## 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, 野村 亮介

### 2021年10月11日(月)

10:30-12:00   オンライン開催

cscK計量に付随する完備スカラー平坦Kähler計量について (Japanese)
[ 講演概要 ]

[ 参考URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2021年07月19日(月)

10:30-12:00   オンライン開催

$\mathbb{C}^n$上の不分岐Riemann領域に対する中間的擬凸性 (Japanese)
[ 講演概要 ]
The talk is based on a joint work with T. Shima and S. Sugiyama.
We characterize the intermediate pseudoconvexity for unramified Riemann domains over $\mathbb{C}^n$ by the continuity property which holds for a class of maps whose projections to $\mathbb{C}^n$ are families of unidirectionally parameterized intermediate dimensional analytic balls written by polynomials of degree $\le 2$.
[ 参考URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2021年07月12日(月)

10:30-12:00   オンライン開催

Parametrization of Weil-Petersson curves on the plane (Japanese)
[ 講演概要 ]
A Weil-Petersson curve is the image of the real line by a quasiconformal homeomorphism of the plane whose complex dilatation is square integrable with respect to the hyperbolic metrics on the upper and the lower half-planes. We consider two parameter spaces of all such curves and show that they are biholomorphically equivalent. As a consequence, we prove that the variant of the Beurling-Ahlfors quasiconformal extension defined by using the heat kernel for the convolution yields a global real-analytic section for the Teichmueller projection to the Weil-Petersson Teichmueller space. This is a joint work with Huaying Wei.
[ 参考URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2021年07月05日(月)

10:30-12:00   オンライン開催

Several stronger concepts of relative K-stability for polarized toric manifolds (Japanese)
[ 講演概要 ]
We study relations between algebro-geometric stabilities for polarized toric manifolds. In this talk, we introduce several strengthenings of relative K-stability such as uniform stability and K-stability tested by more objects than test configurations, and show that these approaches are all equivalent. As a consequence, we solve a uniform version of the Yau-Tian-Donaldson conjecture for Calabi's extremal Kähler metrics in the toric setting. This talk is based on a joint work with Shunsuke Saito.
[ 参考URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2021年06月28日(月)

10:30-12:00   オンライン開催

Orevkov's theorem, Bézout's theorem, and the converse of Brolin's theorem (Japanese)
[ 講演概要 ]
The converse of Brolin's theorem was a problem on characterizing polynomials among rational functions (on the complex projective line) in terms of the equilibrium measures canonically associated to rational functions. We would talk about a history on the studies of this problem, its optimal solution, and a proof outline. The proof is reduced to Bézout's theorem from algebraic geometry, thanks to Orevkov's irreducibility theorem on polynomial lemniscates. This talk is based on joint works with Małgorzata Stawiska (Mathematical Reviews).
[ 参考URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2021年06月14日(月)

10:30-12:00   オンライン開催

Projective K3 surfaces containing Levi-flat hypersurfaces (Japanese)
[ 講演概要 ]
In May 2017, I reported on the gluing construction of a K3 surface at Seminar on Geometric Complex Analysis.
Here, by the gluing construction of a K3 surface, I mean the construction of a K3 surface by holomorphically gluing two open complex surfaces which are the complements of tubular neighborhoods of elliptic curves included in the blow-ups of the projective planes by nine points.
As of 2017, it was an open problem whether a projective K3 surface can be obtained by the gluing construction. Recently, I and Takato Uehara found a very concrete way to construct a projective K3 surface by the gluing method. As a corollary, we obtained the existence of non-Kummer projective K3 surface with compact Levi-flat hypersurfaces.
In this talk, I will explain the detail of the concrete gluing construction of such a K3 surface.
[ 参考URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2021年06月07日(月)

10:30-12:00   オンライン開催

Calabi-Yau structure and Bargmann type transformation on the Cayley projective plane (Japanese)
[ 講演概要 ]
In this talk, I would like to discuss a problem of the geometric quantization for the Cayley projective plane. Our purposes are to show the existence of a Calabi-Yau structure on the punctured cotangent bundle of the Cayley projective plane, and to construct a Bargmann type transformation between a space of holomorphic functions on the bundle and the $L_2$-space on the Cayley projective space. The transformation gives a quantization of the geodesic flow in terms of one parameter group of elliptic Fourier integral operators. This talk is based on a joint work with Kenro Furutani (Osaka City University Advanced Mathematical Institute): arXiv:2101.07505.
[ 参考URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2021年05月31日(月)

