## Seminar on Geometric Complex Analysis

Seminar information archive ～04/19｜Next seminar｜Future seminars 04/20～

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

**Seminar information archive**

### 2006/11/27

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

Logarithmic connections along Saito free divisors

**Aleksandr G. Aleksandrov**(Institute for Control Sciences, Moscow)Logarithmic connections along Saito free divisors

[ Abstract ]

We develop an approach to the study of meromorphic connections with logarithmic poles along a Saito free divisor. In particular, basic properties of Christoffel symbols of such connections are established. We also compute the set of all integrable meromorphic connections with logarithmic poles and describe the corresponding spaces of horizontal sections for some examples of Saito free divisors including the discriminants of the minimal versal deformations of $A_2$- and of $A_3$-singularities, and a divisor in $\mathbf{C}^3$ which appeared in a work of M. Sato in the context of the theory of prehomogeneous spaces.

We develop an approach to the study of meromorphic connections with logarithmic poles along a Saito free divisor. In particular, basic properties of Christoffel symbols of such connections are established. We also compute the set of all integrable meromorphic connections with logarithmic poles and describe the corresponding spaces of horizontal sections for some examples of Saito free divisors including the discriminants of the minimal versal deformations of $A_2$- and of $A_3$-singularities, and a divisor in $\mathbf{C}^3$ which appeared in a work of M. Sato in the context of the theory of prehomogeneous spaces.

### 2006/11/20

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

Advances and examples in the value distribution theory

**野口潤次郎**(東大数理)Advances and examples in the value distribution theory

### 2006/11/13

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

Sasaki-Futaki invariant and existence of Einstein metrics on toric Sasaki manifolds

**小野 肇**(東京工業大学)Sasaki-Futaki invariant and existence of Einstein metrics on toric Sasaki manifolds

### 2006/11/06

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

On the extension of twisted pluricanonical forms

**Mihai Paun**(Université Henri Poincaré Nancy)On the extension of twisted pluricanonical forms

### 2006/10/23

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

Classification of hypersurface simple K3 singularities -- 95 and others

**泊 昌孝**(日本大学文理学部)Classification of hypersurface simple K3 singularities -- 95 and others

### 2006/10/16

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

Differentiability of the volume of divisors and Khovanskii-Teissier inequalities

**Sebastien Boucksom**(東大数理 JSPS研究員)Differentiability of the volume of divisors and Khovanskii-Teissier inequalities

### 2006/07/10

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

Characterization of domains in $C^n$ by their noncompact automorphism groups

**Do Duc Thai**(Hanoi教育大)Characterization of domains in $C^n$ by their noncompact automorphism groups

[ Abstract ]

In this talk, the characterization of domains in $C^n$ by their noncompact automorphism groups are given. By this characterization, the Bedford-Pinchuk theorem is true for any domain (not necessary bounded) in $C^n$.

In this talk, the characterization of domains in $C^n$ by their noncompact automorphism groups are given. By this characterization, the Bedford-Pinchuk theorem is true for any domain (not necessary bounded) in $C^n$.

### 2006/07/03

14:00-15:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Complex Semi-Abelian Varieties II --- Compactifications and etc.

**Jörg Winkelmann**(Université Henri Poincaré Nancy)Complex Semi-Abelian Varieties II --- Compactifications and etc.

### 2006/06/26

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

Toward construction of Green current for modular cycles in modular varieties

**織田孝幸**(東大数理)Toward construction of Green current for modular cycles in modular varieties

### 2006/06/19

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

Deformations and smoothing of (generalized) holomorphic symplectic structures

**後藤竜司**(大阪大学)Deformations and smoothing of (generalized) holomorphic symplectic structures

### 2006/06/12

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

The Rumin complex and Hamiltonian mechanism

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~hirachi/scv/akahori.pdf

**赤堀隆夫**(兵庫県立大学)The Rumin complex and Hamiltonian mechanism

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~hirachi/scv/akahori.pdf

### 2006/06/05

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

Integral formulas for infinite dimensional domains with arbitrary boundary

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~hirachi/scv/Bauer.pdf

**Wolfram Bauer**(東京理科大)Integral formulas for infinite dimensional domains with arbitrary boundary

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~hirachi/scv/Bauer.pdf

### 2006/05/29

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

Uniformisation of Holomorphic Foliations by Curves II

**Marco Brunella**(Bourgogne)Uniformisation of Holomorphic Foliations by Curves II

### 2006/05/29

13:30-15:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Hodge theory with bounds and its application to foliations

