## Seminar on Geometric Complex Analysis

Seminar information archive ～09/23｜Next seminar｜Future seminars 09/24～

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

Organizer(s) | Kengo Hirachi, Shigeharu Takayama |

**Seminar information archive**

### 2023/07/10

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

Degenerations of Riemann surfaces and small eigenvalues of the Laplacian (日本語)

https://u-tokyo-ac-jp.zoom.us/meeting/register/tZEqceqsrTIjEtRxenSMdPogvCxlWzAogj5A

**Ken-Ichi Yoshikawa**(Kyoto University)Degenerations of Riemann surfaces and small eigenvalues of the Laplacian (日本語)

[ Abstract ]

In this talk, we consider a proper surjective holomorphic map from a smooth projective surface to a compact Riemann surface. Near a singular fiber, this is viewed as a one-parameter degeneration of compact Riemann surfaces. We fix a Kähler metric on the projective surface and consider the Kähler metric on the fibers induced from this metric. In this setting, for each regular fiber, we can consider the Laplacian acting on the functions on the fiber. It is known that for any k, the k-th eigenvalue of the Laplacian extends to a continuous function on the base curve. In particular, if the singular fiber is not irreducible, some eigenvalues of the Laplacian of the regular fiber converge to zero as the regular fiber approaches to the singular fiber. We call such eigenvalues small eigenvalues. In this talk, when the singular fiber is reduced, we will explain the asymptotic behavior of the product of all small eigenvalues of the Laplacian.

[ Reference URL ]In this talk, we consider a proper surjective holomorphic map from a smooth projective surface to a compact Riemann surface. Near a singular fiber, this is viewed as a one-parameter degeneration of compact Riemann surfaces. We fix a Kähler metric on the projective surface and consider the Kähler metric on the fibers induced from this metric. In this setting, for each regular fiber, we can consider the Laplacian acting on the functions on the fiber. It is known that for any k, the k-th eigenvalue of the Laplacian extends to a continuous function on the base curve. In particular, if the singular fiber is not irreducible, some eigenvalues of the Laplacian of the regular fiber converge to zero as the regular fiber approaches to the singular fiber. We call such eigenvalues small eigenvalues. In this talk, when the singular fiber is reduced, we will explain the asymptotic behavior of the product of all small eigenvalues of the Laplacian.

https://u-tokyo-ac-jp.zoom.us/meeting/register/tZEqceqsrTIjEtRxenSMdPogvCxlWzAogj5A

### 2023/07/03

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

Hyperbolicity and fundamental groups of complex quasi-projective varieties

https://u-tokyo-ac-jp.zoom.us/meeting/register/tZEqceqsrTIjEtRxenSMdPogvCxlWzAogj5A

**Katsutoshi Yamanoi**(Osaka University)Hyperbolicity and fundamental groups of complex quasi-projective varieties

[ Abstract ]

This talk is based on a joint work with Benoit Cadorel and Ya Deng. arXiv:2212.12225

[ Reference URL ]This talk is based on a joint work with Benoit Cadorel and Ya Deng. arXiv:2212.12225

https://u-tokyo-ac-jp.zoom.us/meeting/register/tZEqceqsrTIjEtRxenSMdPogvCxlWzAogj5A

### 2023/06/26

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

Miyaoka type inequality for terminal weak Fano varieties

https://u-tokyo-ac-jp.zoom.us/meeting/register/tZEqceqsrTIjEtRxenSMdPogvCxlWzAogj5A

**Masataka IWAI**(Osaka Univeristy)Miyaoka type inequality for terminal weak Fano varieties

[ Abstract ]

In this talk, we show that $c_2(X)c_1(X)^{n-2}$ is positive for any $n$-dimensional terminal weak Fano varieties $X$. As a corollary, we obtain some inequalities (Miyaoka type inequalities) with respect to $c_2(X)c_1(X)^{n-2}$ and $c_1(X)^{n}$. This is joint work with Chen Jiang and Haidong Liu (arXiv:2303.00268).

[ Reference URL ]In this talk, we show that $c_2(X)c_1(X)^{n-2}$ is positive for any $n$-dimensional terminal weak Fano varieties $X$. As a corollary, we obtain some inequalities (Miyaoka type inequalities) with respect to $c_2(X)c_1(X)^{n-2}$ and $c_1(X)^{n}$. This is joint work with Chen Jiang and Haidong Liu (arXiv:2303.00268).

https://u-tokyo-ac-jp.zoom.us/meeting/register/tZEqceqsrTIjEtRxenSMdPogvCxlWzAogj5A

### 2023/06/19

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

A new construction method for 3-dimensional indefinite Zoll manifolds

https://u-tokyo-ac-jp.zoom.us/meeting/register/tZEqceqsrTIjEtRxenSMdPogvCxlWzAogj5A

**Nobuhiro Honda**(Tokyo University of Technology)A new construction method for 3-dimensional indefinite Zoll manifolds

[ Abstract ]

The Penrose correspondence gives correspondences between special geometric structures on manifolds and complex manifolds, one of which is between Einstein-Weyl structures on 3-manifolds and complex surfaces. The latter complex surfaces are called mini-Twister spaces. In this talk, I will show that compact mini-Zeister spaces can be constructed in a natural way from hyperelliptic curves of arbitrary species, and that the resulting 3-manifolds have a remarkable geometric property called the Zoll property, which means that all geodesics are closed. A typical example is a sphere. The three-dimensional Einstein-Weyl manifold obtained in this study is indefinite, and the geodesics considered are spatial. These Einstein-Weyl manifolds can be regarded as generalizations of those given in arXiv:2208.13567.

