## Seminar on Geometric Complex Analysis

Seminar information archive ～05/28｜Next seminar｜Future seminars 05/29～

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**

### 2020/12/14

10:30-12:00 Online

On Levi flat hypersurfaces with transversely affine foliation

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

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

[ Abstract ]

In this talk, we discuss the classification problem of Levi flat hypersurfaces in complex surfaces by restricting ourselves to the case that the Levi foliation is transversely affine. After presenting known examples, we give a proof for the non-existence of real analytic Levi flat hypersurface whose complement is 1-convex and Levi foliation is transversely affine in a compact Kähler surface. This is a joint work with Severine Biard (arXiv:2011.06379).

[ Reference URL ]In this talk, we discuss the classification problem of Levi flat hypersurfaces in complex surfaces by restricting ourselves to the case that the Levi foliation is transversely affine. After presenting known examples, we give a proof for the non-existence of real analytic Levi flat hypersurface whose complement is 1-convex and Levi foliation is transversely affine in a compact Kähler surface. This is a joint work with Severine Biard (arXiv:2011.06379).

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

### 2020/11/30

10:30-12:00 Online

On asymptotic base loci of relative anti-canonical divisors

[ Reference URL ]

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

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

[ Reference URL ]

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

### 2020/11/09

10:30-12:00 Online

Meromorphic continuation of local zeta functions and nonpolar singularities

[ Reference URL ]

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

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

[ Reference URL ]

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

### 2020/10/26

10:30-12:00 Online

Spectral convergence in geometric quantization

[ Reference URL ]

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

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

[ Reference URL ]

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

### 2020/10/19

10:30-12:00 Online

On projective manifolds with pseudo-effective tangent bundle

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

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

[ Abstract ]

In this talk, I would like to discuss projective manifolds whose tangent bundle is pseudo-effective or admits a positively curved singular metric. I will explain a structure theorem for such manifolds and the classification in the two-dimensional case, comparing our theory with classical results for nef tangent bundle or non-negative bisectional curvature. Related open problems will be discussed if time permits.

This is joint work with Genki Hosono (Tohoku University) and Masataka Iwai (Osaka City University).

[ Reference URL ]In this talk, I would like to discuss projective manifolds whose tangent bundle is pseudo-effective or admits a positively curved singular metric. I will explain a structure theorem for such manifolds and the classification in the two-dimensional case, comparing our theory with classical results for nef tangent bundle or non-negative bisectional curvature. Related open problems will be discussed if time permits.

This is joint work with Genki Hosono (Tohoku University) and Masataka Iwai (Osaka City University).

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

### 2020/10/12

10:30-12:00 Online

Two topics on psedoconvex domains (Japanese)

[ Reference URL ]

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

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

[ Reference URL ]

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

### 2020/07/13

10:30-12:00 Online

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

[ Reference URL ]

https://forms.gle/vSFPoVR6ugrkTGhX7

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

[ Reference URL ]

https://forms.gle/vSFPoVR6ugrkTGhX7

### 2020/07/06

10:30-12:00 Online

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

https://forms.gle/vSFPoVR6ugrkTGhX7

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

[ Abstract ]

We propose a general definition of Nakano semi-positivity of singular Hermitian metrics on holomorphic vector bundles. By using this positivity notion, we establish $L^2$-estimates for holomorphic vector bundles with Nakano positive singular Hermitian metrics. We also show vanishing theorems, which generalize both Nakano type and Demailly-Nadel type vanishing theorems.

