本文印刷
全画面プリント
複素解析幾何セミナー
過去の記録 ～03/06｜次回の予定｜今後の予定 03/07～
| 開催情報 | 月曜日 10:30～12:00 数理科学研究科棟(駒場) 128号室 |
|---|---|
| 担当者 | 平地 健吾, 高山 茂晴 |
過去の記録
2017年11月13日(月)
10:30-12:00 数理科学研究科棟(駒場) 128号室
Georg Schumacher 氏 (Philipps-Universität Marburg)
Relative Canonical Bundles for Families of Calabi-Yau Manifolds
Georg Schumacher 氏 (Philipps-Universität Marburg)
Relative Canonical Bundles for Families of Calabi-Yau Manifolds
[ 講演概要 ]
We consider holomorphic families of Calabi-Yau manifolds (here being defined by the vanishing of the first real Chern class). We study induced hermitian metrics on the relative canonical bundle, which are related to the Weil-Petersson form on the base. Under a certain condition the total space possesses a Kähler form, whose restriction to fibers are equal to the Ricci flat metrics. Furthermore we prove an extension theorem for the Weil-Petersson form and give applications.
We consider holomorphic families of Calabi-Yau manifolds (here being defined by the vanishing of the first real Chern class). We study induced hermitian metrics on the relative canonical bundle, which are related to the Weil-Petersson form on the base. Under a certain condition the total space possesses a Kähler form, whose restriction to fibers are equal to the Ricci flat metrics. Furthermore we prove an extension theorem for the Weil-Petersson form and give applications.
2017年10月30日(月)
10:30-12:00 数理科学研究科棟(駒場) 128号室
加藤 昌英 氏 (上智大学)
Odd dimensional complex analytic Kleinian groups
加藤 昌英 氏 (上智大学)
Odd dimensional complex analytic Kleinian groups
[ 講演概要 ]
In this talk, I would explain an idea to construct a higher dimensional analogue of the classical Kleinian group theory. For a group $G$ of a certain class of discrete subgroups of $\mathrm{PGL}(2n+2,\mathbf{C})$ which act on $\mathbf{P}^{2n+1}$, there is a canonical way to define the region of discontinuity, on which $G$ acts properly discontinuously. General principle in the discussion is to regard $\mathbf{P}^{n}$ in $\mathbf{P}^{2n+1}$ as a single point. We can consider the quotient space of the discontinuity region by the action of $G$. Though the Ahlfors finiteness theorem is hopeless because of a counter example, the Klein combination theorem and the handle attachment can be defined similarly. Any compact quotients which appear here are non-Kaehler. In the case $n=1$, we explain a new example of compact quotients which is related to a classical Kleinian group.
In this talk, I would explain an idea to construct a higher dimensional analogue of the classical Kleinian group theory. For a group $G$ of a certain class of discrete subgroups of $\mathrm{PGL}(2n+2,\mathbf{C})$ which act on $\mathbf{P}^{2n+1}$, there is a canonical way to define the region of discontinuity, on which $G$ acts properly discontinuously. General principle in the discussion is to regard $\mathbf{P}^{n}$ in $\mathbf{P}^{2n+1}$ as a single point. We can consider the quotient space of the discontinuity region by the action of $G$. Though the Ahlfors finiteness theorem is hopeless because of a counter example, the Klein combination theorem and the handle attachment can be defined similarly. Any compact quotients which appear here are non-Kaehler. In the case $n=1$, we explain a new example of compact quotients which is related to a classical Kleinian group.
2017年10月23日(月)
10:30-12:00 数理科学研究科棟(駒場) 128号室
台風のため11月27日に延期となりました
細野 元気 氏 (東京大学)
On the proof of the optimal $L^2$ extension theorem by Berndtsson-Lempert and related results
台風のため11月27日に延期となりました
細野 元気 氏 (東京大学)
On the proof of the optimal $L^2$ extension theorem by Berndtsson-Lempert and related results
[ 講演概要 ]
We will present the recent progress on the Ohsawa-Takegoshi $L^2$ extension theorem. A version of the Ohsawa-Takegoshi $L^2$ extension with a optimal estimate has been proved by Blocki and Guan-Zhou. After that, by Berndtsson-Lempert, a new proof of the optimal $L^2$ extension theorem was given. In this talk, we will show an optimal $L^2$ extension theorem for jets of holomorphic functions by the Berndtsson-Lempert method. We will also explain the recent result about jet extensions by McNeal-Varolin. Their proof is also based on Berndtsson-Lempert, but there are some differences.
