## Seminar on Geometric Complex Analysis

Seminar information archive ～10/10｜Next seminar｜Future seminars 10/11～

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**

### 2011/11/28

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

An ampleness criterion with the extendability of singular positive metrics (JAPANESE)

**Shin-ichi Matsumura**(University of Tokyo)An ampleness criterion with the extendability of singular positive metrics (JAPANESE)

[ Abstract ]

Coman, Guedj and Zeriahi proved that, for an ample line bundle $L$ on a projective manifold $X$, any singular positive metric on the line bundle $L|_{V}$ along a subvariety $V \subset X$ can be extended to a global singular positive metric of $L$. In this talk, we prove that the extendability of singular positive metrics on a line bundle along a subvariety implies the ampleness of the line bundle.

Coman, Guedj and Zeriahi proved that, for an ample line bundle $L$ on a projective manifold $X$, any singular positive metric on the line bundle $L|_{V}$ along a subvariety $V \subset X$ can be extended to a global singular positive metric of $L$. In this talk, we prove that the extendability of singular positive metrics on a line bundle along a subvariety implies the ampleness of the line bundle.

### 2011/11/21

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

Techniques of computations of Dolbeault cohomology of solvmanifolds (JAPANESE)

**Hisashi Kasuya**(University of Tokyo)Techniques of computations of Dolbeault cohomology of solvmanifolds (JAPANESE)

### 2011/11/16

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Disc functionals and Siciak-Zaharyuta extremal functions on singular varieties (ENGLISH)

**Franc Forstneric**(University of Ljubljana)Disc functionals and Siciak-Zaharyuta extremal functions on singular varieties (ENGLISH)

[ Abstract ]

A disc functional on a complex space, $X$, is a function P that assign a real number $P(f)$ (possibly minus infinity) to every analytic disc $f$ in $X$. An examples is the Poisson functional $P_u$ of an upper semicontinuous function $u$ on $X$: in that case $P_u(f)$ is the average of u over the boundary curve of the disc $f$. Other natural examples include the Lelong and the Riesz functionals. The envelope of a disc functional $P$ is a function on $X$ associating to every point $x$ of $X$ the infimum of the values $P(f)$ over all analytic discs $f$ in $X$ satisfying $f(0)=x$. The main point of interest is that the envelopes of many natural disc functionals are plurisubharmonic functions solving certain extremal problems. In the classical case when $X=\mathbf{C}^n$ this was first discovered by E. Poletsky in the early 1990's. In this talk I will discuss recent results on plurisubharmonicity of envelopes of all the classical disc functional mentioned above on locally irreducible complex spaces. In the second part of the talk I will give formulas expressing the classical Siciak-Zaharyuta maximal function of an open set in an affine algebraic variety as the envelope of certain disc functionals. We establish plurisubharmonicity of envelopes of certain classical disc functionals on locally irreducible complex spaces, thereby generalizing the corresponding results for complex manifolds. We also find new formulae expressing the Siciak-Zaharyuta extremal function of an open set in a locally irreducible affine algebraic variety as the envelope of certain disc functionals, similarly to what has been done for open sets in $\mathbf{C}^n$ by Lempert and by Larusson and Sigurdsson.

A disc functional on a complex space, $X$, is a function P that assign a real number $P(f)$ (possibly minus infinity) to every analytic disc $f$ in $X$. An examples is the Poisson functional $P_u$ of an upper semicontinuous function $u$ on $X$: in that case $P_u(f)$ is the average of u over the boundary curve of the disc $f$. Other natural examples include the Lelong and the Riesz functionals. The envelope of a disc functional $P$ is a function on $X$ associating to every point $x$ of $X$ the infimum of the values $P(f)$ over all analytic discs $f$ in $X$ satisfying $f(0)=x$. The main point of interest is that the envelopes of many natural disc functionals are plurisubharmonic functions solving certain extremal problems. In the classical case when $X=\mathbf{C}^n$ this was first discovered by E. Poletsky in the early 1990's. In this talk I will discuss recent results on plurisubharmonicity of envelopes of all the classical disc functional mentioned above on locally irreducible complex spaces. In the second part of the talk I will give formulas expressing the classical Siciak-Zaharyuta maximal function of an open set in an affine algebraic variety as the envelope of certain disc functionals. We establish plurisubharmonicity of envelopes of certain classical disc functionals on locally irreducible complex spaces, thereby generalizing the corresponding results for complex manifolds. We also find new formulae expressing the Siciak-Zaharyuta extremal function of an open set in a locally irreducible affine algebraic variety as the envelope of certain disc functionals, similarly to what has been done for open sets in $\mathbf{C}^n$ by Lempert and by Larusson and Sigurdsson.

