複素解析幾何セミナー

過去の記録 ~02/18次回の予定今後の予定 02/19~

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, 細野 元気

過去の記録

2019年01月28日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
大野乾太郎 氏 (東京大学)
Minimizing CM degree and slope stability of projective varieties (JAPANESE)
[ 講演概要 ]
Chow-Mumford (CM) line bundle is considered to play an important role in moduli problem for K-stable Fano varieties. In this talk, we consider a minimization problem of the degree of the CM line bundle among all possible fillings of a polarized family over a punctured curve. We show that such minimization implies the slope semistability of the fiber if the central fiber is smooth.

2019年01月21日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
Nicholas James McCleerey 氏 (Northwestern University)
POLAR TRANSFORM AND LOCAL POSITIVITY FOR CURVES (ENGLISH)
[ 講演概要 ]
Using the duality of positive cones, we show that applying the polar transform from convexanalysis to local positivity invariants for divisors gives interesting and new local positivity invariantsfor curves. These new invariants have nice properties similar to those for divisors. In particular, thisenables us to give a characterization of the divisorial components of the non-K¨ahler locus of a big class. This is joint worth with Jian Xiao.

2018年12月17日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
神本丈 氏 (九州大学)
Newton polyhedra and order of contact on real hypersurfaces (JAPANESE)
[ 講演概要 ]
This talk will concern some issues on order of contact on real hypersurfaces, which was introduced by D'Angelo. To be more precise, a sufficient condition for the equality of regular type and singular type is given. This condition is written by using the Newton polyhedron of a defining function. Our result includes earlier known results concerning convex domains, pseudoconvex Reinhardt domains and pseudoconvex domains whose regular types are 4. Furthermore, under the above condition, the values of the types can be directly seen in a simple geometrical information from the Newton polyhedron.

The technique of using Newton polyhedra has many significant applications in singularity theory. In particular, this technique has been great success in the study of the Lojasiewicz exponent. Our study about the types is analogous to some works on the Lojasiewicz exponent.

2018年12月03日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
細野元気 氏 (東京大学)
多変数関数論における変動理論 (JAPANESE)
[ 講演概要 ]
関数論において、領域の擬凸変動に関する様々な量の劣調和性が知られている。例えば、山口によるRobin定数の変動、米谷-山口によるBergman核の変動が知られている。また、Bergman核の変動理論のある種の一般化として、Berndtssonにより、$L^2$正則関数のなす空間の変動に関する正曲率性が知られている。これらの理論は$L^2$拡張定理とも深い関係が知られており、その意味でも興味深い。本講演では、これらの理論に関して知られている結果を紹介し、Robin定数の変動問題の多変数化として東川擬距離の変動問題についての考察を行う。

2018年11月26日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
糟谷久矢 氏 (大阪大学)
DGA-Models of variations of mixed Hodge structures (JAPANESE)
[ 講演概要 ]
Mixed Hodge structureは(Projectiveとは限らない)代数多様体のコホモロジー等に現れる非常に重要な構造です。Variations of mixed Hodge structures(VMHS)とは複素多様体をパラメーターとして複素幾何学的に良い振る舞いをしながら変化するMixed Hodge structureたちのことです。今回のお話ではこのVMHSの代数的なモデルについて考えてみたいと思いいます。具体的にはMorganの Mixed Hodge diagramと呼ばれるケーラー多様体のde Rham複体(あるいは対数的 de Rham複体)を積構造込みで模した代数的な対象に対して、”(Unipotent)VMHSのようなもの"を定義します。このVMHSのようなものは純粋に代数的に定義されたものであるため、本来のVMHSのようにベースとなる空間のパラメーターごとにMixed Hodge structureをとる(ファイバーをとる)ことを自然にはできません。本講演ではこの"VMHSのようなもの”からいかにファイバーを取るかということをメインテーマにしてお話ししたいと思います。さらに時間があれば、本結果の幾何学的応用についてもお話ししたいと思います。特に今回の結果によりMorganのMixed Hodge structureとHainのMixed Hodge structureの深い関係が見えることをお話ししたいと思います。