We will present the recent progress on the Ohsawa-Takegoshi $L^2$ extension theorem. A version of the Ohsawa-Takegoshi $L^2$ extension with a optimal estimate has been proved by Blocki and Guan-Zhou. After that, by Berndtsson-Lempert, a new proof of the optimal $L^2$ extension theorem was given. In this talk, we will show an optimal $L^2$ extension theorem for jets of holomorphic functions by the Berndtsson-Lempert method. We will also explain the recent result about jet extensions by McNeal-Varolin. Their proof is also based on Berndtsson-Lempert, but there are some differences.
2017年10月16日(月)
10:30-12:00 数理科学研究科棟(駒場) 128号室
須川 敏幸 氏 (東北大学)
Characterizations of hyperbolically $k$-convex domains in terms of hyperbolic metric
須川 敏幸 氏 (東北大学)
Characterizations of hyperbolically $k$-convex domains in terms of hyperbolic metric
[ 講演概要 ]
It is known that a plane domain $X$ with hyperbolic metric $h_X=h_X(z)|dz|$ of constant curvature $-4$ is (Euclidean) convex if and only if $h_X(z)d_X(z)\ge 1/2$, where $d_X(z)$ denotes the Euclidean distance from a point $z$ in $X$ to the boundary of $X$. We will consider spherical and hyperbolic versions of this result. More generally, we consider hyperbolic $k$-convexity (in the sense of Mejia and Minda) in the same line. A key is to observe a detailed behaviour of the hyperbolic density $h_X(z)$ near the boundary.
It is known that a plane domain $X$ with hyperbolic metric $h_X=h_X(z)|dz|$ of constant curvature $-4$ is (Euclidean) convex if and only if $h_X(z)d_X(z)\ge 1/2$, where $d_X(z)$ denotes the Euclidean distance from a point $z$ in $X$ to the boundary of $X$. We will consider spherical and hyperbolic versions of this result. More generally, we consider hyperbolic $k$-convexity (in the sense of Mejia and Minda) in the same line. A key is to observe a detailed behaviour of the hyperbolic density $h_X(z)$ near the boundary.
2017年10月02日(月)
10:30-12:00 数理科学研究科棟(駒場) 128号室
千葉 優作 氏 (お茶の水女子大学)
The extension of holomorphic functions on a non-pluriharmonic locus
千葉 優作 氏 (お茶の水女子大学)
The extension of holomorphic functions on a non-pluriharmonic locus
[ 講演概要 ]
Let $n \geq 4$ and let $\Omega$ be a bounded hyperconvex domain in $\mathbb{C}^{n}$. Let $\varphi$ be a negative exhaustive smooth plurisubharmonic function on $\Omega$. In this talk, we show that any holomorphic function defined on a connected open neighborhood of the support of $(i\partial \overline{\partial}\varphi)^{n-3}$ can be extended to the holomorphic function on $\Omega$.
Let $n \geq 4$ and let $\Omega$ be a bounded hyperconvex domain in $\mathbb{C}^{n}$. Let $\varphi$ be a negative exhaustive smooth plurisubharmonic function on $\Omega$. In this talk, we show that any holomorphic function defined on a connected open neighborhood of the support of $(i\partial \overline{\partial}\varphi)^{n-3}$ can be extended to the holomorphic function on $\Omega$.
2017年09月25日(月)
10:30-12:00 数理科学研究科棟(駒場) 128号室
Christophe Mourougane 氏 (Université de Rennes 1)
Asymptotics of $L^2$ and Quillen metrics in degenerations of Calabi-Yau varieties
Christophe Mourougane 氏 (Université de Rennes 1)
Asymptotics of $L^2$ and Quillen metrics in degenerations of Calabi-Yau varieties
[ 講演概要 ]
It is a joint work with Dennis Eriksson and Gerard Freixas i Montplet.