Translated with www.DeepL.com/Translator (free version)

[ Reference URL ]The Penrose correspondence gives correspondences between special geometric structures on manifolds and complex manifolds, one of which is between Einstein-Weyl structures on 3-manifolds and complex surfaces. The latter complex surfaces are called mini-Twister spaces. In this talk, I will show that compact mini-Zeister spaces can be constructed in a natural way from hyperelliptic curves of arbitrary species, and that the resulting 3-manifolds have a remarkable geometric property called the Zoll property, which means that all geodesics are closed. A typical example is a sphere. The three-dimensional Einstein-Weyl manifold obtained in this study is indefinite, and the geodesics considered are spatial. These Einstein-Weyl manifolds can be regarded as generalizations of those given in arXiv:2208.13567.

Translated with www.DeepL.com/Translator (free version)

https://u-tokyo-ac-jp.zoom.us/meeting/register/tZEqceqsrTIjEtRxenSMdPogvCxlWzAogj5A

### 2023/05/29

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

On dynamical degrees of birational maps

https://u-tokyo-ac-jp.zoom.us/meeting/register/tZEqceqsrTIjEtRxenSMdPogvCxlWzAogj5A

**Takato UEHARA**(Okayama University)On dynamical degrees of birational maps

[ Abstract ]

A birational map on a projective surface defines its dynamical degree, which measures the complexity of dynamical behavior of the map. The set of dynamical degrees, called the dynamical spectrum, has properties similar to that of volumes of hyperbolic 3-manifolds, shown by Thurston. In this talk, we will explain the properties of the dynamical spectrum.

[ Reference URL ]A birational map on a projective surface defines its dynamical degree, which measures the complexity of dynamical behavior of the map. The set of dynamical degrees, called the dynamical spectrum, has properties similar to that of volumes of hyperbolic 3-manifolds, shown by Thurston. In this talk, we will explain the properties of the dynamical spectrum.

https://u-tokyo-ac-jp.zoom.us/meeting/register/tZEqceqsrTIjEtRxenSMdPogvCxlWzAogj5A

### 2023/05/22

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

A residue formula for meromorphic connections and applications to stable sets of foliations

https://u-tokyo-ac-jp.zoom.us/meeting/register/tZEqceqsrTIjEtRxenSMdPogvCxlWzAogj5A

**Masanari Adachi**(Shizuoka Univeristy)A residue formula for meromorphic connections and applications to stable sets of foliations

[ Abstract ]

We discuss a proof for Brunella’s conjecture: a codimension one holomorphic foliation on a compact complex manifold of dimension > 2 has no exceptional minimal set if its normal bundle is ample. The main idea is the localization of the first Chern class of the normal bundle of the foliation via a holomorphic connection. Although this localization was done via that of the first Atiyah class in our previous proof, we shall explain that this can be shown more directly by a residue formula. If time permits, we also discuss a nonexistence result of Levi flat hypersurfaces with transversely affine Levi foliation. This talk is based on joint works

with S. Biard and J. Brinkschulte.

[ Reference URL ]We discuss a proof for Brunella’s conjecture: a codimension one holomorphic foliation on a compact complex manifold of dimension > 2 has no exceptional minimal set if its normal bundle is ample. The main idea is the localization of the first Chern class of the normal bundle of the foliation via a holomorphic connection. Although this localization was done via that of the first Atiyah class in our previous proof, we shall explain that this can be shown more directly by a residue formula. If time permits, we also discuss a nonexistence result of Levi flat hypersurfaces with transversely affine Levi foliation. This talk is based on joint works

with S. Biard and J. Brinkschulte.

https://u-tokyo-ac-jp.zoom.us/meeting/register/tZEqceqsrTIjEtRxenSMdPogvCxlWzAogj5A

### 2023/05/15

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

$\mathcal{I}'$-curvatures and the Hirachi conjecture (Japanese)

https://u-tokyo-ac-jp.zoom.us/meeting/register/tZEqceqsrTIjEtRxenSMdPogvCxlWzAogj5A

**Yuya Takeuchi**(Tsukuba Univeristy)$\mathcal{I}'$-curvatures and the Hirachi conjecture (Japanese)

[ Abstract ]

Hirachi conjecture deals with a relation between the integrals of local pseudo-Hermitian invariants and global CR invariants. This is a CR analogue of the Deser-Schwimmer conjceture, which was proved by Alexakis. In this talk, I would like to explain some results on the Hirachi conjecture. In particular, I'll introduce the $\mathcal{I}'$-curvatures and prove that these produce counterexamples to the Hirachi conjecture in higher dimensions. This talk is based on joint work with Jeffrey S. Case.

[ Reference URL ]Hirachi conjecture deals with a relation between the integrals of local pseudo-Hermitian invariants and global CR invariants. This is a CR analogue of the Deser-Schwimmer conjceture, which was proved by Alexakis. In this talk, I would like to explain some results on the Hirachi conjecture. In particular, I'll introduce the $\mathcal{I}'$-curvatures and prove that these produce counterexamples to the Hirachi conjecture in higher dimensions. This talk is based on joint work with Jeffrey S. Case.

https://u-tokyo-ac-jp.zoom.us/meeting/register/tZEqceqsrTIjEtRxenSMdPogvCxlWzAogj5A

### 2023/05/08

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

Non-Kähler Hodge theory and resolutions of cyclic orbifolds (日本語)

https://u-tokyo-ac-jp.zoom.us/meeting/register/tZEqceqsrTIjEtRxenSMdPogvCxlWzAogj5A

**Hisashi Kasuya**(Osaka Univeristy)Non-Kähler Hodge theory and resolutions of cyclic orbifolds (日本語)

[ Abstract ]

This talk is based on the joint works with Jonas Stelzig (LMU München). We discuss the Hodge theory of non-Kähler compact complex manifolds. In this term, we think several types of compact complex manifolds and compact Kähler manifolds are considered as the "simplest”. We give a way of constructing simply connected compact complex non-Kähler manifolds of certain types by using resolutions of cyclic orbifolds.