### 2011/11/07

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

Oka's extra-zero problem and related topics (JAPANESE)

**Junjiro Nocuchi**(University of Tokyo)Oka's extra-zero problem and related topics (JAPANESE)

[ Abstract ]

The main part of this talk is a joint work with my colleagues, M. Abe and S. Hamano. After the solution of Cousin II problem by K. Oka III in 1939, he thought an extra-zero problem in 1945 (his posthumous paper) asking if it is possible to solve an arbitrarily given Cousin II problem adding some extra-zeros whose support is disjoint from the given one. Some special case was affirmatively confirmed in dimension two and a counter-example in dimension three or more was obtained. We will give a complete solution of this problem with examples and to discuss some new questions. An example on a toric variety of which idea is based on K. Stein's paper in 1941 has some special interest and will be discussed. I would like also to discuss some analytic intersections form the viewpoint of Nevanlinna theory.

The main part of this talk is a joint work with my colleagues, M. Abe and S. Hamano. After the solution of Cousin II problem by K. Oka III in 1939, he thought an extra-zero problem in 1945 (his posthumous paper) asking if it is possible to solve an arbitrarily given Cousin II problem adding some extra-zeros whose support is disjoint from the given one. Some special case was affirmatively confirmed in dimension two and a counter-example in dimension three or more was obtained. We will give a complete solution of this problem with examples and to discuss some new questions. An example on a toric variety of which idea is based on K. Stein's paper in 1941 has some special interest and will be discussed. I would like also to discuss some analytic intersections form the viewpoint of Nevanlinna theory.

### 2011/10/31

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

Classification of Moishezon twistor spaces on 4CP^2 (JAPANESE)

**Nobuhiro Honda**(Tohoku Univeristy)Classification of Moishezon twistor spaces on 4CP^2 (JAPANESE)

### 2011/10/17

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

Compact locally homogeneous Kähler manifolds $\Gamma\backslash G/K$ (JAPANESE)

**Yoshinobu Kamishima**(Tokyo Metropolitan University)Compact locally homogeneous Kähler manifolds $\Gamma\backslash G/K$ (JAPANESE)

### 2011/07/11

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

Kobayashi hyperbolic imbeddings into toric varieties (JAPANESE)

**Yusaku Chiba**(University of Tokyo)Kobayashi hyperbolic imbeddings into toric varieties (JAPANESE)

### 2011/07/04

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

Toward a Hirzebruch-Riemann-Roch formula in CR geometry (ENGLISH)

**Raphael Ponge**(University of Tokyo)Toward a Hirzebruch-Riemann-Roch formula in CR geometry (ENGLISH)

### 2011/06/27

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

Vanishing cycles for the entire functions of type $A_{1/2\infty}$ and $D_{1/2\infty}$ (JAPANESE)

**Kyoji Saito**(IPMU, University of Tokyo)Vanishing cycles for the entire functions of type $A_{1/2\infty}$ and $D_{1/2\infty}$ (JAPANESE)

[ Abstract ]

We introduce two elementary transcendental functions $f_{A_{1/2\infty}}$ and $f_{D_{1/2\infty}}$ of two variables. They have countably infinitely many critical points. Then, the vanishing cycles associated with the critical points form Dynkin diagrams of type $A_{1/2\infty}$ and $D_{1/2\infty}$. We calculate the spectral decomposition of the monodromy transformation by embedding the lattice of vanishing cycles into a Hilbert space. All these stories are connected with a new understanding of KP and KdV integral hierarchy. But the relationship is not yet clear.

We introduce two elementary transcendental functions $f_{A_{1/2\infty}}$ and $f_{D_{1/2\infty}}$ of two variables. They have countably infinitely many critical points. Then, the vanishing cycles associated with the critical points form Dynkin diagrams of type $A_{1/2\infty}$ and $D_{1/2\infty}$. We calculate the spectral decomposition of the monodromy transformation by embedding the lattice of vanishing cycles into a Hilbert space. All these stories are connected with a new understanding of KP and KdV integral hierarchy. But the relationship is not yet clear.

### 2011/06/20

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

Domains which satisfy the Oka-Grauert principle in a Stein space (JAPANESE)

**Makoto Abe**(Hiroshima University)Domains which satisfy the Oka-Grauert principle in a Stein space (JAPANESE)

### 2011/06/13

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

On the complement of effective divisors with semipositive normal bundle (JAPANESE)

**Takeo Ohsawa**(Nagoya Univeristy)On the complement of effective divisors with semipositive normal bundle (JAPANESE)

### 2011/06/06

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

An invariant surface of a fixed indeterminate point for rational mappings (JAPANESE)

**Tomoko Shinohara**(Tokyo Metropolitan College of Industrial Technology)An invariant surface of a fixed indeterminate point for rational mappings (JAPANESE)

### 2011/05/30

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

On the Forelli-Rudin construction and explicit formulas of the Bergman kernels (JAPANESE)

**Atusi Yamamori**(Meiji University)On the Forelli-Rudin construction and explicit formulas of the Bergman kernels (JAPANESE)

### 2011/05/16

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

Linearity of order isomorphisms of regular convex cones (JAPANESE)

**Chifune Kai**(Kanazawa Univeristy)Linearity of order isomorphisms of regular convex cones (JAPANESE)

### 2011/05/09

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

Order of meromorphic maps and rationality of the image space (JAPANESE)

