## Seminar on Geometric Complex Analysis

Seminar information archive ～02/07｜Next seminar｜Future seminars 02/08～

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, Ryosuke Nomura |

**Seminar information archive**

### 2014/05/19

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

On degenerations of Ricci-flat Kähler manifolds (JAPANESE)

**Shigeharu Takayama**(University of Tokyo)On degenerations of Ricci-flat Kähler manifolds (JAPANESE)

### 2014/05/12

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

Resolution of singularities via Newton polyhedra and its application to analysis (JAPANESE)

**Joe Kamimoto**(Kyushu university)Resolution of singularities via Newton polyhedra and its application to analysis (JAPANESE)

[ Abstract ]

In the 1970s, A. N. Varchenko precisely investigated the leading term of the asymptotic expansion of an oscillatory integral with real analytic phase by using the geometry of the Newton polyhedron of the phase. Since his study, the importance of the resolution of singularities by means of Newton polyhedra has been strongly recognized. The purpose of this talk is to consider studies around this theme and to explain their relationship with some problems in several complex variables.

In the 1970s, A. N. Varchenko precisely investigated the leading term of the asymptotic expansion of an oscillatory integral with real analytic phase by using the geometry of the Newton polyhedron of the phase. Since his study, the importance of the resolution of singularities by means of Newton polyhedra has been strongly recognized. The purpose of this talk is to consider studies around this theme and to explain their relationship with some problems in several complex variables.

### 2014/04/28

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

On the existence problem of Kähler-Ricci solitons (JAPANESE)

**Sunsuke Saito**(The University of Tokyo)On the existence problem of Kähler-Ricci solitons (JAPANESE)

### 2014/04/21

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

Lagrangian mean curvature flows and some examples (JAPANESE)

**Hikaru Yamamoto**(The University of Tokyo)Lagrangian mean curvature flows and some examples (JAPANESE)

### 2014/04/14

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

Alternative proof of the geometric vrsion of Lemma on logarithmic derivatives (JAPANESE)

**Katsutoshi Yamanoi**(Tokyo Institute of Technology)Alternative proof of the geometric vrsion of Lemma on logarithmic derivatives (JAPANESE)

### 2014/01/27

11:00-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Logarithmic 1-forms and distributions of entire curves and integral points (JAPANESE)

**Junjiro Noguchi**(The University of Tokyo)Logarithmic 1-forms and distributions of entire curves and integral points (JAPANESE)

[ Abstract ]

The Log-Bloch-Ochiai Theorem says, in the most general form so far, that every entire curve in a Zariski open $X$ of a compact Kahler manifold $\bar{X}$ must be degenerate, if $\bar{q}(X)> \dim X$ ([NW02] Noguchi-Winkelmann, Math.\ Z. 239, 2002). If $X$ is defined a quasi-projective algebraic variety defined over a number field, then there is no Zariski dense $(S, D)$-integral subset in $X$ ($D=\partial X=\bar{X}\subset X$). We discuss this kind of properties more.

In the talk we will fix an error in an application in [NW02], and we will show

Theorem 1. (i) Let $M$ be a complex projective algebraic manifold, and let $D=\sum_{j=1}^l D_j$ be a sum of divisors on $M$ which are independent in supports. If $l> \dim M+r(\{D_j\})-q(M)$, then every entire curve $f:\mathbf{C} \to M\setminus D$ must be degenerate.

(ii) Let $M$ and $D_j$ be defined over a number field. If $l> \dim M+r(\{D_j\})-q(M)$, then there is no Zariski-dense $(S,D)$-integral subset of $M\setminus D$.

For the finiteness we obtain

Theorem 2. Let the notation be as above.

(i) If $l \geq 2 \dim M+r(\{D_j\})$, then $M\setminus D$ is completehyperbolic and hyperbolically embedded into $M$.

(ii) Let $M$ and $D_j$ be defined over a number field. If $l> 2\dim M+r(\{D_j\})$, then every $(S,D)$-integral subset of $M\setminus D$ is finite.

Precise definitions will be given in the talk. We will also discuss an application of Theorem 1 (ii) to generalize Siegel's Theorem on integral points on affine curves,

recent due to A. Levin.

The Log-Bloch-Ochiai Theorem says, in the most general form so far, that every entire curve in a Zariski open $X$ of a compact Kahler manifold $\bar{X}$ must be degenerate, if $\bar{q}(X)> \dim X$ ([NW02] Noguchi-Winkelmann, Math.\ Z. 239, 2002). If $X$ is defined a quasi-projective algebraic variety defined over a number field, then there is no Zariski dense $(S, D)$-integral subset in $X$ ($D=\partial X=\bar{X}\subset X$). We discuss this kind of properties more.