2018年11月19日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
Gerard Freixas i Montplet 氏 (Centre National de la Recherche Scientifique)
BCOV invariants of Calabi-Yau varieties (ENGLISH)
[ 講演概要 ]
The BCOV invariant of Calabi-Yau threefolds was introduced by Fang-Lu-Yoshikawa, themselves inspired by physicists Bershadsky-Cecotti-Ooguri-Vafa. It is a real number, obtained from a combination of holomorphic analytic torsion, and suitably normalized so that it only depends on the complex structure of the threefold. It is conjecturaly expected to encode genus 1 Gromov-Witten invariants of a mirror Calabi-Yau threefold. In order to confirm this prediction for a remarkable example, Fang-Lu-Yoshikawa studied the asymptotic behavior for degenerating families of Calabi-Yau threefolds acquiring at most ordinary double point (ODP) singularities. Their methods rely on former results by Yoshikawa on the singularities of Quillen metrics, together with more classical arguments in the theory of degenerations of Hodge structures and Hodge metrics. In this talk I will present joint work with Dennis Eriksson (Chalmers) and Christophe Mourougane (Rennes), where we extend the construction of the BCOV invariant to any dimension and we give precise asymptotic formulas for degenerating families of Calabi-Yau manifolds. Under several hypothesis on the geometry of the singularities acquired, our general formulas drastically simplify and prove some conjectures or predictions in the literature (Liu-Xia for semi-stable minimal families in dimension 3, Klemm-Pandharipande for ODP singularities in dimension 4, etc.). For this, we slightly improve Yoshikawa's results on the singularities of Quillen metrics, and we also provide a complement to Schmid's asymptotics of Hodge metrics when the monodromy transformations are non-unipotent.

2018年11月05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)
On the quasiconformal equivalence of Dynamical Cantor sets (JAPANESE)
[ 講演概要 ]
Let $E$ be a Cantor set in the Riemann sphere $\widehat{\mathbb C}$, that is, a totally disconnected perfect set in $\widehat{\mathbb C}$.
The standard middle one-thirds Cantor set $\mathcal{C}$ is a typical example.
We consider the complement $X_{E}:=\widehat{\mathbb C}\setminus E$ of the Cantor set $E$.
It is an open Riemann surface with uncountable many boundary components.
We are interested in the quasiconformal equivalence of such surfaces.

In this talk, we discuss the quasiconformal equivalence for the complements of Cantor sets given by dynamical systems.

2018年10月29日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
松村慎一 氏 (東北大学)
On morphisms of compact Kaehler manifolds with semi-positive holomorphic sectional curvature (JAPANESE)
[ 講演概要 ]
In this talk, we consider a smooth projective variety $X$ with semi-positive holomorphic "sectional" curvature, motivated by generalizing Howard-Smyth-Wu's structure theorem and Mok's result for compact Kaehler manifold with semi-positive "bisectional" curvature.
We prove that, if $X$ admits a holomorphic maximally rationally connected fibration $X ¥to Y$, then the morphism is always smooth (that is, a submersion), that the image $Y$ admits a finite ¥'etale cover $T ¥to Y$ by a complex
torus $T$, and further that all the fibers $F$ are isomorphic.
This gives a structure theorem for $X$ when $X$ is a surface.
Moreover we show that $X$ is rationally connected, if the holomorphic sectional curvature is quasi-positive.
This result gives a generalization of Yau's conjecture.

2018年10月22日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
足立真訓 氏 (静岡大学)
On certain hyperconvex manifolds without non-constant bounded holomorphic functions (JAPANESE)
[ 講演概要 ]
For each compact Riemann surface of genus > 1, we can construct a Riemann sphere bundle over the given Riemann surface using the projective structure induced by its uniformization.
The total space of this bundle is divided into two 1-convex domains by a closed Levi-flat real hypersurface. Although these two domains are not biholomorphic, we see that they have several function theoretic properties in common. In this talk, I would like to explain these common properties: hyperconvexity and expressions for certain Green function, and Liouville property and growth estimates of holomorphic functions.

2018年10月15日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
堀田一敬 氏 (山口大学)
Recent problems on Loewner theory (JAPANESE)
[ 講演概要 ]
Loewner Theory, which goes back to the parametric representation of univalent functions introduced by Loewner in 1923, has recently undergone significant development in various directions, including Schramm’s stochastic version of the Loewner differential equation and the new intrinsic approach suggested by Bracci, Contreras, Diaz-Madrigal and Gumenyuk.

In this talk, we firstly review the theory of Loewner equations in classical and modern treatments. Then we discuss some recent problems on the theory:
(i) Quasiconformal extensions of Loewner chains;
(ii) Hydrodynamics of multiple SLE.

