## Seminar on Geometric Complex Analysis

Seminar information archive ～02/07｜Next seminar｜Future seminars 02/08～

Date, time & place | Monday 10:30 - 12:00 128Room #128 (Graduate School of Math. Sci. Bldg.) |
---|---|

Organizer(s) | Kengo Hirachi, Shigeharu Takayama, Ryosuke Nomura |

**Seminar information archive**

### 2021/05/10

10:30-12:00 Online

強擬凹複素曲面の境界に現れる接触構造 (Japanese)

https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

**Naohiko Kasuya**(Hokkaido University)強擬凹複素曲面の境界に現れる接触構造 (Japanese)

[ Abstract ]

強擬凸複素曲面の境界は3次元強擬凸CR多様体であり、正の接触構造を誘導する。BogomolovとDe Oliveiraは強擬凸複素曲面の境界に現れる接触構造はStein fillableであること（CR構造としては、Stein fillableなものに変形同値であること）を示した。

一方、強擬凹複素曲面の境界には負の3次元接触構造が現れる。本講演では、任意の負の3次元閉接触多様体が強擬凹複素曲面の境界として実現可能であることを示す。証明は、EliashbergによるStein manifoldの構成法を参考にして強擬凹境界への正則ハンドルの接着手法を確立することによってなされる。

尚、本講演内容はDaniele Zuddas氏（トリエステ大学）との共同研究である。

[ Reference URL ]強擬凸複素曲面の境界は3次元強擬凸CR多様体であり、正の接触構造を誘導する。BogomolovとDe Oliveiraは強擬凸複素曲面の境界に現れる接触構造はStein fillableであること（CR構造としては、Stein fillableなものに変形同値であること）を示した。

一方、強擬凹複素曲面の境界には負の3次元接触構造が現れる。本講演では、任意の負の3次元閉接触多様体が強擬凹複素曲面の境界として実現可能であることを示す。証明は、EliashbergによるStein manifoldの構成法を参考にして強擬凹境界への正則ハンドルの接着手法を確立することによってなされる。

尚、本講演内容はDaniele Zuddas氏（トリエステ大学）との共同研究である。

https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2021/04/26

10:30-12:00 Online

多様体の留数 (Japanese)

https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

**Jun O'Hara**(Chiba University)多様体の留数 (Japanese)

[ Abstract ]

$M$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。

参考資料：https://sites.google.com/site/junohara/ ダウンロード 「多様体のエネルギーと留数」（少し古い）, arXiv:2012.01713

[ Reference URL ]$M$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。

参考資料：https://sites.google.com/site/junohara/ ダウンロード 「多様体のエネルギーと留数」（少し古い）, arXiv:2012.01713

https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2021/04/19

10:30-12:00 Online

カスプと有理同値 (Japanese)

https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

**Shouhei Ma**(Tokyo Institute of Technology)カスプと有理同値 (Japanese)

[ Abstract ]

標題の「カスプ」とはいわゆるモジュラー多様体の（ベイリー・ボレル）コンパクト化の境界成分のことである。

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。

代数サイクルの観点からこの現象の高次元版をいくつか古典的な系列のモジュラー多様体の（ベイリー・ボレル、トロイダル）コンパクト化に対して調べたので、それについて報告する。

[ Reference URL ]標題の「カスプ」とはいわゆるモジュラー多様体の（ベイリー・ボレル）コンパクト化の境界成分のことである。

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。

代数サイクルの観点からこの現象の高次元版をいくつか古典的な系列のモジュラー多様体の（ベイリー・ボレル、トロイダル）コンパクト化に対して調べたので、それについて報告する。

https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2021/01/25

10:30-12:00 Online

Existence of a complete holomorphic vector field via the Kähler-Einstein metric

https://zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

**Young-Jun Choi**(Pusan National University)Existence of a complete holomorphic vector field via the Kähler-Einstein metric

[ Abstract ]

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.

[ Reference URL ]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.

https://zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2021/01/18

10:30-12:00 Online

The hydrodynamic period matrices and closings of an open Riemann surface of finite genus

https://zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

**HAMANO Sachiko**(Osaka City University)The hydrodynamic period matrices and closings of an open Riemann surface of finite genus

[ Abstract ]

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.

[ Reference URL ]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.

https://zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2020/12/21

10:30-12:00 Online

On a mixed Monge-Ampère operator for quasiplurisubharmonic functions

https://zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

**Martin Sera**(KUAS)On a mixed Monge-Ampère operator for quasiplurisubharmonic functions

[ Abstract ]

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.