[ Reference URL ]This talk is based on the joint works with Jonas Stelzig (LMU München). We discuss the Hodge theory of non-Kähler compact complex manifolds. In this term, we think several types of compact complex manifolds and compact Kähler manifolds are considered as the "simplest”. We give a way of constructing simply connected compact complex non-Kähler manifolds of certain types by using resolutions of cyclic orbifolds.

https://u-tokyo-ac-jp.zoom.us/meeting/register/tZEqceqsrTIjEtRxenSMdPogvCxlWzAogj5A

### 2023/04/24

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

On site+Zoom

Guan's theorems on optimal strong openness and concavity of minimal $L^2$ integrals (日本語)

https://u-tokyo-ac-jp.zoom.us/meeting/register/tZEqceqsrTIjEtRxenSMdPogvCxlWzAogj5A

On site+Zoom

**Takeo Ohsawa**(Nogoya Universiry)Guan's theorems on optimal strong openness and concavity of minimal $L^2$ integrals (日本語)

[ Abstract ]

Motivated by a question of approximating plurisubharmonic (=psh) functions by those with tame singularities, Demailly and Kollar asked several basic questions on the singularities of psh functions. Guan solved two of them effectively in a paper published in 2019. One of their corollaries says the following.

THEOREM. Let $\Omega$ be a pseudoconvex domain in $\mathbb{C}^n$ and let $\varphi$ be a negative psh function on $\Omega$ such that $\int_\Omega{e^{-\varphi}}<\infty$. Then, $e^{-p\varphi}\in L^1_{\text{loc}}$ around $x$ for any $x\in\Omega$ and $p>1$ satisfying the inequality $$

\frac{p}{p-1}>\frac{\int_\Omega{e^{-\varphi}}}{K_\Omega(x)},

$$ where $K_\Omega$ denotes the diagonalized Bergman kernel of $\Omega$.

This remarkable result is a consequence of a basic property of the minimal $L^2$ integrals (=MLI). The main purpose of the talk is to give an outline of the proof of Theorem by explaining the relation between several notions including the MLI which measure the singularities of psh functions. It will also be mentioned that the proof of Theorem is essentially based on the optimal Ohsawa-Takegoshi type extension theorem, which leads to a concavity property of MLI. Recent papers by Guan and his students will be reviewed, too.

[ Reference URL ]Motivated by a question of approximating plurisubharmonic (=psh) functions by those with tame singularities, Demailly and Kollar asked several basic questions on the singularities of psh functions. Guan solved two of them effectively in a paper published in 2019. One of their corollaries says the following.

THEOREM. Let $\Omega$ be a pseudoconvex domain in $\mathbb{C}^n$ and let $\varphi$ be a negative psh function on $\Omega$ such that $\int_\Omega{e^{-\varphi}}<\infty$. Then, $e^{-p\varphi}\in L^1_{\text{loc}}$ around $x$ for any $x\in\Omega$ and $p>1$ satisfying the inequality $$

\frac{p}{p-1}>\frac{\int_\Omega{e^{-\varphi}}}{K_\Omega(x)},

$$ where $K_\Omega$ denotes the diagonalized Bergman kernel of $\Omega$.

This remarkable result is a consequence of a basic property of the minimal $L^2$ integrals (=MLI). The main purpose of the talk is to give an outline of the proof of Theorem by explaining the relation between several notions including the MLI which measure the singularities of psh functions. It will also be mentioned that the proof of Theorem is essentially based on the optimal Ohsawa-Takegoshi type extension theorem, which leads to a concavity property of MLI. Recent papers by Guan and his students will be reviewed, too.

https://u-tokyo-ac-jp.zoom.us/meeting/register/tZEqceqsrTIjEtRxenSMdPogvCxlWzAogj5A

### 2023/02/13

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

On a presentation to introduce function theory of several variables (Japanese)

https://forms.gle/hYT2hVhDE3q1wDSh6

**Junjiro Noguchi**(The University of Tokyo)On a presentation to introduce function theory of several variables (Japanese)

[ Abstract ]

微積分は，主に1変数の理論を講義するが，後半で多変数の内容を入れる．同じ様に，複素解析（函数論）でも，一変数の後につなぎよく，多変数の講義を段差なく行えるようにしたい．

モデルケースとして'リーマンの写像定理'がある．現在多くの教科書に書かれているモンテルの定理による初等的な証明(1922, Fejér--Riesz)まで，もとのリーマンの学位論文(1851)から約70年の歳月がかかている．

岡理論・多変数関数論基礎についてみると，Oka IX (1953)より本年でやはり70年たつが，あまり'初等化'の方面へは進展していないように思う．こここでは，学部の複素解析のコースで'リーマンの写像定理'の後に，段差無く完全証明付きで岡理論・多変数関数論基礎を講義する展開を考える．

初等化には，岡のオリジナル法(1943未発表, IX 1953)を第1連接定理に基づき展開するのが適当であることを紹介したい．学部講義の数学内容に日本人による成果が入ることで，学生のモチベーションに好効果を与えるであろうことも期待したい．

時間が許せば，擬凸問題解決の岡のオリジナル法と別証明とされるGrauertの証明との間のFredholm定理をめぐる類似性についても述べたい．

[ Reference URL ]微積分は，主に1変数の理論を講義するが，後半で多変数の内容を入れる．同じ様に，複素解析（函数論）でも，一変数の後につなぎよく，多変数の講義を段差なく行えるようにしたい．

