Geometry Seminar

Speaker:Li Ziyu (YMSC), Zhou Shengxuan (Peking Univ.), Ye Yanan (Peking Univ.)
Organizer:Akito Futaki
Time:Saturday, 13:00 - 17:15 (PM), September 23, 2023
Venue:Jing Zhai 304

Upcoming Talks: (updated)

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.

Past Talks:  

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.