複素解析幾何セミナー

過去の記録 ~07/24次回の予定今後の予定 07/25~

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

2016年05月30日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
大沢 健夫 氏 (名古屋大学)
レビ平坦面の幾何と$\overline{\partial}$-方程式 (JAPANESE)
[ 講演概要 ]
複素多様体上の局所擬凸領域がいつ正則凸になるかは複素解析における基本的な未解決問題である。現状は最終的な解決には程遠いが、射影空間やトーラスなどの場合にはよくわかっている。この問題に関しては、幾何学的な諸条件によって$\overline{\partial}$-方程式が解けたり解けなかったりする状況が詳しくわかってくると面白いのだが、その一つの成功例がレビ平坦面の理論である。レビ平坦面はコンパクトな複素多様体を実一次元分膨らませたようなもので、直積のような自明なものを除けば、トーラス上の非正則凸な擬凸領域の境界としてこのような構造が初めて現れた。以来、レビ平坦面の例が他にもいろいろあることが判明し、その結果分類問題が発生した。これも最終的な解決には程遠いのだが、幾つかの結果は$\overline{\partial}$-方程式の可解性に関する研究の果実となっており、「複素解析幾何らしさ」を持っている。集中講義のマクラとしてこの辺をサーベイしてみたい。

2016年05月23日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
鍋島 克輔 氏 (徳島大学)
A computation method for algebraic local cohomology and its applications (JAPANESE)
[ 講演概要 ]
Local cohomology was introduced by A. Grothendieck. Subsequent development to a great extent has been motivated by Grothendieck's ideas. Nowadays, local cohomology is a key ingredient in algebraic geometry, commutative algebra, topology and D-modules, and is a fundamental tool for applications in several fields.
In this talk, an algorithmic method to compute algebraic local cohomology classes (with parameters), supported at a point, associated with a given zero-dimensional ideal, is considered in the context of symbolic computation. There are several applications of the method. For example, the method can be used to analyze properties of singularities and deformations of Artin algebra. As the applications, methods for computing standard bases of zero-dimensional ideals and solving ideal membership problems, are also introduced.

2016年05月16日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
泊 昌孝 氏 (日本大学)
2次元正規小平特異点の正規化接錐の被約性による特徴づけと、特異点解消および極大イデアル因子の性質 (JAPANESE)
[ 講演概要 ]
曲線の退化に埋め込める特異点としてKarrasにより1970年代に導入さた小平特異点のうち、基本因子の次数についてのトップタイプにあたるものを、正規化接錐の被約性により代数的に特徴づけることができた。これは「例外集合の交点形式が十分に負ならば特異点は小平になる」という認識を与える定理でもあり、90年代からの都丸氏によるこのクラスの研究の自然な拡張になっている。一般の特異点のこのクラスへの近似問題を通じて、かつて論じた「星型特異点の極大イデアルサイクルと基本サイクルの同一視問題」へ超曲面特異点による反例が発見された。これは、ある種のコホモロジー対応の単射性を崩す例でもある。昨年秋の学会以来、いくつかの機会に発表をしてきたこれらのトピックスをまとめて紹介したい。

2016年05月09日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
厚地 淳 氏 (慶應義塾大学)
Nevanlinna type theorems for meromorphic functions on negatively curved Kähler manifolds (JAPANESE)
[ 講演概要 ]
We discuss a generalization of classical Nevanlinna theory to meromorphic functions on complete Kähler manifolds. Several generalization of domains of functions are known in Nevanlinna theory, especially the results due to W.Stoll are well-known. In general Kähler case the remainder term of the second main theorem of Nevanlinna theory usually takes a complicated form. It seems that we have to modify classical
methods in order to simplify the second main theorem. We will use heat diffusion to do that and show some defect relations. We would also like to give some Liouville type theorems for holomorphic maps by using similar heat diffusion methods.

2016年04月25日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
山盛 厚伺 氏 (台湾中央研究院)
The representative domain and its applications (JAPANESE)
[ 講演概要 ]
Bergman introduced the notion of a representative domain to choose a nice holomorphic equivalence class of domains. In this talk, I will explain that the representative domain is also useful to obtain an analogue of Cartan's linearity theorem for some special class of domains.