モデルケースとして'リーマンの写像定理'がある．現在多くの教科書に書かれているモンテルの定理による初等的な証明(1922, Fejér--Riesz)まで，もとのリーマンの学位論文(1851)から約70年の歳月がかかている．

岡理論・多変数関数論基礎についてみると，Oka IX (1953)より本年でやはり70年たつが，あまり'初等化'の方面へは進展していないように思う．こここでは，学部の複素解析のコースで'リーマンの写像定理'の後に，段差無く完全証明付きで岡理論・多変数関数論基礎を講義する展開を考える．

初等化には，岡のオリジナル法(1943未発表, IX 1953)を第1連接定理に基づき展開するのが適当であることを紹介したい．学部講義の数学内容に日本人による成果が入ることで，学生のモチベーションに好効果を与えるであろうことも期待したい．

時間が許せば，擬凸問題解決の岡のオリジナル法と別証明とされるGrauertの証明との間のFredholm定理をめぐる類似性についても述べたい．

https://forms.gle/hYT2hVhDE3q1wDSh6

### 2023/01/16

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

Holomorphic foliation associated with a semi-positive class of numerical dimension one (Japanese)

https://forms.gle/hYT2hVhDE3q1wDSh6

**Takayuki Koike**(Osaka Metropolitan University)Holomorphic foliation associated with a semi-positive class of numerical dimension one (Japanese)

[ Abstract ]

Let $X$ be a compact Kähler manifold and $\alpha$ be a Dolbeault cohomology class of bidegree $(1,1)$ on $X$.

When the numerical dimension of $\alpha$ is one and $\alpha$ admits at least two smooth semi-positive representatives, we show the existence of a family of real analytic Levi-flat hypersurfaces in $X$ and a holomorphic foliation on a suitable domain of $X$ along whose leaves any semi-positive representative of $\alpha$ is zero.

As an application, we give the affirmative answer to a conjecture on the relation between the semi-positivity of the line bundle $[Y]$ and the analytic structure of a neighborhood of $Y$ for a smooth connected hypersurface $Y$ of $X$.

As an application, we give the affirmative answer to a conjecture on the relation between the semi-positivity of the line bundle $[Y]$ and the analytic structure of a neighborhood of $Y$ for a smooth connected hypersurface $Y$ of $X$.

[ Reference URL ]Let $X$ be a compact Kähler manifold and $\alpha$ be a Dolbeault cohomology class of bidegree $(1,1)$ on $X$.

When the numerical dimension of $\alpha$ is one and $\alpha$ admits at least two smooth semi-positive representatives, we show the existence of a family of real analytic Levi-flat hypersurfaces in $X$ and a holomorphic foliation on a suitable domain of $X$ along whose leaves any semi-positive representative of $\alpha$ is zero.

As an application, we give the affirmative answer to a conjecture on the relation between the semi-positivity of the line bundle $[Y]$ and the analytic structure of a neighborhood of $Y$ for a smooth connected hypersurface $Y$ of $X$.

As an application, we give the affirmative answer to a conjecture on the relation between the semi-positivity of the line bundle $[Y]$ and the analytic structure of a neighborhood of $Y$ for a smooth connected hypersurface $Y$ of $X$.

https://forms.gle/hYT2hVhDE3q1wDSh6

### 2022/12/12

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

$L^2$-extension index and its applications (Japanese)

https://forms.gle/hYT2hVhDE3q1wDSh6

**Takahiro Inayama**(Tokyo University of Science)$L^2$-extension index and its applications (Japanese)

[ Abstract ]

In this talk, we introduce a new concept of $L^2$-extension indices. By using this notion, we propose a new way to study the positivity of curvature. We prove that there is an equivalence between how sharp the $L^2$-extension is and how positive the curvature is. As applications, we study Prekopa-type theorems and the positivity of a certain direct image sheaf.

[ Reference URL ]In this talk, we introduce a new concept of $L^2$-extension indices. By using this notion, we propose a new way to study the positivity of curvature. We prove that there is an equivalence between how sharp the $L^2$-extension is and how positive the curvature is. As applications, we study Prekopa-type theorems and the positivity of a certain direct image sheaf.

https://forms.gle/hYT2hVhDE3q1wDSh6

### 2022/12/05

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

On sharper estimates of Ohsawa--Takegoshi $L^2$-extension theorem in higher dimensional case (Japanese)

https://forms.gle/hYT2hVhDE3q1wDSh6

**Shota Kikuchi**(National Institute of Technology, Suzuka College)On sharper estimates of Ohsawa--Takegoshi $L^2$-extension theorem in higher dimensional case (Japanese)

[ Abstract ]

Hosono proposed an idea of getting an $L^2$-estimate sharper than the one of Berndtsson--Lempert type $L^2$-extension theorem by allowing constants depending on weight functions in $\mathbb{C}$.

In this talk, I explain the details of "sharper estimates" and the higher dimensional case of it. Also, I explain my recent studies related to it.

[ Reference URL ]Hosono proposed an idea of getting an $L^2$-estimate sharper than the one of Berndtsson--Lempert type $L^2$-extension theorem by allowing constants depending on weight functions in $\mathbb{C}$.

In this talk, I explain the details of "sharper estimates" and the higher dimensional case of it. Also, I explain my recent studies related to it.

https://forms.gle/hYT2hVhDE3q1wDSh6

### 2022/11/21

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

Resolution of singularities for $C^{\infty}$ functions and meromorphy of local zeta functions (Japanese)

https://forms.gle/hYT2hVhDE3q1wDSh6

**Joe Kamimoto**(Kyushu University)Resolution of singularities for $C^{\infty}$ functions and meromorphy of local zeta functions (Japanese)

[ Abstract ]

In this talk, we attempt to resolve the singularities of the zero variety of a $C^{\infty}$ function of two variables as much as possible by using ordinary blowings up. As a result, we formulate an algorithm to locally express the zero variety in the “almost” normal crossings form, which is close to the normal crossings form but may include flat functions. As an application, we investigate analytic continuation of local zeta functions associated with $C^{\infty}$ functions of two variables.

