## Seminar on Geometric Complex Analysis

Seminar information archive ～05/21｜Next seminar｜Future seminars 05/22～

Date, time & place | Monday 10:30 - 12:00 128Room #128 (Graduate School of Math. Sci. Bldg.) |
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Organizer(s) | Kengo Hirachi, Shigeharu Takayama |

**Seminar information archive**

### 2017/04/17

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Dense holomorphic curves in spaces of holomorphic maps

**Yuta Kusakabe**(Osaka University)Dense holomorphic curves in spaces of holomorphic maps

[ Abstract ]

We study when there exists a dense holomorphic curve in a space of holomorphic maps from a Stein space. Our results state that for any bounded convex domain $\Omega \Subset \mathbb{C}^n$ and any connected complex manifold $Y$, the space $\mathcal{O}(\Omega,Y)$ contains a dense holomorphic disc, and that $Y$ is an Oka manifold if and only if for any Stein space $X$ there exists a dense entire curve in every path component of $\mathcal{O}(X,Y)$. The latter gives a new characterization of Oka manifolds. As an application of the former, we construct universal maps from bounded convex domains to any connected complex manifold.

We study when there exists a dense holomorphic curve in a space of holomorphic maps from a Stein space. Our results state that for any bounded convex domain $\Omega \Subset \mathbb{C}^n$ and any connected complex manifold $Y$, the space $\mathcal{O}(\Omega,Y)$ contains a dense holomorphic disc, and that $Y$ is an Oka manifold if and only if for any Stein space $X$ there exists a dense entire curve in every path component of $\mathcal{O}(X,Y)$. The latter gives a new characterization of Oka manifolds. As an application of the former, we construct universal maps from bounded convex domains to any connected complex manifold.

### 2017/04/10

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Slice theorem for CR structures near the sphere and its applications

**Kengo Hirachi**(The University of Tokyo)Slice theorem for CR structures near the sphere and its applications

[ Abstract ]

We formulate a slice theorem for CR structures by following Bland-Duchamp and give some applications to the rigidity theorems.

We formulate a slice theorem for CR structures by following Bland-Duchamp and give some applications to the rigidity theorems.

### 2017/03/06

10:00-11:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Projective and c-projective metric geometries: why they are so similar (ENGLISH)

**Vladimir Matveev**(University of Jena)Projective and c-projective metric geometries: why they are so similar (ENGLISH)

[ Abstract ]

I will show an unexpected application of the standard techniques of integrable systems in projective and c-projective geometry (I will explain what they are and why they were studied). I will show that c-projectively equivalent metrics on a closed manifold generate a commutative isometric $\mathbb{R}^k$-action on the manifold. The quotients of the metrics w.r.t. this action are projectively equivalent, and the initial metrics can be uniquely reconstructed by the quotients. This gives an almost algorithmic way to obtain results in c-projective geometry starting with results in much better developed projective geometry. I will give many application of this algorithmic way including local description, proof of Yano-Obata conjecture for metrics of arbitrary signature, and describe the topology of closed manifolds admitting strictly nonproportional c-projectively equivalent metrics.

Most results are parts of two projects: one is joint with D. Calderbank, M. Eastwood and K. Neusser, and another is joint with A. Bolsinov and S. Rosemann.

I will show an unexpected application of the standard techniques of integrable systems in projective and c-projective geometry (I will explain what they are and why they were studied). I will show that c-projectively equivalent metrics on a closed manifold generate a commutative isometric $\mathbb{R}^k$-action on the manifold. The quotients of the metrics w.r.t. this action are projectively equivalent, and the initial metrics can be uniquely reconstructed by the quotients. This gives an almost algorithmic way to obtain results in c-projective geometry starting with results in much better developed projective geometry. I will give many application of this algorithmic way including local description, proof of Yano-Obata conjecture for metrics of arbitrary signature, and describe the topology of closed manifolds admitting strictly nonproportional c-projectively equivalent metrics.

Most results are parts of two projects: one is joint with D. Calderbank, M. Eastwood and K. Neusser, and another is joint with A. Bolsinov and S. Rosemann.

### 2017/02/13

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

A Characterization of regular points by Ohsawa-Takegoshi Extension Theorem (ENGLISH)

**Qi'an Guan**(Peking University)A Characterization of regular points by Ohsawa-Takegoshi Extension Theorem (ENGLISH)

[ Abstract ]

In this talk, we will present that the germ of a complex analytic set at the origin in $\mathbb{C}^n$ is regular if and only if the related Ohsawa-Takegoshi extension theorem holds. We also present a necessary condition of the $L^2$ extension of bounded holomorphic sections from singular analytic sets.

This is joint work with Dr. Zhenqian Li.

In this talk, we will present that the germ of a complex analytic set at the origin in $\mathbb{C}^n$ is regular if and only if the related Ohsawa-Takegoshi extension theorem holds. We also present a necessary condition of the $L^2$ extension of bounded holomorphic sections from singular analytic sets.