In the talk we will fix an error in an application in [NW02], and we will show

Theorem 1. (i) Let $M$ be a complex projective algebraic manifold, and let $D=\sum_{j=1}^l D_j$ be a sum of divisors on $M$ which are independent in supports. If $l> \dim M+r(\{D_j\})-q(M)$, then every entire curve $f:\mathbf{C} \to M\setminus D$ must be degenerate.

(ii) Let $M$ and $D_j$ be defined over a number field. If $l> \dim M+r(\{D_j\})-q(M)$, then there is no Zariski-dense $(S,D)$-integral subset of $M\setminus D$.

For the finiteness we obtain

Theorem 2. Let the notation be as above.

(i) If $l \geq 2 \dim M+r(\{D_j\})$, then $M\setminus D$ is completehyperbolic and hyperbolically embedded into $M$.

(ii) Let $M$ and $D_j$ be defined over a number field. If $l> 2\dim M+r(\{D_j\})$, then every $(S,D)$-integral subset of $M\setminus D$ is finite.

Precise definitions will be given in the talk. We will also discuss an application of Theorem 1 (ii) to generalize Siegel's Theorem on integral points on affine curves,

recent due to A. Levin.

### 2014/01/20

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

タイヒミュラー距離の幾何学とその応用 (JAPANESE)

**Hideki Miyachi**(Osaka University)タイヒミュラー距離の幾何学とその応用 (JAPANESE)

### 2013/12/16

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

Shilov boundaries of the pluricomplex Green function's level sets (JAPANESE)

**Yusaku Tiba**(Tokyo Institute of Technology)Shilov boundaries of the pluricomplex Green function's level sets (JAPANESE)

[ Abstract ]

In this talk, we study a relation between the Shilov boundaries of the pluricomplex Green function's level sets and supports of Monge-Ampére type currents.

In this talk, we study a relation between the Shilov boundaries of the pluricomplex Green function's level sets and supports of Monge-Ampére type currents.

### 2013/12/09

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

Donaldson-Tian-Yau 予想と K-安定性について (JAPANESE)

**Toshiki Mabuchi**(Osaka University)Donaldson-Tian-Yau 予想と K-安定性について (JAPANESE)

### 2013/12/02

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

A smoothing property of the Bergman projection (ENGLISH)

**Anne-Katrin Herbig**(Nagoya University)A smoothing property of the Bergman projection (ENGLISH)

[ Abstract ]

Let $D$ be a bounded domain with smooth boundary in complex space of dimension $n$. Suppose its Bergman projection $B$ maps the Sobolev space of order $k$ continuously into the one of order $m$. Then the following smoothing result holds: the full Sobolev norm of $Bf$ of order $k$ is controlled by $L^2$-derivatives of $f$ taken along a single, distinguished direction (of order up to $m$). This talk is based on joint work with J. D. McNeal and E. J. Straube.

Let $D$ be a bounded domain with smooth boundary in complex space of dimension $n$. Suppose its Bergman projection $B$ maps the Sobolev space of order $k$ continuously into the one of order $m$. Then the following smoothing result holds: the full Sobolev norm of $Bf$ of order $k$ is controlled by $L^2$-derivatives of $f$ taken along a single, distinguished direction (of order up to $m$). This talk is based on joint work with J. D. McNeal and E. J. Straube.

### 2013/11/25

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

On hyperkaehler metrics on holomorphic cotangent bundles on complex reductive Lie groups (JAPANESE)

**Kota Hattori**(The University of Tokyo)On hyperkaehler metrics on holomorphic cotangent bundles on complex reductive Lie groups (JAPANESE)

[ Abstract ]

There exists a complete hyperkaehler metric on the holomorphic cotangent bundle on each complex reductive Lie group. It was constructed by Kronheimer, using hyperkaehler quotient method. In this talk I explain how to describe the Kaehler potentials of these metrics.

There exists a complete hyperkaehler metric on the holomorphic cotangent bundle on each complex reductive Lie group. It was constructed by Kronheimer, using hyperkaehler quotient method. In this talk I explain how to describe the Kaehler potentials of these metrics.