Our first motivation is to give a metric analogue of Kodaira's canonical bundle formula for elliptic surfaces, in the case of families of Calabi-Yau varieties. We consider degenerations of complex projective Calabi-Yau varieties and study the singularities of $L^2$, Quillen and BCOV metrics on Hodge and determinant bundles. The dominant and subdominant terms in the expansions of the metrics close to non-smooth fibres are shown to be related to well-known topological invariants of singularities, such as limit Hodge structures, vanishing cycles and log-canonical thresholds.
It is a joint work with Dennis Eriksson and Gerard Freixas i Montplet.
Our first motivation is to give a metric analogue of Kodaira's canonical bundle formula for elliptic surfaces, in the case of families of Calabi-Yau varieties. We consider degenerations of complex projective Calabi-Yau varieties and study the singularities of $L^2$, Quillen and BCOV metrics on Hodge and determinant bundles. The dominant and subdominant terms in the expansions of the metrics close to non-smooth fibres are shown to be related to well-known topological invariants of singularities, such as limit Hodge structures, vanishing cycles and log-canonical thresholds.
2017年07月03日(月)
10:30-12:00 数理科学研究科棟(駒場) 128号室
長友 康行 氏 (明治大学)
Holomorphic isometric embeddings into Grassmannians of rank $2$
長友 康行 氏 (明治大学)
Holomorphic isometric embeddings into Grassmannians of rank $2$
[ 講演概要 ]
We suppose that Grassmannians are equipped with the standard Kähler metrics of Fubini-Study type. This means that the ${\it universal \ quotient}$ bundles over Grassmannians are provided with not only fibre metrics but also connections. Such connections are called the ${\it canonical}$ connection.
First of all, we classify $\text{SU}(2)$ equivariant holomorphic embeddings of the complex projective line into complex Grassmannians of $2$-planes. To do so, we focus our attention on the pull-back connection of the canonical connection, which is an $\text{SU}(2)$ invariant connection by our hypothesis. We use ${\it extensions}$ of vector bundles to classify $\text{SU}(2)$ invariant connections on vector bundles of rank $2$ over the complex projective line. Since the extensions are in one-to-one correspondence with $H^1(\mathbf CP^1;\mathcal O(-2))$, the moduli space of non-trivial invariant connections modulo gauge transformations is identified with the quotient space of $H^1(\mathbf CP^1;\mathcal O(-2))$ by $S^1$-action. The positivity of the mean curvature of the pull-back connection implies that the moduli spaces of $\text{SU}(2)$ equivariant holomorphic embeddings are the open intervals $(0,l)$, where $l$ depends only on the ${\it degree}$ of the map.
Next, we describe moduli spaces of holomorphic isometric embeddings of the complex projective line into complex quadric hypersurfaces of the projective spaces. A harmonic map from a Riemannian manifold into a Grassmannian is characterized by the universal quotient bundle, a space of sections of the bundle and the Laplace operator. This characterization can be considered as a generalization of Theorem of Takahashi on minimal immersions into a sphere (J.Math.Soc.Japan 18 (1966)). Due to this, we can generalize do Carmo-Wallach theory. We apply a generalized do Carmo-Wallach theory to obtain the moduli spaces. This method also gives a description of the moduli space of Einstein-Hermitian harmonic maps with constant Kähler angles of the complex projective line into complex quadrics. It turns out that the moduli space is diffeomorphic to the moduli of holomorphic isometric embeddings of the same degree.
We suppose that Grassmannians are equipped with the standard Kähler metrics of Fubini-Study type. This means that the ${\it universal \ quotient}$ bundles over Grassmannians are provided with not only fibre metrics but also connections. Such connections are called the ${\it canonical}$ connection.