This is joint work with Dr. Zhenqian Li.

### 2017/01/23

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

A unified proof of Cousin I, II and d-bar equation on domains of holomorphy (JAPANESE)

**Junjiro Noguchi**(The University of Tokyo)A unified proof of Cousin I, II and d-bar equation on domains of holomorphy (JAPANESE)

[ Abstract ]

Oka's J\^oku-Ik\^o says that holomorphic functions on a complex submanifold of a polydisk extend holomorphically to the whole polydisk. By making use of Oka's J\^oku-Ik\^o we give a titled proof with introducing an argument that represents one of the three cases.

The proof is a modification of the cube dimension induction, used in the proof of Oka's Syzygy for coherent sheaves.

Oka's J\^oku-Ik\^o says that holomorphic functions on a complex submanifold of a polydisk extend holomorphically to the whole polydisk. By making use of Oka's J\^oku-Ik\^o we give a titled proof with introducing an argument that represents one of the three cases.

The proof is a modification of the cube dimension induction, used in the proof of Oka's Syzygy for coherent sheaves.

### 2017/01/16

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

A geometric second main theorem (ENGLISH)

**Dinh Tuan Huynh**(Osaka University)A geometric second main theorem (ENGLISH)

[ Abstract ]

Using Ahlfors’ theory of covering surfaces, we establish a Cartan’s type Second Main Theorem in the complex projective plane with 1–truncated counting functions for entire holomorphic curves which cluster on an algebraic curve.

Using Ahlfors’ theory of covering surfaces, we establish a Cartan’s type Second Main Theorem in the complex projective plane with 1–truncated counting functions for entire holomorphic curves which cluster on an algebraic curve.

### 2016/12/12

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

(JAPANESE)

**Yu Kawakami**(Kanazawa University)(JAPANESE)

### 2016/12/05

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

(JAPANESE)

**Takahiro Oba**(Tokyo Institute of Technology )(JAPANESE)

### 2016/11/28

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

(JAPANESE)

**Satoshi Nakamura**(Tohoku University)(JAPANESE)

### 2016/11/21

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

(JAPANESE)

**Toshihiro Nose**(Fukuoka Institute of Technology)(JAPANESE)

### 2016/11/14

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

(JAPANESE)

**Sachiko Hamano**(Osaka City University)(JAPANESE)

### 2016/11/07

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

(JAPANESE)

**Hideyuki Ishi**(Nagoya University)(JAPANESE)

### 2016/10/31

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

(JAPANESE)

**Yutaka Ishii**(Kyushu University)(JAPANESE)

### 2016/10/24

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Structure and equivalence of a class of tube domains with solvable groups of automorphisms (JAPANESE)

**Satoru Shimizu**(Tohoku University)Structure and equivalence of a class of tube domains with solvable groups of automorphisms (JAPANESE)

[ Abstract ]

In the study of the holomorphic equivalence problem for tube domains, it is fundamental to investigate tube domains with polynomial infinitesimal automorphisms. To apply Lie group theory to the holomorphic equivalence problem for such tube domains $T_\Omega$, investigating certain solvable subalgebras of $\frak g(T_{\Omega})$ plays an important role, where $\frak g(T_{\Omega})$ is the Lie algebra of all complete polynomial vector fields on $T_\Omega$. Related to this theme, we discuss the structure and equivalence of a class of tube domains with solvable groups of automorphisms. Besides, we give a concrete example of a tube domain whose automorphism group is solvable and contains nonaffine automorphisms.

In the study of the holomorphic equivalence problem for tube domains, it is fundamental to investigate tube domains with polynomial infinitesimal automorphisms. To apply Lie group theory to the holomorphic equivalence problem for such tube domains $T_\Omega$, investigating certain solvable subalgebras of $\frak g(T_{\Omega})$ plays an important role, where $\frak g(T_{\Omega})$ is the Lie algebra of all complete polynomial vector fields on $T_\Omega$. Related to this theme, we discuss the structure and equivalence of a class of tube domains with solvable groups of automorphisms. Besides, we give a concrete example of a tube domain whose automorphism group is solvable and contains nonaffine automorphisms.

### 2016/10/17

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

(JAPANESE)

**Takaaki Nomura**(Kyushu University)(JAPANESE)

### 2016/10/03

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Visualizing the radial Loewner flow and the evolution family (JAPANESE)

**Hirokazu Shimauchi**(Yamanashi Eiwa College)Visualizing the radial Loewner flow and the evolution family (JAPANESE)

### 2016/06/27

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

On a higher codimensional analogue of Ueda theory and its applications (JAPANESE)

**Takayuki Koike**(Kyoto University)On a higher codimensional analogue of Ueda theory and its applications (JAPANESE)

[ Abstract ]

Let $Y$ be a compact complex manifold embedded in a complex manifold with unitary flat normal bundle. As a higher-codimensional generalization of Ueda's theory, we investigate the analytic structure of a neighborhood of $Y$. As an application, we give a criterion for the existence of a smooth Hermitian metric with semi-positive curvature on a nef line bundle.