10:30-12:00   オンライン開催

Nonnegativity of the CR Paneitz operator for embeddable CR manifolds (Japanese)
[ 講演概要 ]
The CR Paneitz operator, which is a fourth-order CR invariant differential operator, plays a crucial role in three-dimensional CR geometry; it is deeply connected to global embeddability and the CR positive mass theorem. In this talk, I will show that the CR Paneitz operator is nonnegative for embeddable CR manifolds. I will also apply this result to some problems in CR geometry. In particular, I will give an affirmative solution to the CR Yamabe problem for embeddable CR manifolds.
[ 参考URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2021年05月24日(月)

10:30-12:00   オンライン開催

Cartan-Hartogs領域の固有正則写像 (Japanese)
[ 講演概要 ]
2つの球の間の固有正則写像は自己同型写像である。球を別の領域にしたらどうなるかを調べたい。球の一般化として複素擬楕円体や有界対称領域が考えられる。これら2つの領域を合わせた領域としてHua領域がある。これは有界対称領域の上に複素擬楕円体が乗っているような領域である。Hua領域の一番簡単な場合としてCartan-Hartogs領域があり、これらの間の固有正則写像の分類問題を考える。分類すると本質的には１種類の写像しかないことが分かる。ここでは2つの多項式写像が自己同型写像の差を省いて一致すれば、Isotoropy写像の差を省いて一致することを使う。
[ 参考URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2021年05月10日(月)

10:30-12:00   オンライン開催

[ 講演概要 ]

[ 参考URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2021年04月26日(月)

10:30-12:00   オンライン開催

[ 講演概要 ]
$M$を多様体、$z$を複素数とし、$M$の二点間の距離の$z$乗を積空間$M\times M$上積分したものを考えると、$z$の実部が大きいところで$z$の正則関数になる。解析接続により複素平面上の有理関数で1位の極のみ持つものが得られる。この有理型関数、特にその留数の性質を紹介する。具体的には、メビウス不変性、留数と似た量（曲面のWillmoreエネルギー、4次元多様体のGraham-Wittenエネルギー、積分幾何で出てくる内在的体積、ラプラシアンのスペクトルなど）との比較、有理型関数・留数による多様体の同定問題などを扱う。

[ 参考URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2021年04月19日(月)

10:30-12:00   オンライン開催

カスプと有理同値 (Japanese)
[ 講演概要 ]

1970年代にマニンとドリンフェルトは合同モジュラー曲線の２つのカスプの差がピカール群において有限位数であることを発見した。

[ 参考URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2021年01月25日(月)

10:30-12:00   オンライン開催
Young-Jun Choi 氏 (Pusan National University)
Existence of a complete holomorphic vector field via the Kähler-Einstein metric
[ 講演概要 ]
A fundamental problem in Several Complex Variables is to classify bounded pseudoconvex domains in the complex Euclidean space with a noncompact automorphism group, especially with a compact quotient. In the results of Wong-Rosay and Frankel, they make use of the "Scaling method'' for obtaining an 1-parameter family of automorphisms, which generates a holomorphic vector field.
In this talk, we discuss the existence of a nowhere vanishing complete holomorphic vector filed on a strongly pseudoconvex manifold admtting a negatively curved Kähler-Einstein metric and discrete sequence of automorphisms by introducing the scaling method on potentials of the Kähler-Einstein metric.
[ 参考URL ]
https://zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2021年01月18日(月)

10:30-12:00   オンライン開催

The hydrodynamic period matrices and closings of an open Riemann surface of finite genus
[ 講演概要 ]
A closing of an open Riemann srface $R$ of finite genus is a shorter name of a closed Riemann surface of the same genus into which $R$ can be embedded by a homology type preserving conformal mapping. We observe the Riemann period matrices of all closings of $R$ in the Siegel upper half space. It is known that every hydrodynamic differential on $R$ yields a closing of $R$ called a hydrodynamic closing. (A hydrodynamic differential is a holomorphic which describes a steady flow on $R$ of an ideal fluid.) We study the period matices induced by hydrodynamic closings of $R$. This is a joint work with Masakazu Shiba.
[ 参考URL ]
https://zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2020年12月21日(月)