First of all, we classify $\text{SU}(2)$ equivariant holomorphic embeddings of the complex projective line into complex Grassmannians of $2$-planes. To do so, we focus our attention on the pull-back connection of the canonical connection, which is an $\text{SU}(2)$ invariant connection by our hypothesis. We use ${\it extensions}$ of vector bundles to classify $\text{SU}(2)$ invariant connections on vector bundles of rank $2$ over the complex projective line. Since the extensions are in one-to-one correspondence with $H^1(\mathbf CP^1;\mathcal O(-2))$, the moduli space of non-trivial invariant connections modulo gauge transformations is identified with the quotient space of $H^1(\mathbf CP^1;\mathcal O(-2))$ by $S^1$-action. The positivity of the mean curvature of the pull-back connection implies that the moduli spaces of $\text{SU}(2)$ equivariant holomorphic embeddings are the open intervals $(0,l)$, where $l$ depends only on the ${\it degree}$ of the map.
Next, we describe moduli spaces of holomorphic isometric embeddings of the complex projective line into complex quadric hypersurfaces of the projective spaces. A harmonic map from a Riemannian manifold into a Grassmannian is characterized by the universal quotient bundle, a space of sections of the bundle and the Laplace operator. This characterization can be considered as a generalization of Theorem of Takahashi on minimal immersions into a sphere (J.Math.Soc.Japan 18 (1966)). Due to this, we can generalize do Carmo-Wallach theory. We apply a generalized do Carmo-Wallach theory to obtain the moduli spaces. This method also gives a description of the moduli space of Einstein-Hermitian harmonic maps with constant Kähler angles of the complex projective line into complex quadrics. It turns out that the moduli space is diffeomorphic to the moduli of holomorphic isometric embeddings of the same degree.
2017年06月26日(月)
10:30-12:00 数理科学研究科棟(駒場) 128号室
二木 昭人 氏 (東京大学)
Volume minimization and obstructions to geometric problems
二木 昭人 氏 (東京大学)
Volume minimization and obstructions to geometric problems
[ 講演概要 ]
We discuss on the volume minimization principle for conformally Kaehler Einstein-Maxwell metrics in the similar spirit as the Kaehler-Ricci solitons and Sasaki-Einstein metrics. This talk is base on a joint work with Hajime Ono.
We discuss on the volume minimization principle for conformally Kaehler Einstein-Maxwell metrics in the similar spirit as the Kaehler-Ricci solitons and Sasaki-Einstein metrics. This talk is base on a joint work with Hajime Ono.
2017年06月19日(月)
10:30-12:00 数理科学研究科棟(駒場) 128号室
竹内 有哉 氏 (東京大学)
$Q$-prime curvature and Sasakian $\eta$-Einstein manifolds
竹内 有哉 氏 (東京大学)
$Q$-prime curvature and Sasakian $\eta$-Einstein manifolds
[ 講演概要 ]
The $Q$-prime curvature is defined for a pseudo-Einstein contact form on a strictly pseudoconvex CR manifold, and its integral, the total $Q$-prime curvature, defines a global CR invariant under some assumptions. In this talk, we will compute the $Q$-prime curvature for Sasakian $\eta$-Einstein manifolds. We will also study the first and the second variation of the total $Q$-prime curvature under deformations of real hypersurfaces at Sasakian $\eta$-Einstein manifolds.
The $Q$-prime curvature is defined for a pseudo-Einstein contact form on a strictly pseudoconvex CR manifold, and its integral, the total $Q$-prime curvature, defines a global CR invariant under some assumptions. In this talk, we will compute the $Q$-prime curvature for Sasakian $\eta$-Einstein manifolds. We will also study the first and the second variation of the total $Q$-prime curvature under deformations of real hypersurfaces at Sasakian $\eta$-Einstein manifolds.