### 2013/11/18

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

無限型リーマン面に対する安定写像類群とモジュライ空間 (JAPANESE)

**Ege Fujikawa**(Chiba University)無限型リーマン面に対する安定写像類群とモジュライ空間 (JAPANESE)

### 2013/11/11

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

Levi-flat real hypersurfaces with Takeuchi 1-complete complements (JAPANESE)

**Masanori Adachi**(Nagoya University)Levi-flat real hypersurfaces with Takeuchi 1-complete complements (JAPANESE)

[ Abstract ]

In this talk, we discuss compact Levi-flat real hypersurfaces with Takeuchi 1-complete complements from several viewpoints. Based on a Bochner-Hartogs type extension theorem for CR sections over these hypersurfaces, we give an example of a compact Levi-flat CR manifold with a positive CR line bundle whose Ohsawa-Sibony's projective embedding map cannot be transversely infinitely differentiable. We also give a geometrical expression of the Diederich-Fornaess exponents of Takeuchi 1-complete defining functions, and discuss a possible dynamical interpretation of them.

In this talk, we discuss compact Levi-flat real hypersurfaces with Takeuchi 1-complete complements from several viewpoints. Based on a Bochner-Hartogs type extension theorem for CR sections over these hypersurfaces, we give an example of a compact Levi-flat CR manifold with a positive CR line bundle whose Ohsawa-Sibony's projective embedding map cannot be transversely infinitely differentiable. We also give a geometrical expression of the Diederich-Fornaess exponents of Takeuchi 1-complete defining functions, and discuss a possible dynamical interpretation of them.

### 2013/10/28

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

Minimal singular metrics of a line bundle admitting no Zariski decomposition (JAPANESE)

**Takayuki Koike**(The University of Tokyo)Minimal singular metrics of a line bundle admitting no Zariski decomposition (JAPANESE)

[ Abstract ]

We give a concrete expression of a minimal singular metric of a big line bundle on a compact Kähler manifold which is the total space of a toric bundle over a complex torus. In this class of manifolds, Nakayama constructed examples which have line bundles admitting no Zariski decomposition even after any proper modifications. As an application, we discuss the Zariski closedness of non-nef loci and the openness conjecture of Demailly and Kollar in this class.

We give a concrete expression of a minimal singular metric of a big line bundle on a compact Kähler manifold which is the total space of a toric bundle over a complex torus. In this class of manifolds, Nakayama constructed examples which have line bundles admitting no Zariski decomposition even after any proper modifications. As an application, we discuss the Zariski closedness of non-nef loci and the openness conjecture of Demailly and Kollar in this class.

### 2013/10/07

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

The limits on boundary of orbifold Kähler-Einstein metrics and orbifold Kähler-Ricci flows over quasi-projective manifolds (JAPANESE)

**Shin Kikuta**(Sophia University)The limits on boundary of orbifold Kähler-Einstein metrics and orbifold Kähler-Ricci flows over quasi-projective manifolds (JAPANESE)

[ Abstract ]

In this talk, we consider a sequence of orbifold Kähler-Einstein metrics of negative Ricci curvature or corresponding orbifold normalized Kähler-Ricci flows on a quasi-projective manifold with ample log-canonical bundle for a simple normal crossing divisor. Tian-Yau, S. Bando and H. Tsuji established that the sequence of orbifold Kähler-Einstein metrics converged to the complete Käler-Einstein metric of negative Ricci curvature on the complement of the boundary divisor. The main purpose of this talk is to show that such a convergence is also true on the boundary for both of the orbifold Kähler-Einstein metrics and the orbifold normalized Kähler-Ricci flows.

In this talk, we consider a sequence of orbifold Kähler-Einstein metrics of negative Ricci curvature or corresponding orbifold normalized Kähler-Ricci flows on a quasi-projective manifold with ample log-canonical bundle for a simple normal crossing divisor. Tian-Yau, S. Bando and H. Tsuji established that the sequence of orbifold Kähler-Einstein metrics converged to the complete Käler-Einstein metric of negative Ricci curvature on the complement of the boundary divisor. The main purpose of this talk is to show that such a convergence is also true on the boundary for both of the orbifold Kähler-Einstein metrics and the orbifold normalized Kähler-Ricci flows.

### 2013/07/08

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

Cohomologies and deformations of solvmanifolds (JAPANESE)

**Hisashi Kasuya**(Tokyo Institute of Technology)Cohomologies and deformations of solvmanifolds (JAPANESE)

[ Abstract ]

$G$を単連結可解リー群とし, $G$はココンパクト離散部分群$\Gamma$を持つとする. この時, コンパクト等質空間$G/\Gamma$をsolvmanifoldと呼ぶ. 本講演では, solvmanifoldのde Rhamコホモロジー, Dolbeaultコホモロジー, Bott-Chernコホモロジーの計算法を紹介する. さらにその計算法を用いた, ホッジ理論と変形理論の研究を紹介する.