**大沢 健夫**(名古屋大学)Hodge theory with bounds and its application to foliations

### 2006/05/22

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

Uniformisation of Holomorphic Foliations by Curves I (Part II on May 29)

**Marco Brunella**(Bourgogne)Uniformisation of Holomorphic Foliations by Curves I (Part II on May 29)

[ Abstract ]

In the first lecture, we give a definition of "leaf" for a singular holomorphic one-dimensional foliation on a projective manifold. The definition is such that the leaves of a foliation glue together in a nice way, giving a "covering tube" which is a sort of semi-global flow box. This is, in some sense, the topological part of the theory. In the second lecture, we prove some convexity property of this covering tube. As a corollary we obtain that, when there are hyperbolic leaves, the leafwise Poincare' metric has some remarkable positivity property. In the third lecture, we study foliations all of whose leaves are parabolic. Using a suitable extension theorem for certain meromorphic maps, we show how to generalise the above positivity property to this degenerate class of foliations.

In the first lecture, we give a definition of "leaf" for a singular holomorphic one-dimensional foliation on a projective manifold. The definition is such that the leaves of a foliation glue together in a nice way, giving a "covering tube" which is a sort of semi-global flow box. This is, in some sense, the topological part of the theory. In the second lecture, we prove some convexity property of this covering tube. As a corollary we obtain that, when there are hyperbolic leaves, the leafwise Poincare' metric has some remarkable positivity property. In the third lecture, we study foliations all of whose leaves are parabolic. Using a suitable extension theorem for certain meromorphic maps, we show how to generalise the above positivity property to this degenerate class of foliations.

### 2006/05/22

15:00-16:30 Room #470 (Graduate School of Math. Sci. Bldg.)

Laminations with Singularities by Riemann Surfaces II

**Nessim Sibony**(Paris Sud)Laminations with Singularities by Riemann Surfaces II

### 2006/05/15

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

Laminations with Singularities by Riemann Surfaces I (Part II on May 22)

**Nessim Sibony**(Paris Sud)Laminations with Singularities by Riemann Surfaces I (Part II on May 22)

[ Abstract ]

The basic example of a lamination, possibly with singularites, by Riemann surfaces, is the closure of a leaf of a holomorphic foliation in the complex projective plane.There are also many examples arising from the theory of iteration of a holomorphic map. The goal is to introduce tools in order to understand the globalproperties of leaves of a holomorphic lamination, mostly in compact Kaehler manifolds. We will develop the following topics.

-Poincare metric on a hyperbolic lamination.

-Positive cycles and positive harmonic currents directed by a lamination.

-Ahlfors construction of positive harmonic currents.

-Cohomological and geometrical intersection of positive harmonic currents.

The basic example of a lamination, possibly with singularites, by Riemann surfaces, is the closure of a leaf of a holomorphic foliation in the complex projective plane.There are also many examples arising from the theory of iteration of a holomorphic map. The goal is to introduce tools in order to understand the globalproperties of leaves of a holomorphic lamination, mostly in compact Kaehler manifolds. We will develop the following topics.

-Poincare metric on a hyperbolic lamination.

-Positive cycles and positive harmonic currents directed by a lamination.

-Ahlfors construction of positive harmonic currents.

-Cohomological and geometrical intersection of positive harmonic currents.

