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Seminar on Geometric Complex Analysis
Seminar information archive ～03/19｜Next seminar｜Future seminars 03/20～
| 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
2013/06/10
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Shin-ichi Matsumura (Kagoshima University)
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.)
Takao Akahori (University of Hyogo)
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.)
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)
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.)
Takeo Ohsawa (Nagoya University)
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.)
Yusuke Okuyama (Kyoto Institute of Technology)
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.)
Yusaku Tiba (Tokyo Institute of Technology)
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.)
Nikolay Shcherbina (University of Wuppertal)
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.)
Akito Futaki (University of Tokyo)
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.
2013/01/28
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Taiji MARUGAME (MS U-Tokyo)
Renormalized Chern-Gauss-Bonnet formula for complete Kaehler-Einstein metrics (JAPANESE)
Taiji MARUGAME (MS U-Tokyo)
Renormalized Chern-Gauss-Bonnet formula for complete Kaehler-Einstein metrics (JAPANESE)
2013/01/21
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Takushi AMEMIYA (MS U-Tokyo)
Value distribution of meromorphic mappings to compact complex manifolds (JAPANESE)
Takushi AMEMIYA (MS U-Tokyo)
Value distribution of meromorphic mappings to compact complex manifolds (JAPANESE)
[ Abstract ]
In a late paper of J. Noguchi and J. Winkelmann they showed the condition of being Kähler or non-Kähler of the image space to make a difference in the value distribution theory of meromorphic mappings into compact complex manifolds. In the present talk, we will discuss the order of meromorphic mappings to a Hopf surface which is more general than dealt with by Noguchi-Winkelmann, and an Inoue surface (they are non-Kähler surfaces). For a general Hopf surface $S$, we prove that there exists a differentiably non-degenerate holomorphic mapping $f:\mathbf{C}^2 \to S$ whose order satisfies $\rho_{f}\leq 1$. For an Inoue surface $S'$, we prove that every non-constant meromorphic mapping $f:\mathbf{C}^n \to S'$ is holomorphic and its order satisfies $\rho_{f}\geq 2$.
In a late paper of J. Noguchi and J. Winkelmann they showed the condition of being Kähler or non-Kähler of the image space to make a difference in the value distribution theory of meromorphic mappings into compact complex manifolds. In the present talk, we will discuss the order of meromorphic mappings to a Hopf surface which is more general than dealt with by Noguchi-Winkelmann, and an Inoue surface (they are non-Kähler surfaces). For a general Hopf surface $S$, we prove that there exists a differentiably non-degenerate holomorphic mapping $f:\mathbf{C}^2 \to S$ whose order satisfies $\rho_{f}\leq 1$. For an Inoue surface $S'$, we prove that every non-constant meromorphic mapping $f:\mathbf{C}^n \to S'$ is holomorphic and its order satisfies $\rho_{f}\geq 2$.
2012/12/17
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Hideaki Ikoma (Kyoto University)
On the existence of strictly effective basis on an arithmetic variety (JAPANESE)
Hideaki Ikoma (Kyoto University)
On the existence of strictly effective basis on an arithmetic variety (JAPANESE)
[ Abstract ]
I would like to talk about some recent work of mine on the asymptotic behavior of the successive minima associated to a graded arithmetic linear series. A complete arithmetic linear series belonging to a hermitian line bundle on an arithmetic variety is defined as the Z-module of the global sections endowed with the supremum-norm, and the successive minima are invariants that measure the size of the sections with small norms.
If time permits, I would like to also explain some close relationship between the results and the general equi-distribution theory of rational points on an arithmetic variety.
I would like to talk about some recent work of mine on the asymptotic behavior of the successive minima associated to a graded arithmetic linear series. A complete arithmetic linear series belonging to a hermitian line bundle on an arithmetic variety is defined as the Z-module of the global sections endowed with the supremum-norm, and the successive minima are invariants that measure the size of the sections with small norms.
If time permits, I would like to also explain some close relationship between the results and the general equi-distribution theory of rational points on an arithmetic variety.
2012/12/10
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Hiroshi Kaneko (Tokyo University of Science)
A Dirichlet space on ends of tree and Dirichlet forms with a nodewise orthogonal property (JAPANESE)
Hiroshi Kaneko (Tokyo University of Science)
A Dirichlet space on ends of tree and Dirichlet forms with a nodewise orthogonal property (JAPANESE)
2012/12/03
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Yu Kawakami (Yamaguchi University)
On the geometric meaning of the maximal number
of exceptional values of Gauss maps for immersed surfaces in space forms
(JAPANESE)
Yu Kawakami (Yamaguchi University)
On the geometric meaning of the maximal number
of exceptional values of Gauss maps for immersed surfaces in space forms
(JAPANESE)
2012/11/26
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Shin Nayatani (Nagoya University)
Quaternionic CR geometry (JAPANESE)
Shin Nayatani (Nagoya University)
Quaternionic CR geometry (JAPANESE)
2012/11/19
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Yohei Komori (Waseda University)
On a degenerate family of Riemann surfaces of genus two over an elliptic curve (JAPANESE)
Yohei Komori (Waseda University)
On a degenerate family of Riemann surfaces of genus two over an elliptic curve (JAPANESE)
[ Abstract ]
We construct a degenerate family of Riemann surfaces of genus two constructed as double branched covering surfaces of a fixed torus. We determine its singular fibers and holomorphic sections.
We construct a degenerate family of Riemann surfaces of genus two constructed as double branched covering surfaces of a fixed torus. We determine its singular fibers and holomorphic sections.
2012/11/12
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
A.G. Aleksandrov (Institute of Control Sciences, Russian Acad. of Sci.)
