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複素解析幾何セミナー
過去の記録 ～12/07｜次回の予定｜今後の予定 12/08～
| 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 |
今後の予定
2024年12月09日(月)
10:30-12:00 数理科学研究科棟(駒場) 128号室
鈴木 良明 氏 (新潟大学)
The spectrum of the Folland-Stein operator on some Heisenberg Bieberbach manifolds (Japanese)
https://forms.gle/gTP8qNZwPyQyxjTj8
鈴木 良明 氏 (新潟大学)
The spectrum of the Folland-Stein operator on some Heisenberg Bieberbach manifolds (Japanese)
[ 講演概要 ]
Heisenberg Bieberbach多様体とは、Heisenberg群とユニタリ群との半直積における離散かつ捩れの無い部分群によってHeisenberg群を割って得られるコンパクト商のことである。この商多様体は、Heisenberg群を自身の離散部分群で割ったコンパクト商(Heisenberg冪零多様体)をさらに有限群で割った空間になっている。この講演では3次元Heisenberg Bieberbach多様体上のFolland-Stein作用素と呼ばれるCR幾何由来の微分作用素の固有値と固有空間について考察する。Heisenberg Bieberbach多様体の被覆空間であるHeisenberg冪零多様体に対しては、2004年にFollandが表現論の手法を用いてFolland-Stein作用素の固有値と固有関数が明示的に求めている。Follandの結果を応用し、3次元Heisenberg Bieberbach多様体のいくつかの例に対してもFolland-Stein作用素の固有値と固有関数を求めることができることを紹介する。特に固有空間の次元も求めることができ、Weylの法則が成り立つことも紹介したい。
[ 参考URL ]Heisenberg Bieberbach多様体とは、Heisenberg群とユニタリ群との半直積における離散かつ捩れの無い部分群によってHeisenberg群を割って得られるコンパクト商のことである。この商多様体は、Heisenberg群を自身の離散部分群で割ったコンパクト商(Heisenberg冪零多様体)をさらに有限群で割った空間になっている。この講演では3次元Heisenberg Bieberbach多様体上のFolland-Stein作用素と呼ばれるCR幾何由来の微分作用素の固有値と固有空間について考察する。Heisenberg Bieberbach多様体の被覆空間であるHeisenberg冪零多様体に対しては、2004年にFollandが表現論の手法を用いてFolland-Stein作用素の固有値と固有関数が明示的に求めている。Follandの結果を応用し、3次元Heisenberg Bieberbach多様体のいくつかの例に対してもFolland-Stein作用素の固有値と固有関数を求めることができることを紹介する。特に固有空間の次元も求めることができ、Weylの法則が成り立つことも紹介したい。
https://forms.gle/gTP8qNZwPyQyxjTj8
2024年12月16日(月)
10:30-12:00 数理科学研究科棟(駒場) 128号室
Laurent Stolovitch 氏 (Universite Cote d'Azur)
CR singularities and dynamical systems (English)
https://forms.gle/gTP8qNZwPyQyxjTj8
Laurent Stolovitch 氏 (Universite Cote d'Azur)
CR singularities and dynamical systems (English)
[ 講演概要 ]
In this talk, we'll survey some recent results done since the seminal work of Moser and Webster about smooth real analytic surfaces in $C^2$ which are totally real everywhere but at a point where the tangent space is a complex line. Such a point is called a singularity of the Cauchy-Riemann structure. We are interested in the holomorphic classification of these surface near the singularity. It happens that there is a deep connection with holomorphic classification of some holomorphic dynamical systems near a fixed point so that new results for the later provide new result for the former.
[ 参考URL ]In this talk, we'll survey some recent results done since the seminal work of Moser and Webster about smooth real analytic surfaces in $C^2$ which are totally real everywhere but at a point where the tangent space is a complex line. Such a point is called a singularity of the Cauchy-Riemann structure. We are interested in the holomorphic classification of these surface near the singularity. It happens that there is a deep connection with holomorphic classification of some holomorphic dynamical systems near a fixed point so that new results for the later provide new result for the former.
https://forms.gle/gTP8qNZwPyQyxjTj8
2024年12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)
Hyperbolicity and sections in a ramified cover over abelian varieties
with trace zero (Japanese)
https://forms.gle/gTP8qNZwPyQyxjTj8
野口 潤次郎 氏 (東京大学)
Hyperbolicity and sections in a ramified cover over abelian varieties
with trace zero (Japanese)
[ 講演概要 ]
We discuss a higher dimensional generalization of the Manin-Grauert Theorem ('63/'65) in relation with the function field analogue of Lang's conjecture on the finiteness of rational points in a Kobayashi hyperbolic algebraic variety over a number field. Let $B$ be a possibly open algebraic curve over $\mathbf{C}$, and let $\pi:X \to B$ be a smooth or normal projective fiber space. In '81 I proved such theorems for $\dim \geq 1$, assuming the ampleness of the cotangent bundle $T^*(X_t)$, and in '85 the Kobayashi hyperbolicity of $X_t$ with some boundary condition (BC) (hyperbolic embedding condition relative over $\bar{B}$).
It is interesting to study if (BC) is really necessary or not. If $\dim X_t=1$, (BC) is automatically satisfied, and if $T^*(X_t)$ is ample, (BC) is not necessary; thus in those cases, (BC) is unnecessary. Lately, Xie-Yuan in arXiv '23 obtained such a result without (BC) for $X$ which is a hyperbolic finite cover of an abelian variety $A/B$.
The aim of this talk is to present a simplified treatment of the Xie-Yuan theorem from the viewpoint of Kobayashi hyperbolic geometry. In particular, if the $K/\mathbf{C}$-trace $Tr(A/B)=0$ with $K=\mathbf{C}(B)$, there are only finitely many $X(K)$-points or sections in $X \to B$. In this case, Bartsch-Javanpeykar in arXiv '24 gave another proof based on Parshin's topological rigidity theorem ('90). We will discuss the proof which is based on the Kobayashi hyperbolicity.
[ 参考URL ]We discuss a higher dimensional generalization of the Manin-Grauert Theorem ('63/'65) in relation with the function field analogue of Lang's conjecture on the finiteness of rational points in a Kobayashi hyperbolic algebraic variety over a number field. Let $B$ be a possibly open algebraic curve over $\mathbf{C}$, and let $\pi:X \to B$ be a smooth or normal projective fiber space. In '81 I proved such theorems for $\dim \geq 1$, assuming the ampleness of the cotangent bundle $T^*(X_t)$, and in '85 the Kobayashi hyperbolicity of $X_t$ with some boundary condition (BC) (hyperbolic embedding condition relative over $\bar{B}$).
It is interesting to study if (BC) is really necessary or not. If $\dim X_t=1$, (BC) is automatically satisfied, and if $T^*(X_t)$ is ample, (BC) is not necessary; thus in those cases, (BC) is unnecessary. Lately, Xie-Yuan in arXiv '23 obtained such a result without (BC) for $X$ which is a hyperbolic finite cover of an abelian variety $A/B$.
The aim of this talk is to present a simplified treatment of the Xie-Yuan theorem from the viewpoint of Kobayashi hyperbolic geometry. In particular, if the $K/\mathbf{C}$-trace $Tr(A/B)=0$ with $K=\mathbf{C}(B)$, there are only finitely many $X(K)$-points or sections in $X \to B$. In this case, Bartsch-Javanpeykar in arXiv '24 gave another proof based on Parshin's topological rigidity theorem ('90). We will discuss the proof which is based on the Kobayashi hyperbolicity.
https://forms.gle/gTP8qNZwPyQyxjTj8
