## 複素解析幾何セミナー

開催情報 月曜日　10:30～12:00　数理科学研究科棟(駒場) 128号室 平地 健吾, 高山 茂晴, 細野 元気

### 2018年11月05日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室

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

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

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

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

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

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

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．

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

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

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

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

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

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

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

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

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

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.

### 2017年12月11日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室

Nishino's rigidity theorem and questions on locally pseudoconvex maps
[ 講演概要 ]
Nishino proved in 1969 that locally Stein maps with fibers $\cong \mathbb{C}$ are locally trivial. Yamaguchi gave an alternate proof of Nishino's theorem which later developed into a the theory of variations of the Bergman kernel. The proofs of Nishino and Yamaguchi will be reviewed and questions suggested by the result will be discussed. A new application of the $L^2$ extension theorem will be also presented in this context.

### 2017年11月27日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室

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.

### 2017年11月20日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室

Relative GIT stabilities of toric Fano manifolds in low dimensions
[ 講演概要 ]
In 2000, Mabuchi extended the notion of Kaehler-Einstein metrics to Fano manifolds with non-vanishing Futaki invariant. Such a metric is called generalized Kaehler-Einstein metric or Mabuchi metric in the literature. Recently this metrics were rediscovered by Yao in the story of Donaldson's infinite dimensional moment map picture. Moreover, he introduced (uniform) relative Ding stability for toric Fano manifolds and showed that the existence of generalized Kaehler-Einstein metrics is equivalent to its uniform relative Ding stability. This equivalence is in the context of the Yau-Tian-Donaldson conjecture. In this talk, we focus on uniform relative Ding stability of toric Fano manifolds. More precisely, we determine all the uniformly relatively Ding stable toric Fano 3- and 4-folds as well as unstable ones. This talk is based on a joint work with Shunsuke Saito and Naoto Yotsutani.

### 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
[ 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.

### 2017年10月30日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室

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.

### 2017年10月23日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室

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.