2017年06月12日(月)
10:30-12:00 数理科学研究科棟(駒場) 128号室
松本 佳彦 氏 (大阪大学)
On Sp(2)-invariant asymptotically complex hyperbolic Einstein metrics on the 8-ball
松本 佳彦 氏 (大阪大学)
On Sp(2)-invariant asymptotically complex hyperbolic Einstein metrics on the 8-ball
[ 講演概要 ]
Following a pioneering work of Pedersen, Hitchin studied SU(2)-invariant asymptotically real/complex hyperbolic (often abbreviated as AH/ACH) solution to the Einstein equation on the 4-dimensional unit open ball. We discuss a similar problem on the 8-ball, on which the quaternionic unitary group Sp(2) acts naturally, focusing on ACH solutions rather than AH ones. The Einstein equation is treated as an asymptotic Dirichlet problem, and the Dirichlet data are Sp(2)-invariant “partially integrable” CR structures on the 7-sphere. A remarkable point is that most of such structures are actually non-integrable. I will present how we can practically compute the formal series expansion of the Einstein ACH metric corresponding to a given Dirichlet data, that is, an invariant partially integrable CR structure on the sphere.
Following a pioneering work of Pedersen, Hitchin studied SU(2)-invariant asymptotically real/complex hyperbolic (often abbreviated as AH/ACH) solution to the Einstein equation on the 4-dimensional unit open ball. We discuss a similar problem on the 8-ball, on which the quaternionic unitary group Sp(2) acts naturally, focusing on ACH solutions rather than AH ones. The Einstein equation is treated as an asymptotic Dirichlet problem, and the Dirichlet data are Sp(2)-invariant “partially integrable” CR structures on the 7-sphere. A remarkable point is that most of such structures are actually non-integrable. I will present how we can practically compute the formal series expansion of the Einstein ACH metric corresponding to a given Dirichlet data, that is, an invariant partially integrable CR structure on the sphere.
2017年05月29日(月)
10:30-12:00 数理科学研究科棟(駒場) 128号室
澤井 洋 氏 (沼津工業高等専門学校)
LCK structures on compact solvmanifolds
澤井 洋 氏 (沼津工業高等専門学校)
LCK structures on compact solvmanifolds
[ 講演概要 ]
A locally conformal Kähler (in short LCK) manifold is said to be Vaisman if Lee form is parallel with respect to Levi-Civita connection. In this talk, we prove that a Vaisman structure on a compact solvmanifolds depends only on the form of the fundamental 2-form, and it do not depends on a complex structure. As an application, we give the structure theorem for Vaisman (completely solvable) solvmanifolds and LCK nilmanifolds. Moreover, we show the existence of LCK solvmanifolds without Vaisman structures.
A locally conformal Kähler (in short LCK) manifold is said to be Vaisman if Lee form is parallel with respect to Levi-Civita connection. In this talk, we prove that a Vaisman structure on a compact solvmanifolds depends only on the form of the fundamental 2-form, and it do not depends on a complex structure. As an application, we give the structure theorem for Vaisman (completely solvable) solvmanifolds and LCK nilmanifolds. Moreover, we show the existence of LCK solvmanifolds without Vaisman structures.
2017年05月22日(月)
10:30-12:00 数理科学研究科棟(駒場) 128号室
小池 貴之 氏 (京都大学)
Complex K3 surfaces containing Levi-flat hypersurfaces
小池 貴之 氏 (京都大学)
Complex K3 surfaces containing Levi-flat hypersurfaces
[ 講演概要 ]
We show the existence of a complex K3 surface $X$ which is not a Kummer surface and has a one-parameter family of Levi-flat hypersurfaces in which all the leaves are dense. We construct such $X$ by patching two open complex surfaces obtained as the complements of tubular neighborhoods of elliptic curves embedded in blow-ups of the projective planes at general nine points.
We show the existence of a complex K3 surface $X$ which is not a Kummer surface and has a one-parameter family of Levi-flat hypersurfaces in which all the leaves are dense. We construct such $X$ by patching two open complex surfaces obtained as the complements of tubular neighborhoods of elliptic curves embedded in blow-ups of the projective planes at general nine points.
