## Seminar on Geometric Complex Analysis

Seminar information archive ～09/24｜Next seminar｜Future seminars 09/25～

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

Organizer(s) | Kengo Hirachi, Shigeharu Takayama, Ryosuke Nomura |

**Seminar information archive**

### 2012/11/12

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Residues of meromorphic differential forms (ENGLISH)

**A.G. Aleksandrov**(Institute of Control Sciences, Russian Acad. of Sci.)Residues of meromorphic differential forms (ENGLISH)

[ Abstract ]

The purpose of the talk is to discuss several interesting aspects

of the classical residue theory originated by H. Poincar\\'e, J. de Rham and J. Leray and their followers. Focus topics of our studies are some of the less known applications, developed by the author in the past few years in complex analysis, topology and geometry of singular varieties and in the theory of differential equations. Almost all considerations are based essentially on properties of a special class of meromorphic differential forms called logarithmic or multi-logarithmic forms.

The purpose of the talk is to discuss several interesting aspects

of the classical residue theory originated by H. Poincar\\'e, J. de Rham and J. Leray and their followers. Focus topics of our studies are some of the less known applications, developed by the author in the past few years in complex analysis, topology and geometry of singular varieties and in the theory of differential equations. Almost all considerations are based essentially on properties of a special class of meromorphic differential forms called logarithmic or multi-logarithmic forms.

### 2012/10/29

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Value distribution of meromorphic functions on foliated manifolds,II (JAPANESE)

**Atsushi Atsuji**(Keio University)Value distribution of meromorphic functions on foliated manifolds,II (JAPANESE)

### 2012/10/22

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

The second main therorem for entire curves into Hilbert modular surfaces (JAPANESE)

**Yusaku Tiba**(Grad. School of Math. Sci., Univ. of Tokyo)The second main therorem for entire curves into Hilbert modular surfaces (JAPANESE)

### 2012/10/15

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

An L^2 estimate on domains and application to Levi-flat surfaces (JAPANESE)

**Takeo Ohsawa**(Nagoya University)An L^2 estimate on domains and application to Levi-flat surfaces (JAPANESE)

### 2012/07/09

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Volume of graded linear series and the existence problem of constant scalar curvature Kaehler metric (JAPANESE)

**Tomoyuki Hisamoto**(Univ. of Tokyo)Volume of graded linear series and the existence problem of constant scalar curvature Kaehler metric (JAPANESE)

[ Abstract ]

We describe the volume of a graded linear series by the Monge-Ampere mass of the associated equilibrium metric. We relate this formula to the question whether the weak geodesic ray associated to a test configuration of given polarized manifold recovers the Donaldson-Futaki invariant.

We describe the volume of a graded linear series by the Monge-Ampere mass of the associated equilibrium metric. We relate this formula to the question whether the weak geodesic ray associated to a test configuration of given polarized manifold recovers the Donaldson-Futaki invariant.

### 2012/06/25

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

CR equivalence problem of CR manifolds with slice structure (JAPANESE)

**Atsushi Hayashimoto**(Nagano National College of Technology)CR equivalence problem of CR manifolds with slice structure (JAPANESE)

### 2012/06/18

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

CR Q-curvature flow and CR Paneitz operator on 3-dimensional CR manifolds (JAPANESE)

**Takanari Saotome**(Academia Sinica)CR Q-curvature flow and CR Paneitz operator on 3-dimensional CR manifolds (JAPANESE)

### 2012/06/11

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Differential forms on complete intersections (ENGLISH)

**Damian BROTBEK**(University of Tokyo)Differential forms on complete intersections (ENGLISH)

[ Abstract ]

Brückmann and Rackwitz proved a vanishing result for particular types of differential forms on complete intersection varieties. We will be interested in the cases not covered by their result. In some cases, we will show how the space $H^0(X,S^{m_1}\Omega_X\otimes \cdots \otimes S^{m_k}\Omega_X)$ depends on the equations defining $X$, and in particular we will prove that the theorem of Brückmann and Rackwitz is optimal. The proofs are based on simple, combinatorial, cohomology computations.

Brückmann and Rackwitz proved a vanishing result for particular types of differential forms on complete intersection varieties. We will be interested in the cases not covered by their result. In some cases, we will show how the space $H^0(X,S^{m_1}\Omega_X\otimes \cdots \otimes S^{m_k}\Omega_X)$ depends on the equations defining $X$, and in particular we will prove that the theorem of Brückmann and Rackwitz is optimal. The proofs are based on simple, combinatorial, cohomology computations.