[ Reference URL ]In this talk, we attempt to resolve the singularities of the zero variety of a $C^{\infty}$ function of two variables as much as possible by using ordinary blowings up. As a result, we formulate an algorithm to locally express the zero variety in the “almost” normal crossings form, which is close to the normal crossings form but may include flat functions. As an application, we investigate analytic continuation of local zeta functions associated with $C^{\infty}$ functions of two variables.

https://forms.gle/hYT2hVhDE3q1wDSh6

### 2022/11/14

15:00-16:30 Online

The double holomorphic tangent space of the Teichmueller spaces (Japanese)

https://forms.gle/hYT2hVhDE3q1wDSh6

**Hideki Miyach**(Kanazawa University)The double holomorphic tangent space of the Teichmueller spaces (Japanese)

[ Abstract ]

The double holomorphic tangent space of a complex manifold is the holomorphic tangent space of the holomorphic tangent bundle of the complex manifold. In this talk, we will give an intrinsic description of the double tangent spaces of the Teichmueller spaces of closed Riemann surfaces of genus at least 2.

[ Reference URL ]The double holomorphic tangent space of a complex manifold is the holomorphic tangent space of the holomorphic tangent bundle of the complex manifold. In this talk, we will give an intrinsic description of the double tangent spaces of the Teichmueller spaces of closed Riemann surfaces of genus at least 2.

https://forms.gle/hYT2hVhDE3q1wDSh6

### 2022/10/31

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

The non-archimedean μ-entropy in toric case (Japanese)

https://forms.gle/hYT2hVhDE3q1wDSh6

**Eiji Inoue**(RIKEN)The non-archimedean μ-entropy in toric case (Japanese)

[ Abstract ]

The non-archimedean μ-entropy is a functional on the space of test configurations of a polarized variety. It plays a key role in μK-stability and can be interpreted as a dual functional to Perelman’s μ-entropy for Kahler metrics. The fundamental question on the non-archimedean μ-entropy is the existence and uniqueness of maximizers. To find its maximizers, it is natural to extend the functional to a suitable completion of the space of test configurations. For general polarized variety, we can realize such completion and extension based on the non-archimedean pluripotential theory.

In the toric case, the torus invariant subspace of the completion is identified with a suitable space of convex functions on the moment polytope and then the non-archimedean μ-entropy is simply expressed by integrations of convex functions on the polytope. I will show a compactness result in the toric case, by which we conclude the existence of maximizers for the toric non-archimedean μ-entropy.

[ Reference URL ]The non-archimedean μ-entropy is a functional on the space of test configurations of a polarized variety. It plays a key role in μK-stability and can be interpreted as a dual functional to Perelman’s μ-entropy for Kahler metrics. The fundamental question on the non-archimedean μ-entropy is the existence and uniqueness of maximizers. To find its maximizers, it is natural to extend the functional to a suitable completion of the space of test configurations. For general polarized variety, we can realize such completion and extension based on the non-archimedean pluripotential theory.

In the toric case, the torus invariant subspace of the completion is identified with a suitable space of convex functions on the moment polytope and then the non-archimedean μ-entropy is simply expressed by integrations of convex functions on the polytope. I will show a compactness result in the toric case, by which we conclude the existence of maximizers for the toric non-archimedean μ-entropy.

https://forms.gle/hYT2hVhDE3q1wDSh6

### 2022/10/24

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

A new approach to the nilpotent orbit theorem via the $L^2$ extension theorem of Ohsawa-Takegoshi type (Japanese)

https://forms.gle/hYT2hVhDE3q1wDSh6

**Taro Fujisawa**(Tokyo Denki University)A new approach to the nilpotent orbit theorem via the $L^2$ extension theorem of Ohsawa-Takegoshi type (Japanese)

[ Abstract ]

I will talk about a new proof of (a part of) the nilpotent orbit theorem for unipotent variations of Hodge structure. This approach is largely inspired by the recent works of Deng and of Sabbah-Schnell. In my proof, the $L^2$ extension theorem of Ohsawa-Takegoshi type plays essential roles.

[ Reference URL ]I will talk about a new proof of (a part of) the nilpotent orbit theorem for unipotent variations of Hodge structure. This approach is largely inspired by the recent works of Deng and of Sabbah-Schnell. In my proof, the $L^2$ extension theorem of Ohsawa-Takegoshi type plays essential roles.

https://forms.gle/hYT2hVhDE3q1wDSh6

### 2022/07/11

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

The CR Killing operator and Bernstein-Gelfand-Gelfand construction in CR geometry (Japanese)

https://forms.gle/hYT2hVhDE3q1wDSh6

**Yoshihiko Matsumoto**(Osaka University)The CR Killing operator and Bernstein-Gelfand-Gelfand construction in CR geometry (Japanese)

[ Abstract ]

In this talk, I introduce the CR Killing operator associated with compatible almost CR structures on contact manifolds, which describes trivial infinitesimal deformations generated by contact Hamiltonian vector fields, and discuss how it can also be reconstructed by the Bernstein-Gelfand-Gelfand construction in the general theory of parabolic geometries. The “modified” adjoint tractor connection defined by Cap (2008) plays a crucial role. If time permits, I’d also like to discuss what this observation might mean in relation to asymptotically complex hyperbolic Einstein metrics, which are bulk geometric structures for compatible almost CR structures at infinity.

