## Seminar on Geometric Complex Analysis

Seminar information archive ～06/17｜Next seminar｜Future seminars 06/18～

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**

### 2024/06/17

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

Oka tubes in holomorphic line bundles (Japanese)

[ Reference URL ]

https://forms.gle/gTP8qNZwPyQyxjTj8

**Yuta Kusakabe**(Kyushu Univ.)Oka tubes in holomorphic line bundles (Japanese)

[ Reference URL ]

https://forms.gle/gTP8qNZwPyQyxjTj8

### 2024/06/10

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

Computing Noetherian operators of polynomial ideals

--How to characterize a polynomial ideal by partial differential operators -- (Japanese)

https://forms.gle/gTP8qNZwPyQyxjTj8

**Katsusuke Nabeshima**(Tokyo Univ. of Science)Computing Noetherian operators of polynomial ideals

--How to characterize a polynomial ideal by partial differential operators -- (Japanese)

[ Abstract ]

Describing ideals in polynomial rings by using systems of differential operators in one of the major approaches to study them. In 1916, F.S. Macaulay brought the notion of an inverse system, a system of differential conditions that describes an ideal. In 1937, W. Groebner mentioned the importance of the Macaulay's inverse system in the study of linear differential equations with constant coefficient, and in 1938, he introduced differential operators to characterize ideals that are primary to a rational maximal ideal. After that the important results and the terminology came from L. Ehrenpreise and V. P. Palamodov in 1961 and 1970, that is the characterization of primary ideals by the differential operators. The differential operators allow one to characterize the primary ideal by differential conditions on the associated characteristic variety. The differential operators are called Noetherian operators.

In this talk, we consider Noetherian operators in the context of symbolic computation. Upon utilizing the theory of holonomic D-modules, we present a new computational method of Noetherian operators associated to a polynomial ideal. The computational method that consists mainly of linear algebra techniques is given for computing them. Moreover, as applications, new computational methods of polynomial ideals are discussed by utilizing the Noetherian operators.

[ Reference URL ]Describing ideals in polynomial rings by using systems of differential operators in one of the major approaches to study them. In 1916, F.S. Macaulay brought the notion of an inverse system, a system of differential conditions that describes an ideal. In 1937, W. Groebner mentioned the importance of the Macaulay's inverse system in the study of linear differential equations with constant coefficient, and in 1938, he introduced differential operators to characterize ideals that are primary to a rational maximal ideal. After that the important results and the terminology came from L. Ehrenpreise and V. P. Palamodov in 1961 and 1970, that is the characterization of primary ideals by the differential operators. The differential operators allow one to characterize the primary ideal by differential conditions on the associated characteristic variety. The differential operators are called Noetherian operators.

In this talk, we consider Noetherian operators in the context of symbolic computation. Upon utilizing the theory of holonomic D-modules, we present a new computational method of Noetherian operators associated to a polynomial ideal. The computational method that consists mainly of linear algebra techniques is given for computing them. Moreover, as applications, new computational methods of polynomial ideals are discussed by utilizing the Noetherian operators.

https://forms.gle/gTP8qNZwPyQyxjTj8

### 2024/05/27

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

Hyperkähler ambient metrics associated with twistor CR manifolds (Japanese)

[ Reference URL ]

https://forms.gle/gTP8qNZwPyQyxjTj8

**Taiji Marugame**(The Univ. of Electro-Communications)Hyperkähler ambient metrics associated with twistor CR manifolds (Japanese)

[ Reference URL ]

https://forms.gle/gTP8qNZwPyQyxjTj8

### 2024/05/20

10:50-12:20 Room #128 (Graduate School of Math. Sci. Bldg.)