2016年04月18日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
小櫃 邦夫 氏 (鹿児島大学)
Weil-Petersson計量の漸近展開についての最近の進展 (JAPANESE)
[ 講演概要 ]
リーマン面のモジュライ空間上のWeil-Petersson計量の境界における漸近展開は、H. Masurが1976年に与えた結果を初めとし、その後Yamada, Wolpert, Obitsu-Wolpertによって改良された。最近、Melrose, X. Zhu, Mazzeo, Swobodaにより、その漸近展開の形が完全に決定された。彼らの仕事を紹介し、残された問題や関連する話題について解説する。

2016年04月11日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
足助 太郎 氏 (東京大学)
Defining the Julia sets on CP^2 (JAPANESE)
[ 講演概要 ]
The Julia sets play a central role in the study of complex dynamical systems as well as Kleinian groups where they appear as limit sets. They are also known to be meaningful for complex foliations without singularities, however still not defined for singular ones. In this talk, I will discuss some expected properties of the Julia sets for singular foliations and difficulties for defining them.

2016年01月25日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
このセミナーは悪天候の影響によりキャンセルになりました。
小櫃 邦夫 氏 (鹿児島大学)
Weil-Petersson 計量の漸近解析についての最近の進展 (Japanese)
[ 講演概要 ]
リーマン面のモジュライ空間上のWeil-Petersson 計量の境界における漸近展開は、H. Masurが1976年に与えた結果を初めとし、その後Yamada, Wolpert, Obitsu-Wolpert によって改良された。最近、Melrose, X. Zhu, Mazzeo, Swoboda により、その漸近展開の形が完全に決定された。彼らの仕事を紹介し、残された問題や関連する話題について解説する。

2016年01月18日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
志賀 啓成 氏 (東京工業大学)
Holomorphic motions and the monodromy (Japanese)
[ 講演概要 ]
Holomorphic motions, which was introduced by Mane, Sad and Sullivan, is a useful tool for Teichmuller theory as well as for complex dynamics. In particular, Slodkowski’s theorem makes a significant contribution to them. The theorem says that every holomorphic motion of a closed set on the Riemann sphere parametrized by the unit disk is extended to a holomorphic motion of the whole Riemann sphere parametrized by the unit disk. In this talk, we consider a generalization of the theorem. If time permits, we will discuss applications of our results.

2015年12月21日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
山ノ井 克俊 氏 (大阪大学)
On pseudo Kobayashi hyperbolicity of subvarieties of abelian varieties
(Japanese)
[ 講演概要 ]
A subvariety of an abelian variety is of general type if and only if it is pseudo Kobayashi hyperbolic. I will discuss the proof of this result.

2015年12月14日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
中田 文憲 氏 (福島大学)
Twistor correspondence for associative Grassmanniann
[ 講演概要 ]
It is well known that the 6-dimensional sphere has a non-integrable almost complex structure which is introduced from the (right) multiplication of imaginary octonians. On this 6-sphere, there is a family of psuedo-holomorphic $\mathbb{C}\mathbb{P}^1$ parameterised by the associative Grassmannian, where the associative Grassmaniann is an 8-dimensional quaternion Kaehler manifold defined as the set of associative 3-planes in the 7-dimensional real vector space of the imaginary octonians. In the talk, we show that this story is quite analogous to the Penrose's twistor correspondence and that the geometric structures on the associative Grassmaniann nicely fit to this construction. This is a joint work with H. Hashimoto, K. Mashimo and M. Ohashi.

2015年12月07日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
巴山 竜来 氏 (専修大学)
Cycle connectivity and pseudoconcavity of flag domains (Japanese)
[ 講演概要 ]
We consider an open real group orbit in a complex flag variety which has no non-constant function. We introduce Huckleberry's results on cycle connectivity and show that it is pseudoconcave if it satisfies a certain condition on the root system of the Lie algebra. In Hodge theory, we are mainly interested in the case where it is a Mumford-Tate domain. We also discuss Hodge theoretical meanings of this work.

2015年11月30日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
Jean-Pierre Demailly 氏 (Univ. de Grenoble I)
Extension of holomorphic functions defined on non reduced analytic subvarieties (English)
[ 講演概要 ]
The goal of this talk will be to discuss $L^2$ extension properties of holomorphic sections of vector bundles satisfying weak semi-positivity properties. Using techniques borrowed from recent proofs of the Ohsawa-Takegoshi extension theorem, we obtain several new surjectivity results for the restriction morphism to a non necessarily reduced subvariety, provided the latter is defined as the zero variety of a multiplier ideal sheaf. These extension results are derived from $L^2$ approximation techniques, and they hold under (probably) optimal curvature conditions.