Let $Y$ be a compact complex manifold embedded in a complex manifold with unitary flat normal bundle. As a higher-codimensional generalization of Ueda's theory, we investigate the analytic structure of a neighborhood of $Y$. As an application, we give a criterion for the existence of a smooth Hermitian metric with semi-positive curvature on a nef line bundle.

### 2016/06/20

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

A transcendental approach to injectivity theorems for log canonical pairs (JAPANESE)

**Shin-ichi Matsumura**(Tohoku University)A transcendental approach to injectivity theorems for log canonical pairs (JAPANESE)

### 2016/06/13

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

(JAPANESE)

**Masanori Adachi**(Tokyo University of Science)(JAPANESE)

### 2016/06/06

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

(JAPANESE)

**Shin Kikuta**(Kogakuin University)(JAPANESE)

### 2016/05/30

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

(JAPANESE)

**Takeo Ohsawa**(Nagoya University)(JAPANESE)

### 2016/05/23

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

A computation method for algebraic local cohomology and its applications (JAPANESE)

**Katsusuke Nabeshima**(The University of Tokushima)A computation method for algebraic local cohomology and its applications (JAPANESE)

[ Abstract ]

Local cohomology was introduced by A. Grothendieck. Subsequent development to a great extent has been motivated by Grothendieck's ideas. Nowadays, local cohomology is a key ingredient in algebraic geometry, commutative algebra, topology and D-modules, and is a fundamental tool for applications in several fields.

In this talk, an algorithmic method to compute algebraic local cohomology classes (with parameters), supported at a point, associated with a given zero-dimensional ideal, is considered in the context of symbolic computation. There are several applications of the method. For example, the method can be used to analyze properties of singularities and deformations of Artin algebra. As the applications, methods for computing standard bases of zero-dimensional ideals and solving ideal membership problems, are also introduced.

Local cohomology was introduced by A. Grothendieck. Subsequent development to a great extent has been motivated by Grothendieck's ideas. Nowadays, local cohomology is a key ingredient in algebraic geometry, commutative algebra, topology and D-modules, and is a fundamental tool for applications in several fields.

In this talk, an algorithmic method to compute algebraic local cohomology classes (with parameters), supported at a point, associated with a given zero-dimensional ideal, is considered in the context of symbolic computation. There are several applications of the method. For example, the method can be used to analyze properties of singularities and deformations of Artin algebra. As the applications, methods for computing standard bases of zero-dimensional ideals and solving ideal membership problems, are also introduced.

### 2016/05/16

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

(JAPANESE)

**Masataka Tomari**(Nihon University)(JAPANESE)

### 2016/05/09

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Nevanlinna type theorems for meromorphic functions on negatively curved Kähler manifolds (JAPANESE)

**Atsushi Atsuji**(Keio University)Nevanlinna type theorems for meromorphic functions on negatively curved Kähler manifolds (JAPANESE)

[ Abstract ]

We discuss a generalization of classical Nevanlinna theory to meromorphic functions on complete Kähler manifolds. Several generalization of domains of functions are known in Nevanlinna theory, especially the results due to W.Stoll are well-known. In general Kähler case the remainder term of the second main theorem of Nevanlinna theory usually takes a complicated form. It seems that we have to modify classical

methods in order to simplify the second main theorem. We will use heat diffusion to do that and show some defect relations. We would also like to give some Liouville type theorems for holomorphic maps by using similar heat diffusion methods.

We discuss a generalization of classical Nevanlinna theory to meromorphic functions on complete Kähler manifolds. Several generalization of domains of functions are known in Nevanlinna theory, especially the results due to W.Stoll are well-known. In general Kähler case the remainder term of the second main theorem of Nevanlinna theory usually takes a complicated form. It seems that we have to modify classical

methods in order to simplify the second main theorem. We will use heat diffusion to do that and show some defect relations. We would also like to give some Liouville type theorems for holomorphic maps by using similar heat diffusion methods.

### 2016/04/25

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

The representative domain and its applications (JAPANESE)

**Atsushi Yamamori**(Academia Sinica)The representative domain and its applications (JAPANESE)

[ Abstract ]

Bergman introduced the notion of a representative domain to choose a nice holomorphic equivalence class of domains. In this talk, I will explain that the representative domain is also useful to obtain an analogue of Cartan's linearity theorem for some special class of domains.

Bergman introduced the notion of a representative domain to choose a nice holomorphic equivalence class of domains. In this talk, I will explain that the representative domain is also useful to obtain an analogue of Cartan's linearity theorem for some special class of domains.