Kähler metrics in the Siegel domain (Japanese)

https://forms.gle/gTP8qNZwPyQyxjTj8

**Lijie Sun**(Yamaguchi Univ.)Kähler metrics in the Siegel domain (Japanese)

[ Abstract ]

The Siegel domain is endowed with an intrinsic Kähler structure, making it an exemplary model for the complex hyperbolic plane. Its boundary, characterized as the one-point compactification of the Heisenberg group, plays an important role in studying the geometry of the Siegel domain. In this talk, using the CR structure of the Heisenberg group we introduce a variety of Kähler structures within the Siegel domain. We conclude by demonstrating that all these metrics are PCR-Kähler equivalent, that is, essentially the same when confined to the CR structure. This talk is based on a joint work with Ioannis Platis and Joonhyung Kim.

[ Reference URL ]The Siegel domain is endowed with an intrinsic Kähler structure, making it an exemplary model for the complex hyperbolic plane. Its boundary, characterized as the one-point compactification of the Heisenberg group, plays an important role in studying the geometry of the Siegel domain. In this talk, using the CR structure of the Heisenberg group we introduce a variety of Kähler structures within the Siegel domain. We conclude by demonstrating that all these metrics are PCR-Kähler equivalent, that is, essentially the same when confined to the CR structure. This talk is based on a joint work with Ioannis Platis and Joonhyung Kim.

https://forms.gle/gTP8qNZwPyQyxjTj8

### 2024/05/13

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

(Japanese)

[ Reference URL ]

https://forms.gle/gTP8qNZwPyQyxjTj8

**Yu Kawakami**(Kanazawa Univ.)(Japanese)

[ Reference URL ]

https://forms.gle/gTP8qNZwPyQyxjTj8

### 2024/04/22

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

Neighborhood of a compact curve whose intersection matrix has a positive eigenvalue (Japanese)

[ Reference URL ]

https://forms.gle/gTP8qNZwPyQyxjTj8

**Takayuki Koike**(Osaka Metropolitan Univ.)Neighborhood of a compact curve whose intersection matrix has a positive eigenvalue (Japanese)

[ Reference URL ]

https://forms.gle/gTP8qNZwPyQyxjTj8

### 2024/04/15

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

Kohn-Rossi cohomology of spherical CR manifolds (Japanese)

[ Reference URL ]

https://forms.gle/gTP8qNZwPyQyxjTj8

**Yuya Takeuchi**(Tsukuba Univ.)Kohn-Rossi cohomology of spherical CR manifolds (Japanese)

[ Reference URL ]

https://forms.gle/gTP8qNZwPyQyxjTj8

### 2023/12/11

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

On Partial deformations and Bers embedding (Japanese)

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

**Ryo Matsuda**(Kyoto Univeristy)On Partial deformations and Bers embedding (Japanese)

[ Abstract ]

The Teichmüller space of the Riemann surface S is the space of deformations of the complex structure of S. For complex analysis on Teich(S), it is biholomorphic embedded into a bounded set of the space of complex Banach spaces, denoted as B(S). This embedding is known as the Bers embedding. Additionally, when S is of infinite type, considering partial deformations can reveal properties of Teich(S). Earle-Gardiner-Lakic prove that asymptotically conformal deformations correspond to subspaces where the norm of the embedding decays at infinity. In this talk, we generalize this result, showing that deformations that become asymptotically conformal at some end correspond to spaces where the norm decays at that end. Finally, using this result and the David map, a generalization of quasiconformal maps, I’ll give that in the Bers boundary of infinite-type Riemann surface satisfying the Shiga condition, Maximal cusps are not dense.

[ Reference URL ]The Teichmüller space of the Riemann surface S is the space of deformations of the complex structure of S. For complex analysis on Teich(S), it is biholomorphic embedded into a bounded set of the space of complex Banach spaces, denoted as B(S). This embedding is known as the Bers embedding. Additionally, when S is of infinite type, considering partial deformations can reveal properties of Teich(S). Earle-Gardiner-Lakic prove that asymptotically conformal deformations correspond to subspaces where the norm of the embedding decays at infinity. In this talk, we generalize this result, showing that deformations that become asymptotically conformal at some end correspond to spaces where the norm decays at that end. Finally, using this result and the David map, a generalization of quasiconformal maps, I’ll give that in the Bers boundary of infinite-type Riemann surface satisfying the Shiga condition, Maximal cusps are not dense.