[ Reference URL ]We propose a general definition of Nakano semi-positivity of singular Hermitian metrics on holomorphic vector bundles. By using this positivity notion, we establish $L^2$-estimates for holomorphic vector bundles with Nakano positive singular Hermitian metrics. We also show vanishing theorems, which generalize both Nakano type and Demailly-Nadel type vanishing theorems.

https://forms.gle/vSFPoVR6ugrkTGhX7

### 2020/06/29

10:30-12:00 Online

Oka properties of complements of holomorphically convex sets

https://forms.gle/vSFPoVR6ugrkTGhX7

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

[ Abstract ]

A complex manifold is called an Oka manifold if the Oka principle for maps from Stein spaces holds. In this talk, we consider the question of when a holomorphically convex set in an Oka manifold has an Oka complement. Our main theorem states that the complement of a compact holomorphically convex set in a Stein manifold with the density property is an Oka manifold.

This gives a positive answer to the well-known long-standing problem in Oka theory whether the complement of a compact polynomially convex set in $\mathbb{C}^{n}$ $(n>1)$ is Oka. The relative version of the main theorem can also be proved.

As an application, we show that the complement $\mathbb{C}^{n}\setminus\mathbb{R}^{k}$ of a totally real affine subspace is Oka if $n>1$ and $(n,k)\neq(2,1),(2,2),(3,3)$.

[ Reference URL ]A complex manifold is called an Oka manifold if the Oka principle for maps from Stein spaces holds. In this talk, we consider the question of when a holomorphically convex set in an Oka manifold has an Oka complement. Our main theorem states that the complement of a compact holomorphically convex set in a Stein manifold with the density property is an Oka manifold.

This gives a positive answer to the well-known long-standing problem in Oka theory whether the complement of a compact polynomially convex set in $\mathbb{C}^{n}$ $(n>1)$ is Oka. The relative version of the main theorem can also be proved.

As an application, we show that the complement $\mathbb{C}^{n}\setminus\mathbb{R}^{k}$ of a totally real affine subspace is Oka if $n>1$ and $(n,k)\neq(2,1),(2,2),(3,3)$.

https://forms.gle/vSFPoVR6ugrkTGhX7

### 2020/06/08

10:30-12:00 Online

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

https://forms.gle/vSFPoVR6ugrkTGhX7

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

[ Abstract ]

The Kobayashi-Hitchin correspondence, proved by Donaldson and Uhlenbeck-Yau by using the nonlinear PDE theory, states that the existence of Hermitian-Einstein metrics on a holomorphic vector bundle is equivalent to an algebro-geometric stability condition. We present some results that exhibit an explicit link between differential and algebraic geometry in the above correspondence, from a variational point of view. The key to such results is an object called the Quot-scheme limit of Fubini-Study metrics, which is used to evaluate certain algebraic 1-parameter subgroups of Hermitian metrics by using the theory of Quot-schemes in algebraic geometry. This method also works for the proof of the correspondence between the balanced metrics and the Gieseker stability, as originally proved by X.W. Wang. Joint work with Julien Keller.

[ Reference URL ]The Kobayashi-Hitchin correspondence, proved by Donaldson and Uhlenbeck-Yau by using the nonlinear PDE theory, states that the existence of Hermitian-Einstein metrics on a holomorphic vector bundle is equivalent to an algebro-geometric stability condition. We present some results that exhibit an explicit link between differential and algebraic geometry in the above correspondence, from a variational point of view. The key to such results is an object called the Quot-scheme limit of Fubini-Study metrics, which is used to evaluate certain algebraic 1-parameter subgroups of Hermitian metrics by using the theory of Quot-schemes in algebraic geometry. This method also works for the proof of the correspondence between the balanced metrics and the Gieseker stability, as originally proved by X.W. Wang. Joint work with Julien Keller.

https://forms.gle/vSFPoVR6ugrkTGhX7

### 2020/05/25

10:30-12:00 Online

Characteristic forms of Cheng-Yau metric and CR invariants

[ Reference URL ]

https://forms.gle/vSFPoVR6ugrkTGhX7

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

[ Reference URL ]

https://forms.gle/vSFPoVR6ugrkTGhX7

### 2020/05/18

10:30-12:00 Online

Higgs bundles and flat connections over compact Sasakian manifolds

https://forms.gle/vSFPoVR6ugrkTGhX7

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

[ Abstract ]