2017年05月15日(月)
10:30-12:00 数理科学研究科棟(駒場) 128号室
服部 広大 氏 (慶應義塾大学)
On the moduli spaces of the tangent cones at infinity of some hyper-Kähler manifolds
服部 広大 氏 (慶應義塾大学)
On the moduli spaces of the tangent cones at infinity of some hyper-Kähler manifolds
[ 講演概要 ]
For a metric space $(X,d)$, the Gromov-Hausdorff limit of $(X, a_n d)$ as $a_n \rightarrow 0$ is called the tangent cone at infinity of $(X,d)$. Although the tangent cone at infinity always exists if $(X,d)$ comes from a complete Riemannian metric with nonnegative Ricci curvature, the uniqueness does not hold in general. Colding and Minicozzi showed the uniqueness under the assumption that $(X,d)$ is a Ricci-flat manifold satisfying some additional conditions.
In this talk, I will explain a example of noncompact complete hyper-Kähler manifold who has several tangent cones at infinity, and determine the moduli space of them.
For a metric space $(X,d)$, the Gromov-Hausdorff limit of $(X, a_n d)$ as $a_n \rightarrow 0$ is called the tangent cone at infinity of $(X,d)$. Although the tangent cone at infinity always exists if $(X,d)$ comes from a complete Riemannian metric with nonnegative Ricci curvature, the uniqueness does not hold in general. Colding and Minicozzi showed the uniqueness under the assumption that $(X,d)$ is a Ricci-flat manifold satisfying some additional conditions.
In this talk, I will explain a example of noncompact complete hyper-Kähler manifold who has several tangent cones at infinity, and determine the moduli space of them.
2017年05月08日(月)
10:30-12:00 数理科学研究科棟(駒場) 128号室
藤澤 太郎 氏 (東京電機大学)
Semipositivity theorems for a variation of Hodge structure
藤澤 太郎 氏 (東京電機大学)
Semipositivity theorems for a variation of Hodge structure
[ 講演概要 ]
I will talk about my recent joint work with Osamu Fujino. The main purpose of our joint work is to generalize the Fujita-Zukcer-Kawamata semipositivity theorem from the analytic viewpoint. In this talk, I would like to illustrate the relation between the two objects, the asymptotic behavior of a variation of Hodge structure and good properties of the induced singular hermitian metric on an invertible subbundle of the Hodge bundle.
I will talk about my recent joint work with Osamu Fujino. The main purpose of our joint work is to generalize the Fujita-Zukcer-Kawamata semipositivity theorem from the analytic viewpoint. In this talk, I would like to illustrate the relation between the two objects, the asymptotic behavior of a variation of Hodge structure and good properties of the induced singular hermitian metric on an invertible subbundle of the Hodge bundle.
2017年04月24日(月)
10:30-12:00 数理科学研究科棟(駒場) 128号室
今野 宏 氏 (明治大学)
Lagrangian Mean Curvature Flows and Moment maps
今野 宏 氏 (明治大学)
Lagrangian Mean Curvature Flows and Moment maps
[ 講演概要 ]
In this talk, we construct various examples of Lagrangian mean curvature flows in Calabi-Yau manifolds, using moment maps for actions of abelian Lie groups on them. The examples include Lagrangian self-shrinkers and translating solitons in the Euclid spaces. We also construct Lagrangian mean curvature flows in non-flat Calabi-Yau manifolds. In particular, we describe Lagrangian mean curvature flows in 4-dimensional Ricci-flat ALE spaces in detail and investigate their singularities.
In this talk, we construct various examples of Lagrangian mean curvature flows in Calabi-Yau manifolds, using moment maps for actions of abelian Lie groups on them. The examples include Lagrangian self-shrinkers and translating solitons in the Euclid spaces. We also construct Lagrangian mean curvature flows in non-flat Calabi-Yau manifolds. In particular, we describe Lagrangian mean curvature flows in 4-dimensional Ricci-flat ALE spaces in detail and investigate their singularities.