10:30-12:00   オンライン開催
Martin Sera 氏 (京都先端科学大学)
On a mixed Monge-Ampère operator for quasiplurisubharmonic functions
[ 講演概要 ]
This reports on a joint work with R. Lärkäng and E. Wulcan. We consider mixed Monge-Ampère products of quasiplurisubharmonic functions with analytic singularities (introduced in a previous work with H. Raufi additionally). These products have the advantage that they preserve mass (a property which is missing for non-pluripolar products).
The main result of the work presented here is that such Monge-Ampère products can be regularized as explicit one parameter limits of mixed Monge-Ampère products of smooth functions, generalizing a result of Andersson-Błocki-Wulcan. We will explain how the theory of residue currents, going back to Coleff-Herrera, Passare and others, plays an important role in the proof.
As a consequence, we get an approximation of Chern and Segre currents of certain singular hermitian metrics on vector bundles by smooth forms in the corresponding Chern and Segre classes.
[ 参考URL ]
https://zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2020年12月14日(月)

10:30-12:00   オンライン開催

On Levi flat hypersurfaces with transversely affine foliation
[ 講演概要 ]
In this talk, we discuss the classification problem of Levi flat hypersurfaces in complex surfaces by restricting ourselves to the case that the Levi foliation is transversely affine. After presenting known examples, we give a proof for the non-existence of real analytic Levi flat hypersurface whose complement is 1-convex and Levi foliation is transversely affine in a compact Kähler surface. This is a joint work with Severine Biard (arXiv:2011.06379).
[ 参考URL ]
https://zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2020年11月30日(月)

10:30-12:00   オンライン開催

On asymptotic base loci of relative anti-canonical divisors
[ 講演概要 ]

この講演では, $-K_{X/Y}$から定まる"asymptotic base loci"という集合を用い て, 上記の結果が巨大や擬有効へ拡張できることを紹介する. また 「$-K_{X/Y}$から定まる"asymptotic base loci"が, 空集合か水平方向に存在す るかのどちらかである」ことも紹介する. この研究は大阪大学の江尻祥氏と東北 大学の松村慎一氏との共同研究である.
[ 参考URL ]
https://zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2020年11月09日(月)

10:30-12:00   オンライン開催

[ 講演概要 ]

[ 参考URL ]
https://zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2020年10月26日(月)

10:30-12:00   オンライン開催

Spectral convergence in geometric quantization
[ 講演概要 ]
シンプレクティック多様体とその上の前量子化束の組に対して、シンプレクティック形式と整合する複素構造の1パラメーター族と正則切断の1パラメーター族を考える。複素構造に対応するケーラー偏極の族が、ラグランジュファイブレーションに対応する実偏極に収束するとき、正則切断の族はボーア・ゾンマーフェルトファイバーに局所化するという現象が、アーベル多様体やトーリック多様体などのいくつかの例で観測されている。本講演では非特異なラグランジュファイブレーションの場合を中心に、前量子化束上のラプラス作用素のスペクトル収束の観点からボーア・ゾンマーフェルトファイバーへの局所化を説明する。本講演の内容は、山下真由子氏（京都大学）との共同研究に基づく。
[ 参考URL ]
https://zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2020年10月19日(月)

10:30-12:00   オンライン開催

On projective manifolds with pseudo-effective tangent bundle
[ 講演概要 ]
In this talk, I would like to discuss projective manifolds whose tangent bundle is pseudo-effective or admits a positively curved singular metric. I will explain a structure theorem for such manifolds and the classification in the two-dimensional case, comparing our theory with classical results for nef tangent bundle or non-negative bisectional curvature. Related open problems will be discussed if time permits.
This is joint work with Genki Hosono (Tohoku University) and Masataka Iwai (Osaka City University).
[ 参考URL ]
https://zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2020年10月12日(月)

10:30-12:00   オンライン開催

[ 講演概要 ]