[ Reference URL ]In this talk, I introduce the CR Killing operator associated with compatible almost CR structures on contact manifolds, which describes trivial infinitesimal deformations generated by contact Hamiltonian vector fields, and discuss how it can also be reconstructed by the Bernstein-Gelfand-Gelfand construction in the general theory of parabolic geometries. The “modified” adjoint tractor connection defined by Cap (2008) plays a crucial role. If time permits, I’d also like to discuss what this observation might mean in relation to asymptotically complex hyperbolic Einstein metrics, which are bulk geometric structures for compatible almost CR structures at infinity.

https://forms.gle/hYT2hVhDE3q1wDSh6

### 2022/07/04

10:30-12:00 Online

Bloch's principle for holomorphic maps into subvarieties of semi-abelian varieties (Japanese)

https://forms.gle/hYT2hVhDE3q1wDSh6

**Katsutoshi Yamanoi**(Osaka University)Bloch's principle for holomorphic maps into subvarieties of semi-abelian varieties (Japanese)

[ Abstract ]

We discuss a generalization of the logarithmic Bloch-Ochiai theorem about entire curves in subvarieties of semi-abelian varieties, in terms of sequences of holomorphic maps from the unit disc.

This generalization implies, among other things, that subvarieties of log general type in semi-abelian varieties are pseudo-Kobayashi hyperbolic.

As another application, we discuss an improvement of a classical theorem due to Cartan in 1920's about the system of nowhere vanishing holomorphic functions on the unit disc satisfying Borel's identity.

[ Reference URL ]We discuss a generalization of the logarithmic Bloch-Ochiai theorem about entire curves in subvarieties of semi-abelian varieties, in terms of sequences of holomorphic maps from the unit disc.

This generalization implies, among other things, that subvarieties of log general type in semi-abelian varieties are pseudo-Kobayashi hyperbolic.

As another application, we discuss an improvement of a classical theorem due to Cartan in 1920's about the system of nowhere vanishing holomorphic functions on the unit disc satisfying Borel's identity.

https://forms.gle/hYT2hVhDE3q1wDSh6

### 2022/06/20

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

Constructions of CR GJMS operators in dimension three (Japanese)

https://forms.gle/hYT2hVhDE3q1wDSh6

**Taiji Marugame**(The University of Electro-Communications)Constructions of CR GJMS operators in dimension three (Japanese)

[ Abstract ]

CR GJMS operators are invariant differential operators on CR manifolds whose leading parts are powers of the sublaplacian. Such operators can be constructed by Fefferman's ambient metric or the Cheng-Yau metric, but the construction is obstructed at a finite order due to the ambiguity of these metrics. Gover-Graham constructed some higher order CR GJMS operators by using tractor calculus and BGG constructions. In particular, they showed that three dimensional CR manifolds admit CR GJMS operators of all orders. In this talk, we give proofs to this fact in two different ways. One is by the use of self-dual Einstein ACH metric and the other is by the Graham-Hirachi inhomogeneous ambient metric adapted to the Fefferman conformal structure. We also state a conjecture on the relationship between these two metrics.

[ Reference URL ]CR GJMS operators are invariant differential operators on CR manifolds whose leading parts are powers of the sublaplacian. Such operators can be constructed by Fefferman's ambient metric or the Cheng-Yau metric, but the construction is obstructed at a finite order due to the ambiguity of these metrics. Gover-Graham constructed some higher order CR GJMS operators by using tractor calculus and BGG constructions. In particular, they showed that three dimensional CR manifolds admit CR GJMS operators of all orders. In this talk, we give proofs to this fact in two different ways. One is by the use of self-dual Einstein ACH metric and the other is by the Graham-Hirachi inhomogeneous ambient metric adapted to the Fefferman conformal structure. We also state a conjecture on the relationship between these two metrics.

https://forms.gle/hYT2hVhDE3q1wDSh6

### 2022/05/30

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

Asymptotic estimates of holomorphic sections on Bohr-Sommerfeld Lagrangian submanifolds (Japanese)

https://forms.gle/hYT2hVhDE3q1wDSh6

**Yusaku Tiba**(Ochanomizu University)Asymptotic estimates of holomorphic sections on Bohr-Sommerfeld Lagrangian submanifolds (Japanese)

[ Abstract ]

In this talk, we study an asymptotic estimate of holomorphic sections of a positive line bundle. Let $M$ be a complex manifold and $L$ be a positive line bundle over $M$ with a Hermitian metric $h$ whose Chern form is a Kähler form $\omega$. Let $X \subset M$ be a Lagrangian submanifold of $(M, \omega)$. When $X$ satisfies the Bohr-Sommerfeld condition, we prove a submean value theorem for holomorphic sections and we give an asymptotic estimate of $\inf_{x \in X}|f(x)|_{h^k}$ for $f \in H^0(M, L^k)$. This estimate provides an analog result about the leading term of the asymptotic series expansion formula of the Bergman kernel function.

[ Reference URL ]In this talk, we study an asymptotic estimate of holomorphic sections of a positive line bundle. Let $M$ be a complex manifold and $L$ be a positive line bundle over $M$ with a Hermitian metric $h$ whose Chern form is a Kähler form $\omega$. Let $X \subset M$ be a Lagrangian submanifold of $(M, \omega)$. When $X$ satisfies the Bohr-Sommerfeld condition, we prove a submean value theorem for holomorphic sections and we give an asymptotic estimate of $\inf_{x \in X}|f(x)|_{h^k}$ for $f \in H^0(M, L^k)$. This estimate provides an analog result about the leading term of the asymptotic series expansion formula of the Bergman kernel function.

https://forms.gle/hYT2hVhDE3q1wDSh6

### 2022/04/18

10:30-12:00 Online

Approximation and bundle convexity on complex manifolds of pseudo convex type (Japanese)

https://forms.gle/hYT2hVhDE3q1wDSh6

**Takeo Ohasawa**(Nagoya University)Approximation and bundle convexity on complex manifolds of pseudo convex type (Japanese)

[ Abstract ]

An approximation theorem will be proved for the space of holomorphic sections of vector bundles on certain Zariski open sets of weakly 1-complete manifolds. As an existence result on such manifolds, a solution of the bundle-valued version of the Levi problem will be given by a variant of a method of Hoermander.