[ Reference URL ]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.

https://zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2020/12/14

10:30-12:00 Online

On Levi flat hypersurfaces with transversely affine foliation

https://zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

**ADACHI Masanori**(Shizuoka University)On Levi flat hypersurfaces with transversely affine foliation

[ Abstract ]

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).

[ Reference URL ]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).

https://zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2020/11/30

10:30-12:00 Online

On asymptotic base loci of relative anti-canonical divisors

[ Reference URL ]

https://zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

**IWAI Masataka**(Osaka City Univ. and Kyoto Univ.)On asymptotic base loci of relative anti-canonical divisors

[ Reference URL ]

https://zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2020/11/09

10:30-12:00 Online

Meromorphic continuation of local zeta functions and nonpolar singularities

[ Reference URL ]

https://zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

**NOSE Toshihiro**(Fukuoka Institute of Technology)Meromorphic continuation of local zeta functions and nonpolar singularities

[ Reference URL ]

https://zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2020/10/26

10:30-12:00 Online

Spectral convergence in geometric quantization

[ Reference URL ]

https://zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

**HATTORI Kota**(Keio University)Spectral convergence in geometric quantization

[ Reference URL ]

https://zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2020/10/19

10:30-12:00 Online

On projective manifolds with pseudo-effective tangent bundle

https://zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

**MATSUMURA, Shin-ichi**(Tohoku University)On projective manifolds with pseudo-effective tangent bundle

[ Abstract ]

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).

[ Reference URL ]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).

https://zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2020/10/12

10:30-12:00 Online

Two topics on psedoconvex domains (Japanese)

[ Reference URL ]

https://zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

**NOGUCHI Junjiro**(University of Tokyo)Two topics on psedoconvex domains (Japanese)

[ Reference URL ]

https://zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2020/07/13

10:30-12:00 Online

$\mu$-cscK metrics and $\mu$K-stability of polarized manifolds

[ Reference URL ]

https://forms.gle/vSFPoVR6ugrkTGhX7

**INOUE Eiji**(University of Tokyo)$\mu$-cscK metrics and $\mu$K-stability of polarized manifolds

[ Reference URL ]

https://forms.gle/vSFPoVR6ugrkTGhX7

### 2020/07/06

10:30-12:00 Online

Nakano positivity of singular Hermitian metrics and vanishing theorems of Demailly-Nadel-Nakano type (Japanese?)

https://forms.gle/vSFPoVR6ugrkTGhX7

**INAYAMA Takahiro**(University of Tokyo)Nakano positivity of singular Hermitian metrics and vanishing theorems of Demailly-Nadel-Nakano type (Japanese?)

[ Abstract ]

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.

[ Reference URL ]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.

https://forms.gle/vSFPoVR6ugrkTGhX7

### 2020/06/29

10:30-12:00 Online

Oka properties of complements of holomorphically convex sets

https://forms.gle/vSFPoVR6ugrkTGhX7

**KUSAKABE Yuta**(Osaka University)Oka properties of complements of holomorphically convex sets

[ Abstract ]

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 $(n,k)\neq(2,1),(2,2),(3,3)$.

[ Reference URL ]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 $(n,k)\neq(2,1),(2,2),(3,3)$.

https://forms.gle/vSFPoVR6ugrkTGhX7

### 2020/06/08

10:30-12:00 Online

Applications of the Quot-scheme limit to variational aspects of the Hermitian-Einstein metric

https://forms.gle/vSFPoVR6ugrkTGhX7

**HASHIMOTO Yoshinori**(Tokyo Institute of Technology)Applications of the Quot-scheme limit to variational aspects of the Hermitian-Einstein metric

[ Abstract ]

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.