2015年11月16日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
宮地 秀樹 氏 (大阪大学)
Towards the complex geometry of Teichmuller space with extremal length (English)
[ 講演概要 ]
In this talk, in aiming for studying a relation between the topological aspect and the complex analytical aspect of Teichmuller space, I will discuss a complex analytic property of extremal length functions. More precisely, I will give a concrete formula of the Levi form of the extremal length functions for ``generic” measured foliations and show that the reciprocal of the extremal length function is plurisuperharmonic. As a corollary, I will give alternate proofs of S. Krushkal results that the distance function for the Teichmuller distance is plurisubharmonic, and Teichmuller space is hyperconvex. If time permits, I will give a topological description of the Levi form with using the Thurston's symplectic form.

2015年11月02日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
下部 博一 氏 (大阪大学)
A class of non-Kahler manifolds (Japanese)
[ 講演概要 ]
We consider a special case of compact complex manifolds which are said to be super strongly Gauduchon manifolds. A super strongly Gauduchon manifold is a complex manifold with a super strongly Gauduchon metric. We mainly consider non-Kähler super strongly Gauduchon manifolds. We give a cohomological condition for a compact complex manifold to have a super strongly Gauduchon metric, and give examples of non-trivial super strongly Gauduchon manifolds from nil-manifolds. We also consider its stability under small deformations and proper modifications of super strongly Gauduchon manifolds.

2015年10月26日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
松本 和子 氏 (東京理科大学)
The Fubini-distance functions to pseudoconvex domains in $\mathbb{C}\mathbb{P}^2$ (Japanese)
[ 講演概要 ]
In this talk, we would like to present two explicit formulas for the Levi forms of the Fubini-Study distance functions to complex or real hypersurfaces in $\mathbb{C}\mathbb{P}^2$. This is the first step for us to approach the non-existence conjecture of Levi-flat real hypersurfaces in $\mathbb{C}\mathbb{P}^2$. We would like to also discuss a certain important quantity found in the formulas.

2015年10月19日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
森脇 淳 氏 (京都大学)
Semiample invertible sheaves with semipositive continuous hermitian metrics (Japanese)
[ 講演概要 ]
Let $(L,h)$ be a pair of a semi ample invertible sheaf and a semipositive continuous hermitian metric on a proper algebraic variety over $C$. In this talk, we would like to present the result that $(L, h)$ has the extension property, answering a generalization of a question of S. Zhang. Moreover, we consider its non-archimedean analogue.

2015年10月05日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
丸亀 泰二 氏 (東京大学)
On the volume expansion of the Blaschke metric on strictly convex domains
[ 講演概要 ]
The Blaschke metric is a projectively invariant metric on a strictly convex domain in a projective manifold, which is a real analogue of the complete Kahler-Einstein metric on strictly pseudoconvex domains. We consider the asymptotic expansion of the volume of subdomains and construct a global conformal invariant of the boundary. We also give some variational formulas under a deformation of the domain.

2015年09月28日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
後藤 竜司 氏 (大阪大学)
Flat structures on moduli spaces of generalized complex surfaces
[ 講演概要 ]
The 2 dimensional complex projective space $P^2$ is rigid as a complex manifold, however $P^2$ admits 2 dimensional moduli spaces of generalized complex structures which has a torsion free flat connection on a open strata. We show that logarithmic generalized complex structure with smooth elliptic curve as type changing loci has unobstructed deformations which are parametrized by an open set of the second de Rham cohomology group of the complement of type changing loci. Then we will construct moduli spaces of generalized del Pezzo surfaces. We further investigate deformations of logarithmic generalized complex structures in the cases of type changing loci with singularities. By using types of singularities, we obtain a stratification of moduli spaces of generalized complex structures on complex surfaces and it turns out that each strata corresponding to nodes admits a flat torsion free connection.