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

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

https://forms.gle/vSFPoVR6ugrkTGhX7

### 2020/02/17

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

Precompactness of the moduli space of pseudo-normed graded algebras

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

[ Abstract ]

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

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

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

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

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

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

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

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

### 2020/01/27

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

Canonical measure and it’s applications

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

[ Abstract ]

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

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

### 2020/01/20

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

Diederich-Fornaess and Steinness indices for abstract CR manifolds

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

[ Abstract ]

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

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

### 2019/12/16

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

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

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

[ Abstract ]

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

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

### 2019/12/11

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

Einstein-Weyl structures (English)

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

[ Abstract ]

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

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

### 2019/12/09

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

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

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

[ Abstract ]

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

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

### 2019/12/02

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

Toward classification of Moishezon twistor spaces

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

[ Abstract ]

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

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

### 2019/11/18

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

j-invariant and Borcherds Phi-function (Japanese)

**Ken-ichi Yoshikawa**(Kyoto Univ.)j-invariant and Borcherds Phi-function (Japanese)

[ Abstract ]

The j-invariant is a modular function on the complex upper half plane inducing an isomorphism between the moduli space of elliptic curves and the complex plane. Besides the j-invariant itself, the difference of j-invariants has also attracted some mathematicians. In this talk, I will explain a factorization of the difference of j-invariants in terms of Borcherds Phi-function, the automorphic form on the period domain for Enriques surfaces characterizing the discriminant divisor. This is a joint work with Shu Kawaguchi and Shigeru Mukai.

The j-invariant is a modular function on the complex upper half plane inducing an isomorphism between the moduli space of elliptic curves and the complex plane. Besides the j-invariant itself, the difference of j-invariants has also attracted some mathematicians. In this talk, I will explain a factorization of the difference of j-invariants in terms of Borcherds Phi-function, the automorphic form on the period domain for Enriques surfaces characterizing the discriminant divisor. This is a joint work with Shu Kawaguchi and Shigeru Mukai.

### 2019/10/28

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

On Kiyoshi Oka's unpublished papers 1943 (Japanese)

**Junjiro Noguchi**(Univ. of Tokyo)On Kiyoshi Oka's unpublished papers 1943 (Japanese)

### 2019/10/21

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

Canonical almost complex structures on ACH Einstein manifolds

**Yoshihiko Matsumoto**(Osaka Univ.)Canonical almost complex structures on ACH Einstein manifolds

[ Abstract ]

Einstein ACH (asymptotically complex hyperbolic) manifolds are seen as a device that establishes a correspondence between CR geometry on the boundary and Riemannian geometry in “the bulk.” This talk concerns an idea of enriching the geometric structure of the bulk by adding some almost complex structure compatible with the metric. I will introduce an energy functional of almost complex structures and discuss an existence result of critical points when the given ACH Einstein metric is a small perturbation of the Cheng-Yau complete K?hler-Einstein metric on a bounded strictly pseudoconvex domain. The renormalized Chern-Gauss-Bonnet formula is also planned to be discussed.

Einstein ACH (asymptotically complex hyperbolic) manifolds are seen as a device that establishes a correspondence between CR geometry on the boundary and Riemannian geometry in “the bulk.” This talk concerns an idea of enriching the geometric structure of the bulk by adding some almost complex structure compatible with the metric. I will introduce an energy functional of almost complex structures and discuss an existence result of critical points when the given ACH Einstein metric is a small perturbation of the Cheng-Yau complete K?hler-Einstein metric on a bounded strictly pseudoconvex domain. The renormalized Chern-Gauss-Bonnet formula is also planned to be discussed.