### 2006/05/08

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

Real-analytic Levi-flats in complex tori

**大沢 健夫**(名古屋大学)Real-analytic Levi-flats in complex tori

### 2006/04/24

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

Monge-Ampére mass at the boundary on some domains with corner

**Jonas Wiklund**(名古屋大学, JSPS fellow)Monge-Ampére mass at the boundary on some domains with corner

[ Abstract ]

The Monge-Ampére operator is a highly non-linear operator that assigns a positive measure to every plurisubharmonic function and the null-measure to every maximal plurisubharmonic measure, whenever it is well defined. We discuss the sweeping out of this measure to the boundary for functions that essentially vanish on the boundary, and show two examples that this boundary measure vanish outside the distinguished boundary. Namely for analytic polyhedrons and for the cross product of two hyperconvex domains. Some related open problems are also mentioned.

The Monge-Ampére operator is a highly non-linear operator that assigns a positive measure to every plurisubharmonic function and the null-measure to every maximal plurisubharmonic measure, whenever it is well defined. We discuss the sweeping out of this measure to the boundary for functions that essentially vanish on the boundary, and show two examples that this boundary measure vanish outside the distinguished boundary. Namely for analytic polyhedrons and for the cross product of two hyperconvex domains. Some related open problems are also mentioned.

### 2006/04/17

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

Dirichlet-to-Neumann map for Poincaré-Einstein metrics

**C. Robin Graham**(University of Washington)Dirichlet-to-Neumann map for Poincaré-Einstein metrics

[ Abstract ]

This talk will describe an analogue of a Dirichlet to Neumann map for Poincaré-Einstein metrics, also known as asymptotically hyperbolic Einstein metrics. An explicit identification of the linearization of the map at the sphere will be given for even interior dimensions, together with applications to the structure of the map near the sphere and to the positive frequency conjecture of LeBrun which was resolved by Biquard.

This talk will describe an analogue of a Dirichlet to Neumann map for Poincaré-Einstein metrics, also known as asymptotically hyperbolic Einstein metrics. An explicit identification of the linearization of the map at the sphere will be given for even interior dimensions, together with applications to the structure of the map near the sphere and to the positive frequency conjecture of LeBrun which was resolved by Biquard.

### 2006/01/30

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

Compact non-kaehler threefolds associated to hyperbolic 3-manifolds

**藤木 明**(大阪大学)Compact non-kaehler threefolds associated to hyperbolic 3-manifolds

### 2006/01/23

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

Stochastic processes and Besov spaces on local field

**金子 宏**(東京理科大)Stochastic processes and Besov spaces on local field

### 2005/12/05

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

Positive cones of hyper-Keahler manifold

**Sebastien Boucksom**(ParisVII / Univ. of Tokyo)Positive cones of hyper-Keahler manifold

### 2005/11/28

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

Schroeder equation and Abel equation

**上田哲生**(京都大学)Schroeder equation and Abel equation

### 2005/11/21

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

On CR-invariant differential operators

**Andreas Cap**(Univ. of Vienna)On CR-invariant differential operators

[ Abstract ]

My talk will be devoted to questions about differential operators which are intrinsic to non--degenerate CR structures of hypersurface type. Restricting to the subclass of spherical CR structures, this question admits an equivalent formulation in terms of representation theory, which leads to several surprising consequences.

Guided by the ideas from representation theory and using the canonical Cartan connection which is available in this situation, one obtains a construction for a large class of such operators, which continues to work for non--spherical structures, and even for a class of almost CR structures. In the end of the talk I will discuss joint work with V. Soucek which shows that in the integrable case many of the operators obtained in this way form complexes.

My talk will be devoted to questions about differential operators which are intrinsic to non--degenerate CR structures of hypersurface type. Restricting to the subclass of spherical CR structures, this question admits an equivalent formulation in terms of representation theory, which leads to several surprising consequences.

Guided by the ideas from representation theory and using the canonical Cartan connection which is available in this situation, one obtains a construction for a large class of such operators, which continues to work for non--spherical structures, and even for a class of almost CR structures. In the end of the talk I will discuss joint work with V. Soucek which shows that in the integrable case many of the operators obtained in this way form complexes.