### 2012/06/04

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Log-plurisubharmonicity of metric deformations induced by Schiffer and harmonic spans. (JAPANESE)

**Sachiko HAMANO**(Fukushima University)Log-plurisubharmonicity of metric deformations induced by Schiffer and harmonic spans. (JAPANESE)

### 2012/05/28

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Local cohomology and hypersurface isolated singularities II (JAPANESE)

**Shinichi TAJIMA**(University of Tsukuba)Local cohomology and hypersurface isolated singularities II (JAPANESE)

[ Abstract ]

局所コホモロジーの孤立特異点への応用として

・$\mu$-constant-deformation の Tjurina 数

・対数的ベクトル場の構造と構成法

・ニュートン非退化な超曲面に対する Kouchnirenko の公式

について述べる.

局所コホモロジーの孤立特異点への応用として

・$\mu$-constant-deformation の Tjurina 数

・対数的ベクトル場の構造と構成法

・ニュートン非退化な超曲面に対する Kouchnirenko の公式

について述べる.

### 2012/05/21

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Local cohomology and hypersurface isolated singularities I (JAPANESE)

**Shinichi TAJIMA**(University of Tsukuba)Local cohomology and hypersurface isolated singularities I (JAPANESE)

### 2012/05/14

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Duality in the unit circle and the ring of p-adic intergers and van der Corput series (JAPANESE)

**Hiroshi KANEKO**(Tokyo University of Science)Duality in the unit circle and the ring of p-adic intergers and van der Corput series (JAPANESE)

### 2012/05/07

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

The second metric variation of the total $Q$-curvature in conformal geometry (JAPANESE)

**Yoshihiko Matsumoto**(University of Tokyo)The second metric variation of the total $Q$-curvature in conformal geometry (JAPANESE)

[ Abstract ]

Branson's $Q$-curvature of even-dimensional compact conformal manifolds integrates to a global conformal invariant called the total $Q$-curvature. While it is topological in two dimensions and is essentially the Weyl action in four dimensions, in the higher dimensional cases its geometric meaning remains mysterious. Graham and Hirachi have shown that the first metric variation of the total $Q$-curvature coincides with the Fefferman-Graham obstruction tensor. In this talk, the second variational formula will be presented, and it will be made explicit especially for conformally Einstein manifolds. The positivity of the second variation will be discussed in connection with the smallest eigenvalue of the Lichnerowicz Laplacian.

Branson's $Q$-curvature of even-dimensional compact conformal manifolds integrates to a global conformal invariant called the total $Q$-curvature. While it is topological in two dimensions and is essentially the Weyl action in four dimensions, in the higher dimensional cases its geometric meaning remains mysterious. Graham and Hirachi have shown that the first metric variation of the total $Q$-curvature coincides with the Fefferman-Graham obstruction tensor. In this talk, the second variational formula will be presented, and it will be made explicit especially for conformally Einstein manifolds. The positivity of the second variation will be discussed in connection with the smallest eigenvalue of the Lichnerowicz Laplacian.

### 2012/04/16

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Fekete configuration, quantitative equidistribution and wanderting critical orbits in non-archimedean dynamics

(JAPANESE)

**Yusuke Okuyama**(Kyoto Institute of Technology)Fekete configuration, quantitative equidistribution and wanderting critical orbits in non-archimedean dynamics

(JAPANESE)

### 2012/04/09

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Effective estimate on the number of deformation types of families of canonically polarized manifolds over curves

(JAPANESE)

**Shigeharu TAKAYAMA**(University of Tokyo)Effective estimate on the number of deformation types of families of canonically polarized manifolds over curves

(JAPANESE)

### 2012/01/30

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

Differential equations as embedding obstructions and vanishing theorems (ENGLISH)

**Damian Brotbek**(University of Tokyo)Differential equations as embedding obstructions and vanishing theorems (ENGLISH)

[ Abstract ]

Given a smooth projective variety $X$ it is natural to wonder what is the smallest integer $N$ such that one can embed $X$ into $\mathbf{P}^N$. In this talk I will first recall what can be said for any smooth projective variety, then I will explain how the existence of some particular differential equations on $X$ yields obstructions to the existence of some projective embeddings.