$G$を単連結可解リー群とし, $G$はココンパクト離散部分群$\Gamma$を持つとする. この時, コンパクト等質空間$G/\Gamma$をsolvmanifoldと呼ぶ. 本講演では, solvmanifoldのde Rhamコホモロジー, Dolbeaultコホモロジー, Bott-Chernコホモロジーの計算法を紹介する. さらにその計算法を用いた, ホッジ理論と変形理論の研究を紹介する.

### 2013/06/17

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

有理曲面上の自己同型写像のエントロピー (JAPANESE)

**Takato Uehara**(Niigata University)有理曲面上の自己同型写像のエントロピー (JAPANESE)

### 2013/06/10

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

A Nadel vanishing theorem for metrics with minimal singularities on big line bundles (JAPANESE)

**Shin-ichi Matsumura**(Kagoshima University)A Nadel vanishing theorem for metrics with minimal singularities on big line bundles (JAPANESE)

[ Abstract ]

In this talk, we study singular metrics with non-algebraic singularities, their multiplier ideal sheaves and a Nadel type vanishing theorem, from the view point of complex geometry. The Nadel vanishing theorem can be seen as an analytic version of the Kawamata-Viehweg vanishing theorem of algebraic geometry. The main purpose of this talk is to establish such a theorem for the multiplier ideal sheaf of a metric with minimal singularities, for the cohomology with values in a big line bundle.

In this talk, we study singular metrics with non-algebraic singularities, their multiplier ideal sheaves and a Nadel type vanishing theorem, from the view point of complex geometry. The Nadel vanishing theorem can be seen as an analytic version of the Kawamata-Viehweg vanishing theorem of algebraic geometry. The main purpose of this talk is to establish such a theorem for the multiplier ideal sheaf of a metric with minimal singularities, for the cohomology with values in a big line bundle.

### 2013/06/03

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

Generalized deformation theory of CR structures (JAPANESE)

**Takao Akahori**(University of Hyogo)Generalized deformation theory of CR structures (JAPANESE)

[ Abstract ]

Let $(M, {}^0 T^{''})$ be a compact strongly pseudo convex CR manifold with dimension $2n-1 \geq 5$, embedded in a complex manifold $N$ as a real hypersurface. In our former papers (T. Akahori, Invent. Math. 63 (1981); T. Akahori, P. M. Garfield, and J. M. Lee, Michigan Math. J. 50 (2002)), we constructed the versal family of CR structures. The purpose of this talk is to show that in more wide scope, our family is versal.

Let $(M, {}^0 T^{''})$ be a compact strongly pseudo convex CR manifold with dimension $2n-1 \geq 5$, embedded in a complex manifold $N$ as a real hypersurface. In our former papers (T. Akahori, Invent. Math. 63 (1981); T. Akahori, P. M. Garfield, and J. M. Lee, Michigan Math. J. 50 (2002)), we constructed the versal family of CR structures. The purpose of this talk is to show that in more wide scope, our family is versal.

### 2013/05/27

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

2次元擬斉次特異点の接層のコホモロジーについて (JAPANESE)

**Tomohiro Okuma**(Yamagata University)2次元擬斉次特異点の接層のコホモロジーについて (JAPANESE)

[ Abstract ]

複素2次元特異点の特異点解消上の接層のコホモロジーの次元は解析的不変量である. セミナーでは, リンクが有理ホモロジー球面であるような2次元擬斉次特異点の場合にはそれが位相的不変量であり, グラフから計算できることを紹介する.

複素2次元特異点の特異点解消上の接層のコホモロジーの次元は解析的不変量である. セミナーでは, リンクが有理ホモロジー球面であるような2次元擬斉次特異点の場合にはそれが位相的不変量であり, グラフから計算できることを紹介する.

### 2013/05/20

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

レヴィ平坦面の分類に関する最近の進展 (JAPANESE)

**Takeo Ohsawa**(Nagoya University)レヴィ平坦面の分類に関する最近の進展 (JAPANESE)

[ Abstract ]

レヴィ平坦面の分類がCP^2の場合にできていないことから、種々の興味深い問題が生じているように思われる。ここではトーラスの場合に観察されたことをホップ曲面に拡げたとき、ホップ曲面においてならレヴィ平坦面の分類が(実解析的な場合に限るが)完全にできることを報告する。

レヴィ平坦面の分類がCP^2の場合にできていないことから、種々の興味深い問題が生じているように思われる。ここではトーラスの場合に観察されたことをホップ曲面に拡げたとき、ホップ曲面においてならレヴィ平坦面の分類が(実解析的な場合に限るが)完全にできることを報告する。

### 2013/05/13

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

Geometry and analysis of isolated essential singularities and their applications (JAPANESE)

**Yusuke Okuyama**(Kyoto Institute of Technology)Geometry and analysis of isolated essential singularities and their applications (JAPANESE)

[ Abstract ]

We establish a rescaling principle for isolated essential singularities of holomorphic curves and quasiregular mappings, and gives several applications of it in the theory of value distribution and dynamics. This is a joint work with Pekka Pankka.