2017年04月17日(月)
10:30-12:00 数理科学研究科棟(駒場) 128号室
日下部 佑太 氏 (大阪大学)
Dense holomorphic curves in spaces of holomorphic maps
日下部 佑太 氏 (大阪大学)
Dense holomorphic curves in spaces of holomorphic maps
[ 講演概要 ]
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年04月10日(月)
10:30-12:00 数理科学研究科棟(駒場) 128号室
平地 健吾 氏 (東京大学)
Slice theorem for CR structures near the sphere and its applications
平地 健吾 氏 (東京大学)
Slice theorem for CR structures near the sphere and its applications
[ 講演概要 ]
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年03月06日(月)
10:00-11:30 数理科学研究科棟(駒場) 128号室
いつもと時間が異なります。
Vladimir Matveev 氏 (University of Jena)
Projective and c-projective metric geometries: why they are so similar (ENGLISH)
いつもと時間が異なります。
Vladimir Matveev 氏 (University of Jena)
Projective and c-projective metric geometries: why they are so similar (ENGLISH)
[ 講演概要 ]
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年02月13日(月)
10:30-12:00 数理科学研究科棟(駒場) 128号室
Qi'an Guan 氏 (北京大学)
A Characterization of regular points by Ohsawa-Takegoshi Extension Theorem (ENGLISH)
Qi'an Guan 氏 (北京大学)
A Characterization of regular points by Ohsawa-Takegoshi Extension Theorem (ENGLISH)
[ 講演概要 ]
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年01月23日(月)
10:30-12:00 数理科学研究科棟(駒場) 128号室
野口 潤次郎 氏 (東京大学)
A unified proof of Cousin I, II and d-bar equation on domains of holomorphy (JAPANESE)
野口 潤次郎 氏 (東京大学)
A unified proof of Cousin I, II and d-bar equation on domains of holomorphy (JAPANESE)
[ 講演概要 ]
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年01月16日(月)
10:30-12:00 数理科学研究科棟(駒場) 128号室
Dinh Tuan Huynh 氏 (大阪大学)
A geometric second main theorem (ENGLISH)
Dinh Tuan Huynh 氏 (大阪大学)
A geometric second main theorem (ENGLISH)
[ 講演概要 ]
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年12月12日(月)
10:30-12:00 数理科学研究科棟(駒場) 128号室
川上 裕 氏 (金沢大学)
完備極小曲面の研究の最近の進展について (JAPANESE)
川上 裕 氏 (金沢大学)
完備極小曲面の研究の最近の進展について (JAPANESE)
[ 講演概要 ]
ユークリッド空間内の完備極小曲面の性質は様々な観点から研究されていますが、複素解析的視点からの研究は近年においてもますます進展しています。そこで本講演では、完備極小曲面の研究の最近の進展について、講演者の成果も交えてお話したいと思います。
ユークリッド空間内の完備極小曲面の性質は様々な観点から研究されていますが、複素解析的視点からの研究は近年においてもますます進展しています。そこで本講演では、完備極小曲面の研究の最近の進展について、講演者の成果も交えてお話したいと思います。
2016年12月05日(月)
10:30-12:00 数理科学研究科棟(駒場) 128号室
大場 貴裕 氏 (東京工業大学)
Stein充填を無限個もつ高次元接触多様体の構成について
(JAPANESE)
大場 貴裕 氏 (東京工業大学)