(1) $\Omega/\mathbf{C}^n$ を不分岐領域とし，$\lambda: \Omega \to [-\infty, \infty)$を多重劣調和関数とする．このとき $\Omega$ がスタインならば$\{ \lambda$ < $c \}$, $c \in \mathbf{R}$, もスタインである. これを準備しておくと擬凸定理の証明で何かと便利である．
(2) Bochnerの管定理の簡短証明．岡の境界距離定理を用いる．この管定理を局所化した柏原の凸錐補題が佐藤超関数論の基礎部分で用いられる．上述の簡短証明のアイデアを用いてその凸錐の図形的情況を詳しく述べる．
[ 参考URL ]
https://zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2020年07月13日(月)

10:30-12:00   オンライン開催

$\mu$-cscK metrics and $\mu$K-stability of polarized manifolds
[ 講演概要 ]
I firstly talk about some backgrounds on the following two frameworks; "cscK metrics & K-stability" and "Kähler-Ricci soliton & modified K-stability", whose intersection is precisely the framework on "Kähler-Einstein metrics & K-stability".
I will introduce a new framework unifying these frameworks, which I call the framework on "$\mu$-cscK metrics and $\mu$K-stability".

There are two divided contents:
1. I explain formulation and first motivation for $\mu$-cscK metrics and give brief remarks on results parallel to those for cscK metrics / Kähler-Ricci solitons. I will illustrate some attractive features/phenomenon special to $\mu$-cscK metrics by examples; "extremal limit" and "phase transition".
(cf. https://arxiv.org/abs/1902.00664)
2. I explain how one should/can formulate/derive/express $\mu$-Futaki invariant of test configurations with general singularities. We also construct a characteristic class for families of polarized schemes, which generalizes the CM line bundle in K-stability. I also give a few words on applications to moduli problem.
(cf. https://arxiv.org/abs/2004.06393)
[ 参考URL ]
https://forms.gle/vSFPoVR6ugrkTGhX7

### 2020年07月06日(月)

10:30-12:00   オンライン開催

Nakano positivity of singular Hermitian metrics and vanishing theorems of Demailly-Nadel-Nakano type (Japanese?)
[ 講演概要 ]
We propose a general definition of Nakano semi-positivity of singular Hermitian metrics on holomorphic vector bundles. By using this positivity notion, we establish $L^2$-estimates for holomorphic vector bundles with Nakano positive singular Hermitian metrics. We also show vanishing theorems, which generalize both Nakano type and Demailly-Nadel type vanishing theorems.
[ 参考URL ]
https://forms.gle/vSFPoVR6ugrkTGhX7

### 2020年06月29日(月)

10:30-12:00   オンライン開催

Oka properties of complements of holomorphically convex sets

[ 講演概要 ]
A complex manifold is called an Oka manifold if the Oka principle for maps from Stein spaces holds. In this talk, we consider the question of when a holomorphically convex set in an Oka manifold has an Oka complement. Our main theorem states that the complement of a compact holomorphically convex set in a Stein manifold with the density property is an Oka manifold. This gives a positive answer to the well-known long-standing problem in Oka theory whether the complement of a compact polynomially convex set in $\mathbb{C}^{n}$ $(n>1)$ is Oka. The relative version of the main theorem can also be proved. As an application, we show that the complement $\mathbb{C}^{n}\setminus\mathbb{R}^{k}$ of a totally real affine subspace is Oka if $n>1$ and $(n,k)\neq(2,1),(2,2),(3,3)$.
[ 参考URL ]
https://forms.gle/vSFPoVR6ugrkTGhX7

### 2020年06月08日(月)

10:30-12:00   オンライン開催

Applications of the Quot-scheme limit to variational aspects of the Hermitian-Einstein metric
[ 講演概要 ]
The Kobayashi-Hitchin correspondence, proved by Donaldson and Uhlenbeck-Yau by using the nonlinear PDE theory, states that the existence of Hermitian-Einstein metrics on a holomorphic vector bundle is equivalent to an algebro-geometric stability condition. We present some results that exhibit an explicit link between differential and algebraic geometry in the above correspondence, from a variational point of view. The key to such results is an object called the Quot-scheme limit of Fubini-Study metrics, which is used to evaluate certain algebraic 1-parameter subgroups of Hermitian metrics by using the theory of Quot-schemes in algebraic geometry. This method also works for the proof of the correspondence between the balanced metrics and the Gieseker stability, as originally proved by X.W. Wang. Joint work with Julien Keller.
[ 参考URL ]
https://forms.gle/vSFPoVR6ugrkTGhX7