### 2019/10/07

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

Cohomology of vector bundles and non-pluriharmonic loci (Japanese)

**Yusaku Chiba**(Ochanomizu Univ.)Cohomology of vector bundles and non-pluriharmonic loci (Japanese)

[ Abstract ]

We study cohomology groups of vector bundles on neighborhoods of a non-pluriharmonic locus in Stein manifolds and in projective manifolds. By using our results, we show variants of the Lefschetz hyperplane theorem. We especially study the examples of non-pluriharmonic loci in smooth toric varieties. I would like to explain the relation of non-pluriharmonic loci and polytopes.

We study cohomology groups of vector bundles on neighborhoods of a non-pluriharmonic locus in Stein manifolds and in projective manifolds. By using our results, we show variants of the Lefschetz hyperplane theorem. We especially study the examples of non-pluriharmonic loci in smooth toric varieties. I would like to explain the relation of non-pluriharmonic loci and polytopes.

### 2019/09/30

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

Rigidity of the directional moduli on pseudoconvex domains fibered by open Riemann surfaces

**Sachiko Hamano**(Osaka City Univ.)Rigidity of the directional moduli on pseudoconvex domains fibered by open Riemann surfaces

[ Abstract ]

G. Schmieder-M. Shiba observed conformal embeddings of a fixed open Riemann surface of positive finite genus into closed Riemann surfaces of the same genus, and they showed the range of each diagonal element of the period matrices. Now we shall consider a smooth deformation of open Riemann surfaces with a complex parameter. In this talk, we show the rigidity of directional moduli induced by elements of the period matrices on pseudoconvex domains fibered by open Riemann surfaces of the same topological type.

G. Schmieder-M. Shiba observed conformal embeddings of a fixed open Riemann surface of positive finite genus into closed Riemann surfaces of the same genus, and they showed the range of each diagonal element of the period matrices. Now we shall consider a smooth deformation of open Riemann surfaces with a complex parameter. In this talk, we show the rigidity of directional moduli induced by elements of the period matrices on pseudoconvex domains fibered by open Riemann surfaces of the same topological type.

### 2019/07/08

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

A Riemann-Roch theorem on a weighted infinite graph (Japanese)

**Hiroshi Kaneko**(Tokyo University of Science)A Riemann-Roch theorem on a weighted infinite graph (Japanese)

[ Abstract ]

A Riemann-Roch theorem on a connected finite graph was initiated by M. Baker and S. Norine, where connected graph with finite vertices was investigated and unit weight was given on each edge and vertex of the graph. Since a counterpart of the lowest exponents of the complex variable in the Laurent series was proposed as divisor for the Riemann-Roch theorem on graph, its relationships with tropical geometry were highlighted earlier than other complex analytical observations on graphs. On the other hand, M. Baker and F. Shokrieh revealed tight relationships between chip-firing games and potential theory on graphs, by characterizing reduced divisors on graphs as the solution to an energy minimization problem. The objective of this talk is to establish a Riemann-Roch theorem on an edge-weighted infinite graph. We introduce vertex weight assigned by the given weights of adjacent edges other than the units for expression of divisors and assume finiteness of total mass of graph. This is a joint work with A. Atsuji.

A Riemann-Roch theorem on a connected finite graph was initiated by M. Baker and S. Norine, where connected graph with finite vertices was investigated and unit weight was given on each edge and vertex of the graph. Since a counterpart of the lowest exponents of the complex variable in the Laurent series was proposed as divisor for the Riemann-Roch theorem on graph, its relationships with tropical geometry were highlighted earlier than other complex analytical observations on graphs. On the other hand, M. Baker and F. Shokrieh revealed tight relationships between chip-firing games and potential theory on graphs, by characterizing reduced divisors on graphs as the solution to an energy minimization problem. The objective of this talk is to establish a Riemann-Roch theorem on an edge-weighted infinite graph. We introduce vertex weight assigned by the given weights of adjacent edges other than the units for expression of divisors and assume finiteness of total mass of graph. This is a joint work with A. Atsuji.