: Optimal Degenerations of K-unstable Fano Threefolds
Abstract: In this talk, we will propose a question of how to explicitly determine the optimal degenerations of the K-unstable Fano manifolds as predicted by the Hamilton-Tian conjecture. We answer this question for a family of K-unstable Fano threefolds (No 2.23 in Mori-Mukai's list), which has discrete automorphism groups and the normalized Kahler-Ricci flow develops Type II singularity. Our approach is based on a new method to check weighted K-stability, which generalizes Abban-Zhuang's theory to give an estimate of the weighted delta invariant by dimension induction. Some speculative relations between the delta invariant and the H invariant will also be discussed. This is based on a joint work with Linsheng Wang.

Saturday, 13:00 - 17:15 (PM), Dec. 30, 2023


13:00 - 14:15 Wang, Linsheng (Nanjing Unv.)
Title: A new example of Fano manifold with Kähler-Ricci soliton.
Abstract: In this talk, I will introduce an effective method to show the existence of the Kähler-Ricci soliton on a given Fano manifold. As an application, we show that any Fano threefold X in the family No.2.28 of Mukai-Mori's list (that is, CP^3 with a smooth plane cubic curve C blowup) admits Kähler-Ricci soliton. Furthermore, we show that the weighted K-stability of the Fano manifolds X is equivalent to the GIT-stability of the plane cubic curves C. This is a joint work with Minghao Miao.

14:30 - 15:45 Kawai, Kotaro (BIMSA)
Title: Mirror of minimal submanifolds and a monotonicity formula.
Abstract: For Hermitian connections on a Hermitian complex line bundle over a Riemannian manifold, we can define the "volume", which can be considered to be the "mirror" of the standard volume for submanifolds. We call the critical points minimal connections. They can be considered as "mirrors" of minimal submanifolds and analogous to Yang-Mills connections.
In this talk, I will introduce some properties of minimal connections and then state a monotonicity formula. As a corollary, we obtain the vanishing theorem for minimal connections in the odd-dimensional case.

16:00 - 17:15 Xie, Song-Yan (Chinese Academy of Sciences )
Title: Entire curves generating all shapes of Nevanlinna currents.
Abstract: Nevanlinna currents were introduced by McQuillan to capture asymptotic behaviors of entire curves. In this talk, we will show that from some single entire curve we can obtain quite distinct Nevanlinna currents.

Past Talks:  

Saturday, 13:00 - 17:15 (PM), September 23, 2023 


13:00 - 14:15  Li, Ziyu (YMSC)
Title: Oka theory in the homotopy-theoretic viewpoint
Abstract: Gromov's 1989 paper is the foundation work of Oka theory, in which he describes many of his insightful ideas. In this paper, Larusson saw the shadow of Quillen's first axiom for a model category, which inspired him use homotopical algebra to study lifting and extension properties of holomorphic maps, such as those given by the Oka Principle. In this talk, we will introduce this abstract process of Oka theory.

14:30 - 15:45  Zhou, Shengxuan (Peking Univ.)
Title: Bergman kernels on degenerations
Abstract: In this talk, we consider the fiberwise Bergman kernel for a flat family of polarized varieties over a Riemann surface. We will explain the continuity of the fiberwise Bergman kernel and give a result on uniform convergence for the Fubini-Study currents. This is a joint work with Linsheng Wang.

16:00 - 17:15  Ye, Yanan (Peking Univ.)
Title: Bismut Einstein metrics on compact complex manifolds
Abstract: We observe that, for a Bismut Einstein metric, the (2,0)-part of Bismut Ricci form is an eigenvector of the Chern Laplacian. With the help of this observation, we prove that a Bismut Einstein metric with non-zero Einstein constant is Kähler Einstein. Additionally, for Bismut Einstein metrics with zero Einstein constant, we prove that they are actually Bismut Ricci flat.

Saturday, 13:00 - 17:30 (PM), June 10, 2023

13:00 - 14:00  Li, Ziyu (YMSC)

Title: A brief introduction to Oka theory.

Abstract: A central question of complex geometry is to understand the space of holomorphic maps X➡Y between a pair of complex manifolds. In contrast to Kobayashi hyperbolicity theory, Oka theory considers complex manifolds Y which admit many holomorphic maps X➡Y from any Stein Manifold X. I will talk about Oka Principle and Oka Manifolds and give some examples.

14:10 - 15:10  Liu, Yaxiong (YMSC)

Title: Negativity of the second Chern character of holomorphic vector bundles.