Stein充填を無限個もつ高次元接触多様体の構成について
(JAPANESE)
[ 講演概要 ]
与えられた接触多様体に対し，それを境界にもつ Stein 領域のことを，その接触多様体の Stein 充填という．これまでに，Stein 充填を無限個もつ 3 次元接触多様体の例は数多く知られている．一方で，5 次元以上の接触多様体でそのような例は知られていない．3 次元の場合には，オープンブック分解や Lefschetz ファイバー空間といった多様体上のファイバー構造を利用して構成されたが，その際に曲面の写像類群が重要な役割を果たしている．高次元の場合も，これらの空間に対応する概念はあるものの，写像類群が高次元多様体のものとなり，その解析は一般には難しい．本講演では，ファイバーとなる高次元多様体として特別なものを選ぶことにより，ファイバー構造を用いて Stein 充填を無限個もつ高次元接触多様体が構成できることを紹介する．
与えられた接触多様体に対し，それを境界にもつ Stein 領域のことを，その接触多様体の Stein 充填という．これまでに，Stein 充填を無限個もつ 3 次元接触多様体の例は数多く知られている．一方で，5 次元以上の接触多様体でそのような例は知られていない．3 次元の場合には，オープンブック分解や Lefschetz ファイバー空間といった多様体上のファイバー構造を利用して構成されたが，その際に曲面の写像類群が重要な役割を果たしている．高次元の場合も，これらの空間に対応する概念はあるものの，写像類群が高次元多様体のものとなり，その解析は一般には難しい．本講演では，ファイバーとなる高次元多様体として特別なものを選ぶことにより，ファイバー構造を用いて Stein 充填を無限個もつ高次元接触多様体が構成できることを紹介する．
2016年11月28日(月)
10:30-12:00 数理科学研究科棟(駒場) 128号室
中村 聡 氏 (東北大学)
偏極トーリック多様体の対数的チャウ準安定性について (JAPANESE)
中村 聡 氏 (東北大学)
偏極トーリック多様体の対数的チャウ準安定性について (JAPANESE)
[ 講演概要 ]
対数的チャウ(準)安定性とは，射影代数多様体とその因子との組に対して定義される代数幾何学的 (特に幾何学的不変式論的) 概念である．本講演では，偏極トーリック多様体とそのトーリック因子の組が準安定であるための障害について得られた結果を紹介する．またその応用として，(1) 対数的チャウ安定性と対数的K-安定性の関係や, (2) いくつかの具体例でこの障害を計算し，Conical Kahler Einstein 計量の存在との関連を紹介する．
対数的チャウ(準)安定性とは，射影代数多様体とその因子との組に対して定義される代数幾何学的 (特に幾何学的不変式論的) 概念である．本講演では，偏極トーリック多様体とそのトーリック因子の組が準安定であるための障害について得られた結果を紹介する．またその応用として，(1) 対数的チャウ安定性と対数的K-安定性の関係や, (2) いくつかの具体例でこの障害を計算し，Conical Kahler Einstein 計量の存在との関連を紹介する．
2016年11月21日(月)
10:30-12:00 数理科学研究科棟(駒場) 128号室
野瀬 敏洋 氏 (福岡工業大学)
局所ゼータ関数の漸近解析と有理型関数としての解析接続について (JAPANESE)
野瀬 敏洋 氏 (福岡工業大学)
局所ゼータ関数の漸近解析と有理型関数としての解析接続について (JAPANESE)
[ 講演概要 ]
１つの滑らかな関数に対して対応する局所ゼータ関数が定義され、一般には右半平面上の正則関数となる。特定の条件の下、例えばこの滑らかな関数が実解析的である場合、局所ゼータ関数は複素平面全体へ有理型関数として解析接続されることが知られている。このような解析接続およびその漸近挙動について調べることは、調和解析における種々の問題(例えば、振動積分の漸近挙動の問題)に密接に関連している。本講演では、実解析性を仮定しない場合の局所ゼータ関数の解析接続について、得られた結果を紹介する。特に最近得られた、局所ゼータ関数が境界点を越えて解析接続されないような例について、その漸近解析と共に報告する。また、局所ゼータ関数と振動積分の関連についても述べる。本講演は、神本丈氏(九州大学)との共同研究に基づく。
１つの滑らかな関数に対して対応する局所ゼータ関数が定義され、一般には右半平面上の正則関数となる。特定の条件の下、例えばこの滑らかな関数が実解析的である場合、局所ゼータ関数は複素平面全体へ有理型関数として解析接続されることが知られている。このような解析接続およびその漸近挙動について調べることは、調和解析における種々の問題(例えば、振動積分の漸近挙動の問題)に密接に関連している。本講演では、実解析性を仮定しない場合の局所ゼータ関数の解析接続について、得られた結果を紹介する。特に最近得られた、局所ゼータ関数が境界点を越えて解析接続されないような例について、その漸近解析と共に報告する。また、局所ゼータ関数と振動積分の関連についても述べる。本講演は、神本丈氏(九州大学)との共同研究に基づく。
