## Seminar on Geometric Complex Analysis

Seminar information archive ～02/26｜Next seminar｜Future seminars 02/27～

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

Organizer(s) | Kengo Hirachi, Shigeharu Takayama |

**Seminar information archive**

### 2015/01/19

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

Hyperbolic span and pseudoconvexity (Japanese)

**Hiroshi Yamaguchi**(Shia University, Prof. emeritus)Hyperbolic span and pseudoconvexity (Japanese)

[ Abstract ]

We show that the hyperbolic span for open torus (which is introduced by M. Shiba in 1993) has the intimate relation with the pseudoconvexity.

We show that the hyperbolic span for open torus (which is introduced by M. Shiba in 1993) has the intimate relation with the pseudoconvexity.

### 2014/12/15

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

The limits of Kähler-Ricci flows

**Hajime Tsuji**(Sophia University)The limits of Kähler-Ricci flows

### 2014/12/08

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

On maximal ideal cycle and fundamental cycle of normal two-dimensional quasi-homogeneous singularities, and singularities with star-shaped resolution (JAPANESE)

**Masatake Tomari**(Nihon University)On maximal ideal cycle and fundamental cycle of normal two-dimensional quasi-homogeneous singularities, and singularities with star-shaped resolution (JAPANESE)

### 2014/12/01

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

Effective and noneffective extension theorems (Japanese)

**Takeo Ohsawa**(Nagoya University)Effective and noneffective extension theorems (Japanese)

[ Abstract ]

As an effective extension theorem, I will review the sharp $L^2$ extension theorem explaining the ideas of its proofs due to Blocki and Guan-Zhou. A new proof using the Poincare metric with be given, too. As a noneffective extension theorem, I will talk about an extension theorem from semipositive divisors. It is obtained as an application of an isomorphism theorem which is essentially contained in my master thesis.

As an effective extension theorem, I will review the sharp $L^2$ extension theorem explaining the ideas of its proofs due to Blocki and Guan-Zhou. A new proof using the Poincare metric with be given, too. As a noneffective extension theorem, I will talk about an extension theorem from semipositive divisors. It is obtained as an application of an isomorphism theorem which is essentially contained in my master thesis.

### 2014/11/17

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

On strong K-stability of polarized algebraic manifolds (JAPANESE)

**Yasufumi Nitta**(Tokyo Institute of Technology)On strong K-stability of polarized algebraic manifolds (JAPANESE)

### 2014/11/10

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

On a convex level set of a plurisubharmonic function and the support of the Monge-Ampere current (JAPANESE)

**Yusaku Tiba**(Tokyo Institute of Technology)On a convex level set of a plurisubharmonic function and the support of the Monge-Ampere current (JAPANESE)

[ Abstract ]

In this talk, we study a geometric property of a continuous plurisubharmonic function which is a solution of the Monge-Ampere equation and has a convex level set. By using our results and Lempert's results, we show a relation between the supports of the Monge-Ampere currents and complex $k$-extreme points of closed balls for the Kobayashi distance in a bounded convex domain in $C^n$.

In this talk, we study a geometric property of a continuous plurisubharmonic function which is a solution of the Monge-Ampere equation and has a convex level set. By using our results and Lempert's results, we show a relation between the supports of the Monge-Ampere currents and complex $k$-extreme points of closed balls for the Kobayashi distance in a bounded convex domain in $C^n$.

### 2014/10/27

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

On the minimality of canonically attached singular Hermitian metrics on certain nef line bundles (JAPANESE)

**Takayuki Koike**(University of Tokyo)On the minimality of canonically attached singular Hermitian metrics on certain nef line bundles (JAPANESE)

[ Abstract ]

We apply Ueda theory to a study of singular Hermitian metrics of a (strictly) nef line bundle $L$. Especially we study minimal singular metrics of $L$, metrics of $L$ with the mildest singularities among singular Hermitian metrics of $L$ whose local weights are plurisubharmonic. In some situations, we determine a minimal singular metric of $L$. As an application, we give new examples of (strictly) nef line bundles which admit no smooth Hermitian metric with semi-positive curvature.

We apply Ueda theory to a study of singular Hermitian metrics of a (strictly) nef line bundle $L$. Especially we study minimal singular metrics of $L$, metrics of $L$ with the mildest singularities among singular Hermitian metrics of $L$ whose local weights are plurisubharmonic. In some situations, we determine a minimal singular metric of $L$. As an application, we give new examples of (strictly) nef line bundles which admit no smooth Hermitian metric with semi-positive curvature.