Given a smooth projective variety $X$ it is natural to wonder what is the smallest integer $N$ such that one can embed $X$ into $\mathbf{P}^N$. In this talk I will first recall what can be said for any smooth projective variety, then I will explain how the existence of some particular differential equations on $X$ yields obstructions to the existence of some projective embeddings.

### 2012/01/23

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

Twistor correspondence for R-invariant indefinite self-dual metric on R^4 (JAPANESE)

**Fuminori Nakata**(Tokyo University of Science)Twistor correspondence for R-invariant indefinite self-dual metric on R^4 (JAPANESE)

### 2012/01/16

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

A ramification theorem for the ratio of canonical forms of flat surfaces in hyperbolic 3-space (JAPANESE)

**Yu Kawakami**(Yamaguchi University)A ramification theorem for the ratio of canonical forms of flat surfaces in hyperbolic 3-space (JAPANESE)

[ Abstract ]

We provide an effective ramification theorem for the ratio of canonical forms of weakly complete flat fronts in the hyperbolic 3-space. As an application, we give a simple proof of the classification of complete nonsingular flat surfaces in the hyperbolic 3-space.

We provide an effective ramification theorem for the ratio of canonical forms of weakly complete flat fronts in the hyperbolic 3-space. As an application, we give a simple proof of the classification of complete nonsingular flat surfaces in the hyperbolic 3-space.

### 2011/12/12

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

Brody curves and mean dimension (JAPANESE)

**Shinichiroh MATSUO**(Kyoto University)Brody curves and mean dimension (JAPANESE)

[ Abstract ]

We study the mean dimensions of the spaces of Brody curves. In particular we give the formula of the mean dimension of the space of Brody curves in the Riemann sphere.

We study the mean dimensions of the spaces of Brody curves. In particular we give the formula of the mean dimension of the space of Brody curves in the Riemann sphere.

### 2011/11/28

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

An ampleness criterion with the extendability of singular positive metrics (JAPANESE)

**Shin-ichi Matsumura**(University of Tokyo)An ampleness criterion with the extendability of singular positive metrics (JAPANESE)

[ Abstract ]

Coman, Guedj and Zeriahi proved that, for an ample line bundle $L$ on a projective manifold $X$, any singular positive metric on the line bundle $L|_{V}$ along a subvariety $V \subset X$ can be extended to a global singular positive metric of $L$. In this talk, we prove that the extendability of singular positive metrics on a line bundle along a subvariety implies the ampleness of the line bundle.

Coman, Guedj and Zeriahi proved that, for an ample line bundle $L$ on a projective manifold $X$, any singular positive metric on the line bundle $L|_{V}$ along a subvariety $V \subset X$ can be extended to a global singular positive metric of $L$. In this talk, we prove that the extendability of singular positive metrics on a line bundle along a subvariety implies the ampleness of the line bundle.

### 2011/11/21

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

Techniques of computations of Dolbeault cohomology of solvmanifolds (JAPANESE)

**Hisashi Kasuya**(University of Tokyo)Techniques of computations of Dolbeault cohomology of solvmanifolds (JAPANESE)

### 2011/11/16

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Disc functionals and Siciak-Zaharyuta extremal functions on singular varieties (ENGLISH)

**Franc Forstneric**(University of Ljubljana)Disc functionals and Siciak-Zaharyuta extremal functions on singular varieties (ENGLISH)

[ Abstract ]

A disc functional on a complex space, $X$, is a function P that assign a real number $P(f)$ (possibly minus infinity) to every analytic disc $f$ in $X$. An examples is the Poisson functional $P_u$ of an upper semicontinuous function $u$ on $X$: in that case $P_u(f)$ is the average of u over the boundary curve of the disc $f$. Other natural examples include the Lelong and the Riesz functionals. The envelope of a disc functional $P$ is a function on $X$ associating to every point $x$ of $X$ the infimum of the values $P(f)$ over all analytic discs $f$ in $X$ satisfying $f(0)=x$. The main point of interest is that the envelopes of many natural disc functionals are plurisubharmonic functions solving certain extremal problems. In the classical case when $X=\mathbf{C}^n$ this was first discovered by E. Poletsky in the early 1990's. In this talk I will discuss recent results on plurisubharmonicity of envelopes of all the classical disc functional mentioned above on locally irreducible complex spaces. In the second part of the talk I will give formulas expressing the classical Siciak-Zaharyuta maximal function of an open set in an affine algebraic variety as the envelope of certain disc functionals. We establish plurisubharmonicity of envelopes of certain classical disc functionals on locally irreducible complex spaces, thereby generalizing the corresponding results for complex manifolds. We also find new formulae expressing the Siciak-Zaharyuta extremal function of an open set in a locally irreducible affine algebraic variety as the envelope of certain disc functionals, similarly to what has been done for open sets in $\mathbf{C}^n$ by Lempert and by Larusson and Sigurdsson.