**Junjiro Noguchi**(University of Tokyo)Order of meromorphic maps and rationality of the image space (JAPANESE)

### 2011/04/25

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

On the classification of CR mappings between generalized pseudoellipsoids (JAPANESE)

**Atsushi Hayashimoto**(Nagano National College of Technology)On the classification of CR mappings between generalized pseudoellipsoids (JAPANESE)

### 2011/04/18

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

Asymptotic cohomology vanishing and a converse of the Andreotti-Grauert theorem on surface (JAPANESE)

**Shinichi Matsumura**(University of Tokyo)Asymptotic cohomology vanishing and a converse of the Andreotti-Grauert theorem on surface (JAPANESE)

### 2011/04/11

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

Algebraic analysis of resolvents and an exact algorithm for computing Spectral decomposition matrices (JAPANESE)

**Shinichi Tajima**(University of Tsukuba)Algebraic analysis of resolvents and an exact algorithm for computing Spectral decomposition matrices (JAPANESE)

### 2011/01/31

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

Varieties with ample cotangent bundle and hyperbolicity (ENGLISH)

**Damian Brotbek**(Rennes Univ.)Varieties with ample cotangent bundle and hyperbolicity (ENGLISH)

[ Abstract ]

Varieties with ample cotangent bundle satisfy many interesting properties and are supposed to be abundant, however relatively few concrete examples are known. In this talk we will construct such examples as complete intersection surfaces in projective space, and explain how this problem is related to the study of hyperbolicity properties for hypersurfaces.

Varieties with ample cotangent bundle satisfy many interesting properties and are supposed to be abundant, however relatively few concrete examples are known. In this talk we will construct such examples as complete intersection surfaces in projective space, and explain how this problem is related to the study of hyperbolicity properties for hypersurfaces.

### 2011/01/24

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

Toward a complex analytic 3-dimensional Kleinian group theory (JAPANESE)

**Masahide Kato**(Sophia Univ.)Toward a complex analytic 3-dimensional Kleinian group theory (JAPANESE)

### 2011/01/17

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

Asymptotics of the Bergman function for semipositive holomorphic line bundles (JAPANESE)

**Toshihiro Nose**(Kyushu Univ.)Asymptotics of the Bergman function for semipositive holomorphic line bundles (JAPANESE)

[ Abstract ]

In this talk, an asymptotic expansion of the Bergman function at a degenerate point is given for high powers of semipositive holomorphic line bundles on compact Kahler manifolds, whose Hermitian metric has some kind of quasihomogeneous properties. In the sence of pointwise asymptotics, This expansion is a generalization of the expansion of Tian-Zelditch-Catlin-Lu in the positive line bundle case.

In this talk, an asymptotic expansion of the Bergman function at a degenerate point is given for high powers of semipositive holomorphic line bundles on compact Kahler manifolds, whose Hermitian metric has some kind of quasihomogeneous properties. In the sence of pointwise asymptotics, This expansion is a generalization of the expansion of Tian-Zelditch-Catlin-Lu in the positive line bundle case.

### 2010/12/20

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

Pseudoconvex domains in Hopf surfaces (JAPANESE)

**Hiroshi Yamaguchi**(Shiga Univ*)Pseudoconvex domains in Hopf surfaces (JAPANESE)

### 2010/12/13

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

An equality estimate for the second main theorem (JAPANESE)

**Katsutoshi Yamanoi**(Tokyo Institute of Technology)An equality estimate for the second main theorem (JAPANESE)

### 2010/12/06

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

Chow semistability of polarized toric manifolds (JAPANESE)

**Hajime Ono**(Tokyo Univ of Science)Chow semistability of polarized toric manifolds (JAPANESE)

### 2010/11/15

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

Excess intersections and residues in improper dimension (JAPANESE)

**Tatsuo Suwa**(Hokkaido Univ*)Excess intersections and residues in improper dimension (JAPANESE)

[ Abstract ]

This talk concerns localization of characteristic classes and associated residues, in the context of intersection theory and residue theory of singular holomorphic foliations. The localization comes from the vanishing of certain characteristic forms, usually caused by the existence of some geometric object, away from the "singular set" of the object. This gives rise to residues in the homology of the singular set and residue theorems relating local and global invariants. In the generic situation, i.e., if the dimension of the singular set is "proper", we have a reasonable understanding of the residues. We indicate how to cope with the problem when the dimension is "excessive" (partly a joint work with F. Bracci).

This talk concerns localization of characteristic classes and associated residues, in the context of intersection theory and residue theory of singular holomorphic foliations. The localization comes from the vanishing of certain characteristic forms, usually caused by the existence of some geometric object, away from the "singular set" of the object. This gives rise to residues in the homology of the singular set and residue theorems relating local and global invariants. In the generic situation, i.e., if the dimension of the singular set is "proper", we have a reasonable understanding of the residues. We indicate how to cope with the problem when the dimension is "excessive" (partly a joint work with F. Bracci).