Abstract: Given a hermitian metric on holomorphic vector bundle, a natural question is under what conditions its higher Chern forms or Chern characters are positive or negative. We will introduce a curvature condition, under which its second Chern character form is negative. As an application, we obtain a vanishing theorem under the condition of second Chern character. This is the joint work with X. Li, Z. Liu and H. Yang.  

15:20 - 16:20  Deng, Jialong (YMSC)

Title: Convergences of metrics and the scalar curvature

Abstract: The Gromov-Hausdorff limit of Riemannian n-manifolds with Ricci curvature bounded below by k is the CD(n,k) space.  An ongoing project is to find a convergence for Riemannian manifolds with the scalar curvature bounded below such that the limit space satisfies the scalar curvature bounded below in the generalized sense.  Several old and new convergences  will be introduced in the talk.

16:30 - 17:30  Kawai, Kotaro (BIMSA)

Title: Mirror of submanifolds and special holonomy.

Abstract: The Strominger-Yau-Zaslow conjecture suggests that it will be important to consider the special Lagrangian torus fibration for the study of mirror symmetry. I will first introduce the real Fourier-Mukai transform, which gives the explicit ``mirror" correspondence on the trivial torus fibration. By this method, we can define notions for Hermitian connections on a Hermitian line bundle arising from submanifolds, such as the "mirror" volume. I will also explain that the "mirror" of a calibrated submanifold in a $G_2$-manifold is also defined, and it indeed has similar properties to a calibrated submanifold (and a $G_2$-instanton). I will state some recent results and problems about minimal connections, which are critical points of the "mirror" volume. A part of this talk is based on the joint work with Hikaru Yamamoto.

Saturday, 13:00 - 17:30 (PM), March 4, 2023

13:00 - 14:15 Li, Ziyu (YMSC)

Title: Nevanlinna Theory and Diophantine Approximation.

Abstract: Nevanlinna Theory is a theory about meromorphic functions, and it has a good analogy with Diophantine Approximation. This analogy is the source of the idea that we hope to establish Nevanlinna theory on many other mathematical objects.

14:30 - 15:45 Liu, Yaxiong (YMSC)

Title: A Le Potier-type isomorphism with multiplier submodule sheaves,

Abstract: We obtain a Le Potier-type isomorphism theorem which relates holomorphic vector bundles with multiplier submodule sheaves associated to strongly Nakano semi-positive singular hermitian metrics to the tautological line bundles with multiplier ideal sheaves. As applications, we obtain a Kollar-type infectivity theorem, a Nadel-type vanishing theorem and a singular holomorphic Morse inequalities for holomorphic vector bundles. This is the joint work with Zhuo Liu, Hui Yang and Xiangyu Zhou.

16:00 - 17:15 Deng, Jialong (YMSC)

Title: What is the scalar curvature  on non-smooth spaces?

Abstract: Since the scalar curvature appears in the term of  Einstein field equations,  the study of it becomes also important in general relativity.  One of the open question about it is  how to define the non-negative scalar curvature on non-smooth spaces.  We will propose two definitions: one is the MV-scalar curvature on a closed topological manifold and the other one is n-volumic scalar curvature on a compact metric measure space.

November 27, 2022 (Sunday)

13:00 - 14:15  Liu, Yaxiong (YMSC)

Title: Valuative stability of polarized varieties and applications.

Abstract: Recently, Dervan-Legendre considered the valuative criterion of polarized varieties. We will study the valuative stability and show that it is an open condition. We would like to study the valuative criterion for the Donaldson's J-equation. Motivated by the beta-invariant of Dervan-Legendre, we introduce a notion, the so-called valuative J-stability and prove that J-stability implies valuative J-stability. If time permits, we show the upper bound of the volume of K-semistable polarized toric varieties as an application of valuative stability.

14:30 - 15:45  Deng, Jialong (YMSC)

Title: The weighted scalar curvature

Abstract: Inspired by the importance  of the Bakry-Emery curvature on a weighted Riemannian manifold $(M^n, g, e^{f}Vol_g)$, we will introduce the weighted scalar curvature on it and then extend some classic results of the scalar curvature to the weighted version.  For example, we will generalize  Schoen-Yau's minimal hypersurface method, Gromov-Lawson's index theory approach and Seiberg-Witten invariants (in four dimensions) to  a weighted Riemannian manifold with positively  weighted scalar curvature.

16:00 - 17:15  Kawai, Kotaro (BIMSA)

Title: Construction of nearly Kahler manifolds by Foscolo and Haskins

Abstract: A Riemannian 6-manifold is called nearly Kahler if its Riemannian cone has holonomy contained in G2. Only known examples were some homogeneous spaces for a long time, but Foscolo and Haskins constructed new cohomogeneity one nearly Kahler manifolds in 2017. I will explain an outline of the construction.