2018年07月23日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
Filippo Bracci 氏 (University of Rome Tor Vergata)
Strange Fatou components of automorphisms of $\mathbb{C}^2$ and Runge embedding of $\mathbb{C} \times \mathbb{C}^*$ into $\mathbb{C}^2$. (ENGLISH)
[ 講演概要 ]
The classification of Fatou components for automorphisms of the complex space of dimension greater than $1$ is an interesting and difficult task. Recent deep results prove that the one-dimensional setting is deeply different from the higher dimensional one. Given an automorphism F of $\mathbb{C}^k$, the first bricks in the theory that one would like to understand are invariant Fatou components, namely, those connected open sets $U$, completely invariant under $F$, where the dynamics of $F$ is not chaotic. Among those, we consider “attracting” Fatou components, that is, those components on which the iterates of $F$ converge to a single point. Attracting Fatou components can be recurrent, if the limit point is inside the component or non-recurrent. Recurrent attracting Fatou components are always biholomorphic to $\mathbb{C}^k$, since it was proved by H. Peters, L. Vivas and E. F. Wold that in such a case the point is an attracting (hyperbolic) fixed point, and the Fatou component coincides with the global basin of attraction. Also, as a consequence of works of Ueda and Peters-Lyubich, it is know that all attracting non-recurrent Fatou components of polynomial automorphisms of $\mathbb{C}^2$ are biholomorphic to $\mathbb{C}^2$. One can quite easily find non-simply connected non-recurrent attracting Fatou components in $\mathbb{C}^3$ (mixing a two- dimensional dynamics with a dynamics with non-isolated fixed points in one- variable). In this talk I will explain how to construct a non-recurrent attracting Fatou component in $\mathbb{C}^2$ which is biholomorphic to $\mathbb{C}\times\mathbb{C}^*$. This“fantastic beast” is obtained by globalizing, using a result of F. Forstneric, a local construction due to the speaker and Zaitsev, which allows to create a global basin of attraction for an automorphism, and a Fatou coordinate on it. The Fatou coordinate turns out to be a fiber bundle map on $\mathbb{C}$, whose fiber is $\mathbb{C}^*$, then the global basin is biholomorphic to $\mathbb{C}\times\mathbb{C}^*$. The most subtle point is to show that such a basin is indeed a Fatou component. This is done exploiting Poschel's results about existence of local Siegel discs and suitable estimates for the Kobayashi distance.

Since attracting Fatou components are Runge, it turns out that this construction gives also an example of a Runge embedding of $\mathbb{C}\times\mathbb{C}^*$ into $\mathbb{C}^2$. Moreover, this example shows an automorphism of $\mathbb{C}^2$ leaving invariant two analytic discs intersecting transversally at the origin.

The talk is based on a joint work with J. Raissy and B. Stensones.

2018年07月09日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
Casey Kelleher 氏 (Princeton University)
Rigidity results for symplectic curvature flow (ENGLISH)
[ 講演概要 ]
We continue studying a parabolic flow of almost Kähler structure introduced by Streets and Tian which naturally extends Kähler-Ricci flow onto symplectic manifolds. In a system consisting primarily of quantities related to the Chern connection we establish clean formulas for the evolutions of canonical objects. Using this we give an extended characterization of fixed points of the flow.

2018年07月02日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
松崎克彦 氏 (早稲田大学)
Rigidity of certain groups of circle homeomorphisms and Teichmueller spaces (JAPANESE)
[ 講演概要 ]
In this talk, I explain a complex analytic method and its applications for the study of quasisymmetric homeomorphisms of the circle by extending them to the unit disk quasi-conformally. In RIMS conference "Open Problems in Complex Geometry'' held in 2010, I gave a talk entitled "Problems on infinite dimensional Teichmueller spaces", and mentioned several problems on the fixed points of group actions on the universal Teichmueller space and its subspaces, and the rigidity of conjugation of certain groups of circle homeomorphisms. I will report on the development of these problems since then.

2018年06月25日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
Stephen McKeown 氏 (Princeton University)
Cornered Asymptotically Hyperbolic Spaces
[ 講演概要 ]
This talk will concern cornered asymptotically hyperbolic spaces, which have a finite boundary in addition to the usual infinite boundary. I will first describe the construction a normal form near the corner for these spaces. Then I will discuss formal existence and uniqueness, near the corner, of asymptotically hyperbolic Einstein metrics, with a CMC-umbilic condition imposed on the finite boundary. This is analogous to the Fefferman-Graham construction for the ordinary, non-cornered setting. Finally, I will present work in progress regarding scattering on such spaces.