[ Reference URL ]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.

https://forms.gle/vSFPoVR6ugrkTGhX7

### 2020/05/25

10:30-12:00 Online

Characteristic forms of Cheng-Yau metric and CR invariants

[ Reference URL ]

https://forms.gle/vSFPoVR6ugrkTGhX7

**MARUGAME Taiji**(Riken AIP - Osaka University)Characteristic forms of Cheng-Yau metric and CR invariants

[ Reference URL ]

https://forms.gle/vSFPoVR6ugrkTGhX7

### 2020/05/18

10:30-12:00 Online

Higgs bundles and flat connections over compact Sasakian manifolds

https://forms.gle/vSFPoVR6ugrkTGhX7

**KASUYA Hisashi**(Osaka University)Higgs bundles and flat connections over compact Sasakian manifolds

[ Abstract ]

It is known that on a compact Kähler manifold, there is a correspondence between semisimple flat vector bundles and polystable higgs bundles with vanishing Chern classes via harmonic metrics (Simpson-Corlette). The purpose of this talk is to give the Sasakian (odd dimensional analogue of Kähler geometry) version of this correspondence. We prove that on a compact Sasakian manifold, there is an correspondence between semisimple flat vector bundles and the polystable basic Higgs bundles with vanishing basic Chern classes. (Joint work with Indranil Biswas, arXiv:1905.06178)

[ Reference URL ]It is known that on a compact Kähler manifold, there is a correspondence between semisimple flat vector bundles and polystable higgs bundles with vanishing Chern classes via harmonic metrics (Simpson-Corlette). The purpose of this talk is to give the Sasakian (odd dimensional analogue of Kähler geometry) version of this correspondence. We prove that on a compact Sasakian manifold, there is an correspondence between semisimple flat vector bundles and the polystable basic Higgs bundles with vanishing basic Chern classes. (Joint work with Indranil Biswas, arXiv:1905.06178)

https://forms.gle/vSFPoVR6ugrkTGhX7

### 2020/02/17

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Precompactness of the moduli space of pseudo-normed graded algebras

**Toshiki Mabuchi**(Osaka Univ.)Precompactness of the moduli space of pseudo-normed graded algebras

[ Abstract ]

Graded algebras (such as canonical rings) coming from the spaces of sections of polarized algebraic varieties are studied by many mathematicians. On the other hand, the pseudo-norm project proposed by S.-T. Yau and C.-Y. Chi gives us a new differential geometric aspect of the Torelli type theorem.

In this talk, we give the details of how the geometry of pseudo-normed graded algebras allows us to obtain a natural compactification of the moduli space of pseudo-normed graded algebras.

(1) For a sequence of pseudo-normed graded algebras (of the same type), the above precompactness gives us some limit different from the Gromov-Hausdorff limit in Riemannian geometry.

(2) As an example of our construction, we have the Deligne-Mumford compactification, in which the notion of the orthogonal direct sum of pseudo-normed spaces comes up naturally. We also have a higher dimensional analogue by using weight filtration.

Graded algebras (such as canonical rings) coming from the spaces of sections of polarized algebraic varieties are studied by many mathematicians. On the other hand, the pseudo-norm project proposed by S.-T. Yau and C.-Y. Chi gives us a new differential geometric aspect of the Torelli type theorem.

In this talk, we give the details of how the geometry of pseudo-normed graded algebras allows us to obtain a natural compactification of the moduli space of pseudo-normed graded algebras.

(1) For a sequence of pseudo-normed graded algebras (of the same type), the above precompactness gives us some limit different from the Gromov-Hausdorff limit in Riemannian geometry.

(2) As an example of our construction, we have the Deligne-Mumford compactification, in which the notion of the orthogonal direct sum of pseudo-normed spaces comes up naturally. We also have a higher dimensional analogue by using weight filtration.

### 2020/01/27

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Canonical measure and it’s applications

**Hajime Tsuji**(Sophia Univ.)Canonical measure and it’s applications

[ Abstract ]

The canonical measure is a natural generalization of K\”ahler-Einstein metrics to the case of projective manifolds with nonnegative Kodaira dimension. In this talk we consider the variation of canonical measures under projective deformations and give some applications.

The canonical measure is a natural generalization of K\”ahler-Einstein metrics to the case of projective manifolds with nonnegative Kodaira dimension. In this talk we consider the variation of canonical measures under projective deformations and give some applications.

### 2020/01/20

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Diederich-Fornaess and Steinness indices for abstract CR manifolds

**Masanori Adachi**(Shizuoka Univ.)Diederich-Fornaess and Steinness indices for abstract CR manifolds

[ Abstract ]

The Diederich-Fornaes and Steinness indices are estimated for weakly pseudoconvex domains in complex manifolds in terms of the D'Angelo 1-form of the boundary CR manifolds. In particular, CR invariance of these indices is shown when the domain is Takeuchi 1-convex. This is a joint work with Jihun Yum (Pusan National University).