Residues of meromorphic differential forms (ENGLISH)
A.G. Aleksandrov (Institute of Control Sciences, Russian Acad. of Sci.)
Residues of meromorphic differential forms (ENGLISH)
[ Abstract ]
The purpose of the talk is to discuss several interesting aspects
of the classical residue theory originated by H. Poincar\\'e, J. de Rham and J. Leray and their followers. Focus topics of our studies are some of the less known applications, developed by the author in the past few years in complex analysis, topology and geometry of singular varieties and in the theory of differential equations. Almost all considerations are based essentially on properties of a special class of meromorphic differential forms called logarithmic or multi-logarithmic forms.
The purpose of the talk is to discuss several interesting aspects
of the classical residue theory originated by H. Poincar\\'e, J. de Rham and J. Leray and their followers. Focus topics of our studies are some of the less known applications, developed by the author in the past few years in complex analysis, topology and geometry of singular varieties and in the theory of differential equations. Almost all considerations are based essentially on properties of a special class of meromorphic differential forms called logarithmic or multi-logarithmic forms.
2012/10/29
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Atsushi Atsuji (Keio University)
Value distribution of meromorphic functions on foliated manifolds,II (JAPANESE)
Atsushi Atsuji (Keio University)
Value distribution of meromorphic functions on foliated manifolds,II (JAPANESE)
2012/10/22
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Yusaku Tiba (Grad. School of Math. Sci., Univ. of Tokyo)
The second main therorem for entire curves into Hilbert modular surfaces (JAPANESE)
Yusaku Tiba (Grad. School of Math. Sci., Univ. of Tokyo)
The second main therorem for entire curves into Hilbert modular surfaces (JAPANESE)
2012/10/15
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Takeo Ohsawa (Nagoya University)
An L^2 estimate on domains and application to Levi-flat surfaces (JAPANESE)
Takeo Ohsawa (Nagoya University)
An L^2 estimate on domains and application to Levi-flat surfaces (JAPANESE)
2012/07/09
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Tomoyuki Hisamoto (Univ. of Tokyo)
Volume of graded linear series and the existence problem of constant scalar curvature Kaehler metric (JAPANESE)
Tomoyuki Hisamoto (Univ. of Tokyo)
Volume of graded linear series and the existence problem of constant scalar curvature Kaehler metric (JAPANESE)
[ Abstract ]
We describe the volume of a graded linear series by the Monge-Ampere mass of the associated equilibrium metric. We relate this formula to the question whether the weak geodesic ray associated to a test configuration of given polarized manifold recovers the Donaldson-Futaki invariant.
We describe the volume of a graded linear series by the Monge-Ampere mass of the associated equilibrium metric. We relate this formula to the question whether the weak geodesic ray associated to a test configuration of given polarized manifold recovers the Donaldson-Futaki invariant.
2012/06/25
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Atsushi Hayashimoto (Nagano National College of Technology)
CR equivalence problem of CR manifolds with slice structure (JAPANESE)
Atsushi Hayashimoto (Nagano National College of Technology)
CR equivalence problem of CR manifolds with slice structure (JAPANESE)
2012/06/18
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Takanari Saotome (Academia Sinica)
CR Q-curvature flow and CR Paneitz operator on 3-dimensional CR manifolds (JAPANESE)
Takanari Saotome (Academia Sinica)
CR Q-curvature flow and CR Paneitz operator on 3-dimensional CR manifolds (JAPANESE)
2012/06/11
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Damian BROTBEK (University of Tokyo)
Differential forms on complete intersections (ENGLISH)
Damian BROTBEK (University of Tokyo)
Differential forms on complete intersections (ENGLISH)
[ Abstract ]
Brückmann and Rackwitz proved a vanishing result for particular types of differential forms on complete intersection varieties. We will be interested in the cases not covered by their result. In some cases, we will show how the space $H^0(X,S^{m_1}\Omega_X\otimes \cdots \otimes S^{m_k}\Omega_X)$ depends on the equations defining $X$, and in particular we will prove that the theorem of Brückmann and Rackwitz is optimal. The proofs are based on simple, combinatorial, cohomology computations.
Brückmann and Rackwitz proved a vanishing result for particular types of differential forms on complete intersection varieties. We will be interested in the cases not covered by their result. In some cases, we will show how the space $H^0(X,S^{m_1}\Omega_X\otimes \cdots \otimes S^{m_k}\Omega_X)$ depends on the equations defining $X$, and in particular we will prove that the theorem of Brückmann and Rackwitz is optimal. The proofs are based on simple, combinatorial, cohomology computations.
2012/06/04
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Sachiko HAMANO (Fukushima University)
Log-plurisubharmonicity of metric deformations induced by Schiffer and harmonic spans. (JAPANESE)
Sachiko HAMANO (Fukushima University)
Log-plurisubharmonicity of metric deformations induced by Schiffer and harmonic spans. (JAPANESE)
2012/05/28
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Shinichi TAJIMA (University of Tsukuba)
Local cohomology and hypersurface isolated singularities II (JAPANESE)
Shinichi TAJIMA (University of Tsukuba)
Local cohomology and hypersurface isolated singularities II (JAPANESE)
[ Abstract ]
局所コホモロジーの孤立特異点への応用として
・$\mu$-constant-deformation の Tjurina 数
・対数的ベクトル場の構造と構成法
・ニュートン非退化な超曲面に対する Kouchnirenko の公式
について述べる.
局所コホモロジーの孤立特異点への応用として
・$\mu$-constant-deformation の Tjurina 数
・対数的ベクトル場の構造と構成法
・ニュートン非退化な超曲面に対する Kouchnirenko の公式
について述べる.