We establish a rescaling principle for isolated essential singularities of holomorphic curves and quasiregular mappings, and gives several applications of it in the theory of value distribution and dynamics. This is a joint work with Pekka Pankka.

### 2013/04/22

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

Kobayashi hyperbolic imbeddings into low degree surfaces in three dimensional projective spaces (JAPANESE)

**Yusaku Tiba**(Tokyo Institute of Technology)Kobayashi hyperbolic imbeddings into low degree surfaces in three dimensional projective spaces (JAPANESE)

[ Abstract ]

We construct smooth irreducible curves of the lowest possible degree in quadric and cubic surfaces whose complements are Kobayashi hyperbolically imbedded into those surfaces. This is a joint work with Atsushi Ito.

We construct smooth irreducible curves of the lowest possible degree in quadric and cubic surfaces whose complements are Kobayashi hyperbolically imbedded into those surfaces. This is a joint work with Atsushi Ito.

### 2013/04/15

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

On defining functions for unbounded pseudoconvex domains (ENGLISH)

**Nikolay Shcherbina**(University of Wuppertal)On defining functions for unbounded pseudoconvex domains (ENGLISH)

[ Abstract ]

We show that every strictly pseudoconvex domain $\Omega$ with smooth boundary in a complex manifold $M$ admits a global defining function, i.e. a smooth plurisubharmonic function $\varphi \colon U \to \mathbf{R}$ defined on an open neighbourhood $U \subset M$ of $\Omega$ such that $\Omega =\{ \varphi < 0 \}$, $d\varphi \not= 0$ on $b\Omega$ and $\varphi$ is strictly plurisubharmonic near $b\Omega$. We then introduce the notion of the kernel $K(\Omega)$ of an arbitrary domain $\Omega \subset M$ as the set of all points where every smooth and bounded from above plurisubharmonic function on $\Omega$ fails to be strictly plurisubharmonic. If $\Omega$ is not relatively compact in $M$, then in general $K(\Omega)$ is nonempty, even in the case when $M$ is Stein. It is shown that every strictly pseudoconvex domain $\Omega \subset M$ with smooth boundary admits a global defining function that is strictly plurisubharmonic precisely in the complement of $K(\Omega)$. We then investigate properties of the kernel. Among other results we prove 1-pseudoconcavity of the kernel, we show that in general the kernel does not possess any analytic structure, and we investigate Liouville type properties of the kernel.

We show that every strictly pseudoconvex domain $\Omega$ with smooth boundary in a complex manifold $M$ admits a global defining function, i.e. a smooth plurisubharmonic function $\varphi \colon U \to \mathbf{R}$ defined on an open neighbourhood $U \subset M$ of $\Omega$ such that $\Omega =\{ \varphi < 0 \}$, $d\varphi \not= 0$ on $b\Omega$ and $\varphi$ is strictly plurisubharmonic near $b\Omega$. We then introduce the notion of the kernel $K(\Omega)$ of an arbitrary domain $\Omega \subset M$ as the set of all points where every smooth and bounded from above plurisubharmonic function on $\Omega$ fails to be strictly plurisubharmonic. If $\Omega$ is not relatively compact in $M$, then in general $K(\Omega)$ is nonempty, even in the case when $M$ is Stein. It is shown that every strictly pseudoconvex domain $\Omega \subset M$ with smooth boundary admits a global defining function that is strictly plurisubharmonic precisely in the complement of $K(\Omega)$. We then investigate properties of the kernel. Among other results we prove 1-pseudoconcavity of the kernel, we show that in general the kernel does not possess any analytic structure, and we investigate Liouville type properties of the kernel.

### 2013/04/08

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

K\\"ahler-Einstein metrics and K stability (JAPANESE)

**Akito Futaki**(University of Tokyo)K\\"ahler-Einstein metrics and K stability (JAPANESE)

[ Abstract ]

I will describe an outline of the proof of the equivalence between the existence of K\\"ahler-Einstein metrics and K-stablity after Chen-Donaldson-Sun and Tian.

I will describe an outline of the proof of the equivalence between the existence of K\\"ahler-Einstein metrics and K-stablity after Chen-Donaldson-Sun and Tian.