The Diederich-Fornaes and Steinness indices are estimated for weakly pseudoconvex domains in complex manifolds in terms of the D'Angelo 1-form of the boundary CR manifolds. In particular, CR invariance of these indices is shown when the domain is Takeuchi 1-convex. This is a joint work with Jihun Yum (Pusan National University).

### 2019/12/16

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

A simplified proof of the optimal L^2 extension theorem and its application (Japanese)

**Genki Hosono**(Tohoku Univ.)A simplified proof of the optimal L^2 extension theorem and its application (Japanese)

[ Abstract ]

I will explain a simplified proof of an optimal version of the Ohsawa-Takegoshi L^2-extension theorem. In the proof, I use a method of Berndtsson-Lempert and skip some argument by the method of McNeal-Varolin. As an application, I will explain a result on extensions from possibly non-reduced varieties.

I will explain a simplified proof of an optimal version of the Ohsawa-Takegoshi L^2-extension theorem. In the proof, I use a method of Berndtsson-Lempert and skip some argument by the method of McNeal-Varolin. As an application, I will explain a result on extensions from possibly non-reduced varieties.

### 2019/12/11

16:00-17:00 Room #156 (Graduate School of Math. Sci. Bldg.)

Einstein-Weyl structures (English)

**Joel Merker**(Paris Sud)Einstein-Weyl structures (English)

[ Abstract ]

On a conformal 3D manifold with electromagnetic field, Einstein-Weyl equations are the counterpart of Einstein's classical field equations. In 1943, Elie Cartan showed, using abstract arguments, that the general solution depends on 4 functions of 2 variables. I will present families of explicit solutions depending on 9 functions of 1 variable, much beyond what was known before. Such solutions are generic in the sense that the Cotton tensor is nonzero. This is joint work with Pawel Nurowski.

On a conformal 3D manifold with electromagnetic field, Einstein-Weyl equations are the counterpart of Einstein's classical field equations. In 1943, Elie Cartan showed, using abstract arguments, that the general solution depends on 4 functions of 2 variables. I will present families of explicit solutions depending on 9 functions of 1 variable, much beyond what was known before. Such solutions are generic in the sense that the Cotton tensor is nonzero. This is joint work with Pawel Nurowski.

### 2019/12/09

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Analytic torsions associated with the Rumin complex on contact spheres (Japanese)

**Akira Kitaoka**(The Univ. of Tokyo)Analytic torsions associated with the Rumin complex on contact spheres (Japanese)

[ Abstract ]

The Rumin complex, which is defined on contact manifolds, is a resolution of the constant sheaf of $\mathbb{R}$ given by a subquotient of the de Rham complex. In this talk, we explicitly write down all eigenvalues of the Rumin Laplacian on the standard contact spheres, and express the analytic torsion functions associated with the Rumin complex in terms of the Riemann zeta function. In particular, we find that the functions vanish at the origin and determine the analytic torsions.

The Rumin complex, which is defined on contact manifolds, is a resolution of the constant sheaf of $\mathbb{R}$ given by a subquotient of the de Rham complex. In this talk, we explicitly write down all eigenvalues of the Rumin Laplacian on the standard contact spheres, and express the analytic torsion functions associated with the Rumin complex in terms of the Riemann zeta function. In particular, we find that the functions vanish at the origin and determine the analytic torsions.

### 2019/12/02

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Toward classification of Moishezon twistor spaces

**Nobuhiro Honda**(Tokyo Tech.)Toward classification of Moishezon twistor spaces

[ Abstract ]

Twistor spaces are complex 3-folds which arise from 4-dimensional conformal geometry. These spaces always have negative Kodaira dimension, and most of them are known to be non-Kahler. But there are a plenty of compact twistor spaces which are Moishezon variety. The topology of such spaces is strongly constrained, and it seems not hopeless to obtain a classification and explicit description of them. I will talk about results in such a direction, which classify such spaces under a simple assumption. No example seems to be known which does not satisfy that assumption.

Twistor spaces are complex 3-folds which arise from 4-dimensional conformal geometry. These spaces always have negative Kodaira dimension, and most of them are known to be non-Kahler. But there are a plenty of compact twistor spaces which are Moishezon variety. The topology of such spaces is strongly constrained, and it seems not hopeless to obtain a classification and explicit description of them. I will talk about results in such a direction, which classify such spaces under a simple assumption. No example seems to be known which does not satisfy that assumption.