[ Reference URL ]An approximation theorem will be proved for the space of holomorphic sections of vector bundles on certain Zariski open sets of weakly 1-complete manifolds. As an existence result on such manifolds, a solution of the bundle-valued version of the Levi problem will be given by a variant of a method of Hoermander.

https://forms.gle/hYT2hVhDE3q1wDSh6

### 2022/01/24

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

Analytic Ax-Schanuel Theorem for semi-abelian varieties and Nevanlinna theory (Japanese)

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

**Junjiro Noguchi**(The University of Tokyo)Analytic Ax-Schanuel Theorem for semi-abelian varieties and Nevanlinna theory (Japanese)

[ Abstract ]

The present study is motivated by $\textit{Schanuel Conjecture}$, which in particular implies the algebraic independence of $e$ and $\pi$. Our aim is to explore, as a transcendental functional analogue of Schanuel Conjecture, the value distribution theory (Nevanlinna theory) of the entire curve $\widehat{\mathrm{ex}}_A f:=(\exp_Af,f):\mathbf{C} \to A \times \mathrm{Lie}(A)$ associated with an entire curve $f: \mathbf{C} \to \mathrm{Lie}(A)$, where $\exp_A:\mathrm{Lie}(A)\to A$ is an exponential map of a semi-abelian variety $A$.

We firstly give a Nevanlinna theoretic proof to the $\textit{analytic Ax-Schanuel Theorem}$ for semi-abelian varieties, which was proved by J. Ax 1972 in the case of formal power series $\mathbf{C}[[t]]$ (Ax-Schanuel Theorem). We assume some non-degeneracy condition for $f$ such that in the case of $A=(\mathbf{C}^*)^n$ and $\mathrm{Lie}((\mathbf{C}^*)^n)=\mathbf{C}^n$, the elements of the vector-valued function $f(z)-f(0)$ are $\mathbf{Q}$-linearly independent. Then by the method of Nevanlinna theory (the Log Bloch-Ochiai Theorem), we prove that $\mathrm{tr.deg}_\mathbf{C}\, \widehat{\mathrm{ex}}_A f \geq n+ 1.$

Secondly, we prove a $\textit{Second Main Theorem}$ for $\widehat{\mathrm{ex}}_A f$ and an algebraic divisor $D$ on $A \times \mathrm{Lie}(A)$ with compactifications $\bar D \subset \bar A \times \overline{\mathrm{Lie}(A)}$ such that

\[

T_{\widehat{\mathrm{ex}}_Af}(r, L({\bar D})) \leq N_1 (r,

(\widehat{\mathrm{ex}}_A f)^* D)+

\varepsilon T_{\exp_Af}(r)+O(\log r) ~~ ||_\varepsilon.

\]

We will also deal with the intersections of $\widehat{\mathrm{ex}}_Af$ with higher codimensional algebraic cycles of $A \times \mathrm{Lie}(A)$ as well as the case of higher jets.

[ Reference URL ]The present study is motivated by $\textit{Schanuel Conjecture}$, which in particular implies the algebraic independence of $e$ and $\pi$. Our aim is to explore, as a transcendental functional analogue of Schanuel Conjecture, the value distribution theory (Nevanlinna theory) of the entire curve $\widehat{\mathrm{ex}}_A f:=(\exp_Af,f):\mathbf{C} \to A \times \mathrm{Lie}(A)$ associated with an entire curve $f: \mathbf{C} \to \mathrm{Lie}(A)$, where $\exp_A:\mathrm{Lie}(A)\to A$ is an exponential map of a semi-abelian variety $A$.

We firstly give a Nevanlinna theoretic proof to the $\textit{analytic Ax-Schanuel Theorem}$ for semi-abelian varieties, which was proved by J. Ax 1972 in the case of formal power series $\mathbf{C}[[t]]$ (Ax-Schanuel Theorem). We assume some non-degeneracy condition for $f$ such that in the case of $A=(\mathbf{C}^*)^n$ and $\mathrm{Lie}((\mathbf{C}^*)^n)=\mathbf{C}^n$, the elements of the vector-valued function $f(z)-f(0)$ are $\mathbf{Q}$-linearly independent. Then by the method of Nevanlinna theory (the Log Bloch-Ochiai Theorem), we prove that $\mathrm{tr.deg}_\mathbf{C}\, \widehat{\mathrm{ex}}_A f \geq n+ 1.$

Secondly, we prove a $\textit{Second Main Theorem}$ for $\widehat{\mathrm{ex}}_A f$ and an algebraic divisor $D$ on $A \times \mathrm{Lie}(A)$ with compactifications $\bar D \subset \bar A \times \overline{\mathrm{Lie}(A)}$ such that

\[

T_{\widehat{\mathrm{ex}}_Af}(r, L({\bar D})) \leq N_1 (r,

(\widehat{\mathrm{ex}}_A f)^* D)+

\varepsilon T_{\exp_Af}(r)+O(\log r) ~~ ||_\varepsilon.

\]

We will also deal with the intersections of $\widehat{\mathrm{ex}}_Af$ with higher codimensional algebraic cycles of $A \times \mathrm{Lie}(A)$ as well as the case of higher jets.