2018年06月11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)
Cohomology of non-pluriharmonic loci (JAPANESE)
[ 講演概要 ]
In this talk, we study a pseudoconvex counterpart of the Lefschetz hyperplane theorem.
We show a relation between the cohomology of a pseudoconvex domain and the cohomology of the non-pluriharmonic locus of an exhaustive plurisubharmonic function.

2018年06月04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)
Picardの大定理とManin-Mumford予想(Raynaudの定理) (JAPANESE)
[ 講演概要 ]
Manin-Mumford予想とは,関数体上のMordell予想が解決された後の1960年代後半にManinとMumfordにより(独立に)提示されたもので1983年にM. Raynaudにより『代数体上定義されたアーベル多様体の代数的部分空間$X$内のトージョン点集合$X_{tor}$の$\mathbb{Z}$-閉包は部分群の平行移動の有限和である』という形で解決された.この結果は内容の深さからか多くの研究者の関心を呼び、その後,一般化や種々の別証明がM. Hindry ('88), E. Hrushovski ('96), Pila-Zannier ('08)等により与えられてきた.最後のPila-Zannierがここでの話に関係する.
本講演では,準アーベル多様体に対し拡張されたPicardの大定理(N. '81)を用いて上記Manin-Mumford予想(Raynaudの定理)を準アーベル多様体の場合に証明する.
Nevanlinna理論とDiophantus幾何については,これまで類似の観点からの議論・成果が多くあったが,今回の結果は証明レベルでの直接的な関係で,この様な関係を講演者は永く求めてきた(missing link).その意味で今般の知見は新しいもものであると思う.両理論の間をモデル理論の"o-minimal sets 理論''が取り持つ点も興味深いところと思う.

2018年05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)
A generalization of Kähler Einstein metrics for Fano manifolds with non-vanishing Futaki invariant (JAPANESE)
[ 講演概要 ]
The existence problem of Kähler Einstein metrics for Fano manifolds was one of the central problems in Kähler Geometry. The vanishing of the Futaki invariant is known as an obstruction to the existence of Kähler Einstein metrics. Generalized Kähler Einstein metrics (GKE for short), introduced by Mabuchi in 2000, is a generalization of Kähler Einstein metrics for Fano manifolds with non-vanishing Futaki invariant. In this talk, we give the followings:
(i) The positivity for the Hessian of the Ricci Calabi functional which characterizes GKE as its critical points, and its application.
(ii) A criterion for the existence of GKE on toric Fano manifolds from view points of an algebraic stability and an analytic stability.

2018年05月21日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
井上瑛二 氏 (東京大学)
Kähler-Ricci soliton, K-stability and moduli space of Fano
manifolds (JAPANESE)
[ 講演概要 ]
Kähler-Ricci soliton is a kind of canonical metrics on Fano manifolds and is a natural generalization of Kähler-Einstein metric in view of Kähler-Ricci flow.

In this talk, I will explain the following good geometric features of Fano manifolds admitting Kähler-Ricci solitons:
1. Volume minimization, reductivity and uniqueness results established by Tian&Zhu.
2. Relation to algebraic (modified) K-stability estabilished by Berman&Witt-Niström and Datar&Székelyhidi.
3. Moment map picture for Kähler-Ricci soliton (‘real side’)
4. Moduli stack (‘virtual side’) and moduli space of them

A result in 1 is indispensable for the formulation in 3 and 4, and explains why we should consider solitons, beyond Einstein metrics. I also show an essential idea in the construction of the moduli space of Fano manifolds admitting Kähler-Ricci solitons and give some remarks on technical key point.

2018年05月14日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
山田澄生 氏 (学習院大学)
Harmonic map and the Einstein equation in five dimension (JAPANESE)
[ 講演概要 ]
We present a new method in constructing 5-dimensional stationary solutions to the vacuum Einstein equation. In 1917, H. Weyl expressed the Schwarzschild black hole solution using a cylindical coordinate system, and consequently realized that the metric is completely determined by a harmonic function. Since then, the relation between harmonic maps and the Einstein equation has been explored mostly by physicists, which they call the sigma model of the Einstein equation. In this talk, after explaining the historical background, we demonstrate that in 5D, the Einstein spacetimes can have a wide range of black hole horizons in their topological types. In particular we establish an existence theorem of harmonic maps, which subsequently leads to constructions of 5D spacetimes with black hole horizons of positive Yamabe types, namely $S^3$, $S^2 \times S^1$, and the lens space $L(p,q)$. This is a joint work with Marcus Khuri and Gilbert Weinstein.