A disc functional on a complex space, $X$, is a function P that assign a real number $P(f)$ (possibly minus infinity) to every analytic disc $f$ in $X$. An examples is the Poisson functional $P_u$ of an upper semicontinuous function $u$ on $X$: in that case $P_u(f)$ is the average of u over the boundary curve of the disc $f$. Other natural examples include the Lelong and the Riesz functionals. The envelope of a disc functional $P$ is a function on $X$ associating to every point $x$ of $X$ the infimum of the values $P(f)$ over all analytic discs $f$ in $X$ satisfying $f(0)=x$. The main point of interest is that the envelopes of many natural disc functionals are plurisubharmonic functions solving certain extremal problems. In the classical case when $X=\mathbf{C}^n$ this was first discovered by E. Poletsky in the early 1990's. In this talk I will discuss recent results on plurisubharmonicity of envelopes of all the classical disc functional mentioned above on locally irreducible complex spaces. In the second part of the talk I will give formulas expressing the classical Siciak-Zaharyuta maximal function of an open set in an affine algebraic variety as the envelope of certain disc functionals. We establish plurisubharmonicity of envelopes of certain classical disc functionals on locally irreducible complex spaces, thereby generalizing the corresponding results for complex manifolds. We also find new formulae expressing the Siciak-Zaharyuta extremal function of an open set in a locally irreducible affine algebraic variety as the envelope of certain disc functionals, similarly to what has been done for open sets in $\mathbf{C}^n$ by Lempert and by Larusson and Sigurdsson.

### 2011/11/07

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

Oka's extra-zero problem and related topics (JAPANESE)

**Junjiro Nocuchi**(University of Tokyo)Oka's extra-zero problem and related topics (JAPANESE)

[ Abstract ]

The main part of this talk is a joint work with my colleagues, M. Abe and S. Hamano. After the solution of Cousin II problem by K. Oka III in 1939, he thought an extra-zero problem in 1945 (his posthumous paper) asking if it is possible to solve an arbitrarily given Cousin II problem adding some extra-zeros whose support is disjoint from the given one. Some special case was affirmatively confirmed in dimension two and a counter-example in dimension three or more was obtained. We will give a complete solution of this problem with examples and to discuss some new questions. An example on a toric variety of which idea is based on K. Stein's paper in 1941 has some special interest and will be discussed. I would like also to discuss some analytic intersections form the viewpoint of Nevanlinna theory.

The main part of this talk is a joint work with my colleagues, M. Abe and S. Hamano. After the solution of Cousin II problem by K. Oka III in 1939, he thought an extra-zero problem in 1945 (his posthumous paper) asking if it is possible to solve an arbitrarily given Cousin II problem adding some extra-zeros whose support is disjoint from the given one. Some special case was affirmatively confirmed in dimension two and a counter-example in dimension three or more was obtained. We will give a complete solution of this problem with examples and to discuss some new questions. An example on a toric variety of which idea is based on K. Stein's paper in 1941 has some special interest and will be discussed. I would like also to discuss some analytic intersections form the viewpoint of Nevanlinna theory.

### 2011/10/31

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

Classification of Moishezon twistor spaces on 4CP^2 (JAPANESE)

**Nobuhiro Honda**(Tohoku Univeristy)Classification of Moishezon twistor spaces on 4CP^2 (JAPANESE)

### 2011/10/17

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

Compact locally homogeneous Kähler manifolds $\Gamma\backslash G/K$ (JAPANESE)

**Yoshinobu Kamishima**(Tokyo Metropolitan University)Compact locally homogeneous Kähler manifolds $\Gamma\backslash G/K$ (JAPANESE)