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

### 2021/12/13

10:30-12:00 Online

A generalized Hermitian curvature flow on almost Hermitian manifolds (Japanese)

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

**Masaya Kawamura**(National Institute of Technology)A generalized Hermitian curvature flow on almost Hermitian manifolds (Japanese)

[ Abstract ]

It is well-known that the Uniformization theorem (any Riemannian metric on a closed 2-manifold is conformal to one of constant curvature) can be proven by using the Ricci flow. J. Streets and G. Tian questioned whether or not a geometric flow can be used to classify non-Kähler complex surfaces as in the case of the Ricci flow. Also they asked if it is possible to prove classification results in higher dimensions by using geometric flows in non-Kähler Hermitian geometry. Streets and Tian considered that these flows should be close to the Kähler-Ricci flow as much as possible. From this point of view, they introduced a geometric flow called the Hermitian curvature flow (HCF) which evolves an initial Hermitian metric in the direction of a Ricci-type tensor of the Chern connection modified with some lower order torsion terms. Streets and Tian also introduced another geometric flow, which is called the pluriclosed flow (PCF), by choosing torsion terms to preserve the pluriclosed condition on Hermitian metrics. Y. Ustinovskiy studied a particular version of the HCF over a compact Hermitian manifold. Ustinovskiy proved that if the initial metric has Griffiths positive (non-negative) Chern curvature, then this property is preserved along the flow.

In recent years, some results concerning geometric flows on complex manifolds have been extended to the almost complex setting. For instance, L. Vezzoni defined a new Hermitian curvature flow on almost Hermitian manifolds for generalizing some studies on the HCF and the Hermitian Hilbert functional. And J. Chu, V. Tosatti and B. Weinkove considered parabolic Monge-Ampère equation on almost Hermitian manifolds, which is equivalent to the almost complex Chern-Ricci flow. T. Zheng characterized the maximal existence time for a solution to the almost complex Chern-Ricci flow.

In this talk, we consider a generalized Hermitian curvature flow in almost Hermitian geometry and introduce that it has some properties such as the long-time existence obstruction, the uniform equivalence between its solution and an almost Hermitian metric, and the preservation result along the flow.

[ Reference URL ]It is well-known that the Uniformization theorem (any Riemannian metric on a closed 2-manifold is conformal to one of constant curvature) can be proven by using the Ricci flow. J. Streets and G. Tian questioned whether or not a geometric flow can be used to classify non-Kähler complex surfaces as in the case of the Ricci flow. Also they asked if it is possible to prove classification results in higher dimensions by using geometric flows in non-Kähler Hermitian geometry. Streets and Tian considered that these flows should be close to the Kähler-Ricci flow as much as possible. From this point of view, they introduced a geometric flow called the Hermitian curvature flow (HCF) which evolves an initial Hermitian metric in the direction of a Ricci-type tensor of the Chern connection modified with some lower order torsion terms. Streets and Tian also introduced another geometric flow, which is called the pluriclosed flow (PCF), by choosing torsion terms to preserve the pluriclosed condition on Hermitian metrics. Y. Ustinovskiy studied a particular version of the HCF over a compact Hermitian manifold. Ustinovskiy proved that if the initial metric has Griffiths positive (non-negative) Chern curvature, then this property is preserved along the flow.

In recent years, some results concerning geometric flows on complex manifolds have been extended to the almost complex setting. For instance, L. Vezzoni defined a new Hermitian curvature flow on almost Hermitian manifolds for generalizing some studies on the HCF and the Hermitian Hilbert functional. And J. Chu, V. Tosatti and B. Weinkove considered parabolic Monge-Ampère equation on almost Hermitian manifolds, which is equivalent to the almost complex Chern-Ricci flow. T. Zheng characterized the maximal existence time for a solution to the almost complex Chern-Ricci flow.

In this talk, we consider a generalized Hermitian curvature flow in almost Hermitian geometry and introduce that it has some properties such as the long-time existence obstruction, the uniform equivalence between its solution and an almost Hermitian metric, and the preservation result along the flow.

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

### 2021/11/29

10:30-12:00 Online

レンズ空間上のRay-Singer捩率とRumin複体のラプラシアン (Japanese)

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

**Akira Kitaoka**(The University of Tokyo)レンズ空間上のRay-Singer捩率とRumin複体のラプラシアン (Japanese)

[ Abstract ]

Rumin複体は、接触多様体に関するBernstein-Gelfand-Gelfand複体(BGG複体)である。BGG複体は、放物型幾何やフィルター付き多様体に対して構成される複体であり、BGG複体のコホモロジーはde Rhamコホモロジーに一致するという事が挙げられる。また、Rumin複体はsub-Riemmann極限を考えた際に自然に現れるという性質を持つ。

De Rham複体を使って定義した概念をRumin複体に置き換えるとどうなるのか、ということを考える。本講演では、この考えを解析的捩率に適応した場合を話す。レンズ空間上のユニモジュラーなホロのミーから誘導される平坦ベクトル束に対して、Rumin複体の解析的捩率の値が、Betti数とRay-Singer捩率を用いて表されることを報告する。

[ Reference URL ]Rumin複体は、接触多様体に関するBernstein-Gelfand-Gelfand複体(BGG複体)である。BGG複体は、放物型幾何やフィルター付き多様体に対して構成される複体であり、BGG複体のコホモロジーはde Rhamコホモロジーに一致するという事が挙げられる。また、Rumin複体はsub-Riemmann極限を考えた際に自然に現れるという性質を持つ。

De Rham複体を使って定義した概念をRumin複体に置き換えるとどうなるのか、ということを考える。本講演では、この考えを解析的捩率に適応した場合を話す。レンズ空間上のユニモジュラーなホロのミーから誘導される平坦ベクトル束に対して、Rumin複体の解析的捩率の値が、Betti数とRay-Singer捩率を用いて表されることを報告する。

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