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

### 2023/11/27

11:00-12:30 Room #128 (Graduate School of Math. Sci. Bldg.)

On a holomorphic tubular neighborhood of a compact complex curve and Brjuno condition (Japanese)

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

**Satoshi Ogawa**(Osaka Metropolitan University)On a holomorphic tubular neighborhood of a compact complex curve and Brjuno condition (Japanese)

[ Abstract ]

Let $C$ be a compact complex curve holomorphically embedded in a non-singular complex surface $M$ with a unitary flat normal bundle $N_{C/M}$ and let $\mathcal{U}$ be a finite open cover of $C$. Gong--Stolovitch posed a sufficient condition for the existence of a holomorphic tubular neighborhood of $C$ in $M$ expressed with operator norms of Čech coboundary maps $\delta$ on $\check{C}^0(\mathcal{U}, \mathcal{O}_C(N_{C/M}^\nu))$ and $\check{C}^0(\mathcal{U}, \mathcal{O}_C(T_C \otimes N_{C/M}^\nu))$.

In this talk, we introduce some estimates of the operator norms of $\delta$. As a result, we see the Brjuno condition appears as a sufficient condition for the existence of a holomorphic tubular neighborhood.

[ Reference URL ]Let $C$ be a compact complex curve holomorphically embedded in a non-singular complex surface $M$ with a unitary flat normal bundle $N_{C/M}$ and let $\mathcal{U}$ be a finite open cover of $C$. Gong--Stolovitch posed a sufficient condition for the existence of a holomorphic tubular neighborhood of $C$ in $M$ expressed with operator norms of Čech coboundary maps $\delta$ on $\check{C}^0(\mathcal{U}, \mathcal{O}_C(N_{C/M}^\nu))$ and $\check{C}^0(\mathcal{U}, \mathcal{O}_C(T_C \otimes N_{C/M}^\nu))$.

In this talk, we introduce some estimates of the operator norms of $\delta$. As a result, we see the Brjuno condition appears as a sufficient condition for the existence of a holomorphic tubular neighborhood.

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

### 2023/10/30

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

The Nonvanishing problem for varieties with nef anticanonical bundle

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

**Shin-Ichi Matsumura**(Tohoku Univeristy)The Nonvanishing problem for varieties with nef anticanonical bundle

[ Abstract ]

In the framework of the minimal model program for generalized pairs, the abundance conjecture does not hold. However, interestingly, the generalized nonvanishing conjecture is expected to hold. This conjecture asks whether the canonical divisor of generalized pairs can be represented by an effective divisor in its numerical class. In this talk, we discuss the nonvanishing conjecture for generalized LC pairs in three dimensions and prove that the conjecture is true for the nef anti-canonical divisors.

This talk is based on joint work with V. Lazic, Th. Peternell, N. Tsakanikas, and Z. Xie.

[ Reference URL ]In the framework of the minimal model program for generalized pairs, the abundance conjecture does not hold. However, interestingly, the generalized nonvanishing conjecture is expected to hold. This conjecture asks whether the canonical divisor of generalized pairs can be represented by an effective divisor in its numerical class. In this talk, we discuss the nonvanishing conjecture for generalized LC pairs in three dimensions and prove that the conjecture is true for the nef anti-canonical divisors.

This talk is based on joint work with V. Lazic, Th. Peternell, N. Tsakanikas, and Z. Xie.

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

### 2023/10/16

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

The limit of Kähler-Ricci flows

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

**Hajime Tsuji**(Sophia University)The limit of Kähler-Ricci flows

[ Abstract ]

In this talk, I would like to present the (normalized) limit of Kähler-Ricci flows for compact Kähler manifolds with intermediate Kodaira dimesion under the condition that the canonical bundle is abundant.

[ Reference URL ]In this talk, I would like to present the (normalized) limit of Kähler-Ricci flows for compact Kähler manifolds with intermediate Kodaira dimesion under the condition that the canonical bundle is abundant.

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

### 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