2015年07月13日(月)

10:30-12:00   数理科学研究科棟(駒場) 126号室
松本 佳彦 氏 (東京工業大学)
$L^2$ cohomology and deformation of Einstein metrics on strictly pseudo convex domains
[ 講演概要 ]
Consider a bounded domain of a Stein manifold, with strictly pseudo convex smooth boundary, endowed with an ACH-Kähler metric (examples being domains of $\mathbb{C}^n$ with their Bergman metrics or Cheng-Yau’s Einstein metrics). We give a vanishing theorem on the $L^2$ $\overline{\partial}$-cohomology group with values in the holomorphic tangent bundle. As an application, Einstein perturbations of the Cheng-Yau metric are discussed.

2015年07月06日(月)

10:30-12:00   数理科学研究科棟(駒場) 126号室
児玉 秋雄 氏
On the structure of holomorphic automorphism groups of generalized complex ellipsoids and generalized Hartogs triangles (JAPANESE)
[ 講演概要 ]
In this talk, we first review the structure of holomorphic automorphism groups of generalized complex ellipsoids and, as an application of this, we clarify completely the structure of generalized Hartogs triangles. Finally, if possible, I will mention some known results on proper holomorphic self-mappings of generalized complex ellipsoids, generalized Hartogs triangles, and discuss a related question to these results.

2015年06月29日(月)

10:30-12:00   数理科学研究科棟(駒場) 126号室
鈴木 雄大 氏 (東京大学)
Cohomology Formula for Obstructions to Asymptotic Chow semistability (JAPANESE)
[ 講演概要 ]
Odaka and Wang proved the intersection formula for the Donaldson-Futaki invariant. We generalize this result for the higher Futaki invariants which are obstructions to asymptotic Chow semistability.

2015年06月22日(月)

10:30-12:00   数理科学研究科棟(駒場) 126号室
田邊 晋 氏 (Université Galatasaray)
Amoebas and Horn hypergeometric functions
[ 講演概要 ]
Since 10 years, the utility of the Horn hypergeometric functions in Algebraic Geometry has been recognized in a small circle of specialists. The main reason for this interest lies in the fact that every period integral of an affine non-degenerate complete intersection variety can be described as a Horn hypergeometric function (HGF). Therefore the monodromy of the middle dimensional homology can be calculated as the monodromy of an Horn HGF’s.
There is a slight difference between the Gel’fand-Kapranov-Zelevinski HGF’s and the Horn HGF’s. The latter may contain so called “persistent polynomial solutions” that cannot be mapped to GKZ HGF’s via a natural isomorphism between two spaces of HGF’s. In this talk, I will review basic facts on the Horn HGF’s. As a main tool to study the topology of the discriminant loci together with the
analytic aspects of the story, amoebas – image by the log map of the discriminant- will be highlighted.
As an application of this theory the following theorem can be established. For a bivariate Horn HGF system, its monodromy invariant space is always one dimensional if and only if its Ore-Sato polygon is either a zonotope or a Minkowski sum of a triangle and some segments.
This is a collaboration with Timur Sadykov.

2015年06月15日(月)

10:30-12:00   数理科学研究科棟(駒場) 126号室
早乙女 飛成 氏
The Lyapunov-Schmidt reduction for the CR Yamabe equation on the Heisenberg group (Japanese)
[ 講演概要 ]
We will study CR Yamabe equation for a CR structure on the Heisenberg group which is deformed from the standard structure. By using Lyapunov-Schmidt reduction, it is shown that the perturbation of the standard CR Yamabe solution is a solution to the deformed CR Yamabe equation, under certain conditions of the deformation.

2015年06月08日(月)

10:30-12:00   数理科学研究科棟(駒場) 126号室
糟谷 久矢 氏 (東京工業大学)
Mixed Hodge structures and Sullivan's minimal models of Sasakian manifolds (Japanese)
[ 講演概要 ]
By the result of Deligne, Griffiths, Morgan and Sullivan, the Malcev completion of the fundamental group of a compact Kahler manifold is quadratically presented. This fact gives good advances in "Kahler group problem" (Which groups can be the fundamental groups of compact Kahler manifolds?) In this talk, we consider the fundamental groups of compact Sasakian manifolds. We show that the Malcev Lie algebra of the fundamental group of a compact 2n+1-dimensional Sasakian manifold with n >= 2 admits a quadratic presentation by using Morgan's bigradings of Sullivan's minimal models of mixed-Hodge diagrams.

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