### 2014/10/20

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

Kodaira dimension of modular variety of type IV (JAPANESE)

**Shouhei Ma**(Tokyo Institute of Technology)Kodaira dimension of modular variety of type IV (JAPANESE)

### 2014/07/14

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

Higher dimensional analogues of fake projective planes (ENGLISH)

**Gopal Prasad**(University of Michigan)Higher dimensional analogues of fake projective planes (ENGLISH)

[ Abstract ]

A fake projective plane is a smooth projective complex algebraic surface which is not isomorphic to the complex projective plane but whose Betti numbers are that of the complex projective plane. The fake projective planes are algebraic surfaces of general type and have smallest possible Euler-Poincare characteristic among them. The first fake projective plane was constructed by D. Mumford using p-adic uniformization, and it was known that there can only be finitely many of them. A complete classification of the fake projective planes was obtained by Sai-Kee Yeung and myself. We showed that there are 28 classes of them, and constructed at least one explicit example in each class. Later, using long computer assisted computations, D. Cartwright and Tim Steger found that the 28 families altogether contain precisely 100 fake projective planes. Using our work, they also found a very interesting smooth projective complex algebraic surface whose Euler-Poincare characteristic is 3 but whose first Betti

number is 2. We have a natural notion of higher dimensional analogues of fake projective planes and to a large extent determined them. My talk will be devoted to an exposition of this work.

A fake projective plane is a smooth projective complex algebraic surface which is not isomorphic to the complex projective plane but whose Betti numbers are that of the complex projective plane. The fake projective planes are algebraic surfaces of general type and have smallest possible Euler-Poincare characteristic among them. The first fake projective plane was constructed by D. Mumford using p-adic uniformization, and it was known that there can only be finitely many of them. A complete classification of the fake projective planes was obtained by Sai-Kee Yeung and myself. We showed that there are 28 classes of them, and constructed at least one explicit example in each class. Later, using long computer assisted computations, D. Cartwright and Tim Steger found that the 28 families altogether contain precisely 100 fake projective planes. Using our work, they also found a very interesting smooth projective complex algebraic surface whose Euler-Poincare characteristic is 3 but whose first Betti

number is 2. We have a natural notion of higher dimensional analogues of fake projective planes and to a large extent determined them. My talk will be devoted to an exposition of this work.

### 2014/06/30

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

Primitive automorphisms of positive entropy of rational and Calabi-Yau threefolds (JAPANESE)

**Keiji Oguiso**(Osaka University)Primitive automorphisms of positive entropy of rational and Calabi-Yau threefolds (JAPANESE)

### 2014/06/23

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

A remark to the division algorithm in the proof of Oka's First Coherence Theorem (JAPANESE)

**Junjiro Noguchi**(University of Tokyo)A remark to the division algorithm in the proof of Oka's First Coherence Theorem (JAPANESE)

[ Abstract ]

The problem is the local finite generation of a relation sheaf $R(f_1, \ldots, f_q)$ in $\mathcal{O}_n=\mathcal{O}_{C^n}$. After $f_j$ reduced to Weierstrass' polynomials in $z_n$, it is the key to apply the induction in $n$ to show that elements of $R(f_1, \ldots, q)$ are expressed by $z_n$-polynomial-like elements of degree at most $p=\max_j\deg f_j$ over $\mathcal{O}_n$. In that proof one is used to use a divison by $f_j$ of $\deg f_j=p$ (Oka '48, Cartan '50, Hörmander, Demailly, . . .). In this talk we shall confirm that the division abve works by making use of $f_k$ of the minimum degree $\min_j \deg f_j$. This proof is natrually compatible with the simple case when some $f_j$ is a unit, and gives some improvement in the degree estimate of generators.

The problem is the local finite generation of a relation sheaf $R(f_1, \ldots, f_q)$ in $\mathcal{O}_n=\mathcal{O}_{C^n}$. After $f_j$ reduced to Weierstrass' polynomials in $z_n$, it is the key to apply the induction in $n$ to show that elements of $R(f_1, \ldots, q)$ are expressed by $z_n$-polynomial-like elements of degree at most $p=\max_j\deg f_j$ over $\mathcal{O}_n$. In that proof one is used to use a divison by $f_j$ of $\deg f_j=p$ (Oka '48, Cartan '50, Hörmander, Demailly, . . .). In this talk we shall confirm that the division abve works by making use of $f_k$ of the minimum degree $\min_j \deg f_j$. This proof is natrually compatible with the simple case when some $f_j$ is a unit, and gives some improvement in the degree estimate of generators.

### 2014/06/16

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

New examples of weighted Bergman kernels on a certain non-homogeneous Siegel domain (JAPANESE)

**Hideyuki Ishi**(Nagoya University)New examples of weighted Bergman kernels on a certain non-homogeneous Siegel domain (JAPANESE)

### 2014/06/09

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

Modified Kähler-Ricci flow on projective bundles (JAPANESE)

**Ryosuke Takahashi**(Nagoya University)Modified Kähler-Ricci flow on projective bundles (JAPANESE)

### 2014/06/02

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

Generalized pseudoellipsoids and proper holomorphic mappings between them (JAPANESE)

**Atsushi Hayashimoto**(Nagano National College of Technology)Generalized pseudoellipsoids and proper holomorphic mappings between them (JAPANESE)

### 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.