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Seminar on Geometric Complex Analysis
Seminar information archive ～03/28｜Next seminar｜Future seminars 03/29～
| 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
2014/10/27
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Takayuki Koike (University of Tokyo)
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.)
Shouhei Ma (Tokyo Institute of Technology)
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.)
Gopal Prasad (University of Michigan)
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.)
Keiji Oguiso (Osaka University)
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.)
Junjiro Noguchi (University of Tokyo)
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.)
Hideyuki Ishi (Nagoya University)
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.)
Ryosuke Takahashi (Nagoya University)
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.)
Atsushi Hayashimoto (Nagano National College of Technology)
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.)
Shigeharu Takayama (University of Tokyo)
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.)
Joe Kamimoto (Kyushu university)
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.)
Sunsuke Saito (The University of Tokyo)
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.)
Hikaru Yamamoto (The University of Tokyo)
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.)
Katsutoshi Yamanoi (Tokyo Institute of Technology)
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.)
Junjiro Noguchi (The University of Tokyo)
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.)
Hideki Miyachi (Osaka University)
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.)
Yusaku Tiba (Tokyo Institute of Technology)
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.)
Toshiki Mabuchi (Osaka University)
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.)
Anne-Katrin Herbig (Nagoya University)
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.)
Kota Hattori (The University of Tokyo)
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.)
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)
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.)
Masanori Adachi (Nagoya University)
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.)
Takayuki Koike (The University of Tokyo)
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.)
Shin Kikuta (Sophia University)
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.)
Hisashi Kasuya (Tokyo Institute of Technology)
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.)
Takato Uehara (Niigata University)
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)