2018年05月07日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
林本厚志 氏 (長野高専)
Proper holomorphic mappings and generalized pseudoellipsoids (JAPANESE)
[ 講演概要 ]
We study the classification of proper holomorphic mappings between generalized pseudoellipsoids of different dimensions.

Huang proved some classification theorems of proper holomorphic mappings between balls of different dimensions, which are called gap theorems. Our present theorems are their weakly pseudoconvex versions.

In the theorem, classified mapping is so-called a variables splitting mapping and each component is derived from a homogeneous proper polynomial mapping between balls.

The essential methods are the ''good'' decompositions of CR vector bundle and reduction the mapping under consideration to the mapping of balls. By this reduction, we can apply Huang's gap theorem.

2018年04月23日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
奥山裕介 氏 (京都工芸繊維大学)
Degeneration and bifurcation of quadratic endomorphisms of $\mathbb{P}^2$ towards a Hénon map (JAPANESE)
[ 講演概要 ]
The space of quadratic holomorphic endomorphisms of $\mathbb{P}^2$ (over $\mathbb{C}$) is canonically identified with the complement of the zero locus of the resultant form on $\mathbb{P}^{17}$, and all Hénon maps, which are (the only) interesting ones among all the quadratic polynomial automorphisms of $\mathbb{C}^2$, live in this zero locus.

We will talk about our joint work with Fabrizio Bianchi (Imperial College, London) on the (algebraic) degeneration of quadratic endomorphisms of $\mathbb{C}^2$ towards Hénon maps in terms of Berteloot-Bianchi-Dupont's bifurcation/unstability theory of holomorphic families of endomorphisms of $\mathbb{P}^k$, which mostly generalizes Mañé-Sad-Sullivan, Lyubich, and DeMarco's seminal and similar theory on $\mathbb{P}^1$.

Some preliminary knowledge on ergodic theory and pluripotential theory would be desirable, but not be assumed.

2018年04月16日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
納谷信 氏 (名古屋大学)
ラプラシアンの第1固有値を最大化する閉曲面上の計量について (JAPANESE)
[ 講演概要 ]
この講演では、閉曲面においてラプラシアンの第1固有値を(面積一定の仮定の下で)最大化する計量について、最近の進展を中心に解説する。まず、そのような問題の出発点となったHersch-Yang-Yauの不等式(1970, 1980)を紹介する。これは第1固有値(と面積の積)が曲面の種数のみに依存する定数で上から押さえられることを示す不等式である。続いて、最大化計量の存在問題に関する最近の進展について、球面内の極小曲面との関わりを交えて概説する。最後に、種数2の場合に最大化計量を明示的に予言するJacobson-Levitin-Nadirashvili-Nigam-Polterovich予想とその肯定的解決(庄田敏宏氏との共同研究)について述べさせていただく。

2018年01月22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)
Recent topics on the study of the Gauss images of minimal surfaces
[ 講演概要 ]
In this talk, we give a survey of recent advances on the study of the images of the Gauss maps of complete minimal surfaces in Euclidean space.

2018年01月15日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
赤川 晋哉 氏 (大阪大学)
Vanishing theorems of $L^2$-cohomology groups on Hessian manifolds
[ 講演概要 ]
A Hessian manifold is a Riemannian manifold whose metric is locally given by the Hessian of a function with respect to flat coordinates. In this talk, we discuss vanishing theorems of $L^2$-cohomology groups on complete Hessian Manifolds coupled with flat line bundles. In particular, we obtain stronger vanishing theorems on regular convex cones with the Cheng-Yau metrics. Further we show that the Cheng-Yau metrics on regular convex cones give rise to harmonic maps to the positive symmetric matrices.

2017年12月18日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
久本 智之 氏 (名古屋大学)
Gradient flow of the Ding functional
[ 講演概要 ]
This is a joint work with T. Collins and R. Takahashi. We introduce the flow in the title to study the stability of a Fano manifold. The first result is the long-time existence of the flow. In the stable case it then converges to the Kähler-Einstein metric. In general the flow is expected to produce the optimally destabilizing degeneration of a Fano manifold. We confirm this expectation in the toric case.

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