## Seminar information archive

Seminar information archive ～02/01｜Today's seminar 02/02 | Future seminars 02/03～

#### Algebraic Geometry Seminar

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

MMP revisited, II (ENGLISH)

**Vladimir Lazić**(Imperial College London)MMP revisited, II (ENGLISH)

[ Abstract ]

I will talk about how finite generation of certain adjoint rings implies everything we currently know about the MMP. This is joint work with A. Corti.

I will talk about how finite generation of certain adjoint rings implies everything we currently know about the MMP. This is joint work with A. Corti.

### 2011/06/24

#### thesis presentations

13:15-14:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Spatial-temporal Modeling and Simulation of Transcription (JAPANESE)

**Yoshihiro OTA**(Graduate School of Mathematical Sciences University of Tokyo)Spatial-temporal Modeling and Simulation of Transcription (JAPANESE)

#### Classical Analysis

15:00-16:30 Room #128 (Graduate School of Math. Sci. Bldg.)

A Schwarz map of Appell's $F_2$ whose monodromy group is

related to the reflection group of type $D_4$ (JAPANESE)

**J. Sekiguchi**(Tokyo University of Agriculture and Technology)A Schwarz map of Appell's $F_2$ whose monodromy group is

related to the reflection group of type $D_4$ (JAPANESE)

[ Abstract ]

The system of differential equations for Appell's hypergeometric function $F_2(a,b,b',c,c';x,y)$ has four fundamental solutions.

Let $u_1,u_2,u_3,u_4$ be such solutions. If the monodromy group of the system is finite, the closure of the image of the Schwarz map $U(x,y)=(u_1(x,y),u_2(x,y),u_3(x,y),u_4(x,y))$

is a hypersurface $S$ of the 3-dimensional projective space ${\\bf P}^3$. Then $S$ is defined by $P(u_1,u_2,u_3,u_4)=0$ for a polynomial $P(t_1,t_2,t_3,t_4)$.

It is M. Kato (Univ. Ryukyus) who determined the parameter

$a,b,b',c,c'$ such that the monodromy group of the system for $F_2(a,b,b',c,c';x,y)$ is finite. It follows from his result that such a group is the semidirect product of an irreducible finite reflection group $G$ of rank four by an abelian group.

In this talk, we treat the system for $F_2(a,b,b',c,c';x,y)$ with

$(a,b,b',c,c')=(1/2,1/6,-1/6,1/3,2/3$. In this case, the monodromy group is the semidirect group of $G$ by $Z/3Z$, where $G$ is the reflection group of type $D_4$. The polynomial $P(t_1,t_2,t_3,t_4)$ in this case is of degree four. There are 16 ordinary singular points in the hypersurface $S$.

In the rest of my talk, I explain the background of the study.

The system of differential equations for Appell's hypergeometric function $F_2(a,b,b',c,c';x,y)$ has four fundamental solutions.

Let $u_1,u_2,u_3,u_4$ be such solutions. If the monodromy group of the system is finite, the closure of the image of the Schwarz map $U(x,y)=(u_1(x,y),u_2(x,y),u_3(x,y),u_4(x,y))$

is a hypersurface $S$ of the 3-dimensional projective space ${\\bf P}^3$. Then $S$ is defined by $P(u_1,u_2,u_3,u_4)=0$ for a polynomial $P(t_1,t_2,t_3,t_4)$.

It is M. Kato (Univ. Ryukyus) who determined the parameter

$a,b,b',c,c'$ such that the monodromy group of the system for $F_2(a,b,b',c,c';x,y)$ is finite. It follows from his result that such a group is the semidirect product of an irreducible finite reflection group $G$ of rank four by an abelian group.

In this talk, we treat the system for $F_2(a,b,b',c,c';x,y)$ with

$(a,b,b',c,c')=(1/2,1/6,-1/6,1/3,2/3$. In this case, the monodromy group is the semidirect group of $G$ by $Z/3Z$, where $G$ is the reflection group of type $D_4$. The polynomial $P(t_1,t_2,t_3,t_4)$ in this case is of degree four. There are 16 ordinary singular points in the hypersurface $S$.

In the rest of my talk, I explain the background of the study.

#### Colloquium

16:30-17:30 Room #123 (Graduate School of Math. Sci. Bldg.)

On logarithmic extension of p-adic differential equations (JAPANESE)

**Jun SHIHO**(Graduate School of Mathematical Sciences, University of Tokyo)On logarithmic extension of p-adic differential equations (JAPANESE)

### 2011/06/22

#### Seminar on Probability and Statistics

15:00-16:10 Room #002 (Graduate School of Math. Sci. Bldg.)

計算機代数を用いた統計的漸近論 (JAPANESE)

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2011/00.html

**KOBAYASHI, Kei**(The Institute of Statistical Mathematics)計算機代数を用いた統計的漸近論 (JAPANESE)

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2011/00.html

### 2011/06/21

#### Numerical Analysis Seminar

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Research on CFD (Computational Fluid Dynamics) and Its Application to Development of Spacecraft and Rockets

(JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

**Keiichi Kitamura**(Japan Aerospace Exploration Agency (JAXA))Research on CFD (Computational Fluid Dynamics) and Its Application to Development of Spacecraft and Rockets

(JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

### 2011/06/20

#### Seminar on Geometric Complex Analysis

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)

### 2011/06/16

#### Operator Algebra Seminars

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Introduction to rigidity theory of von Neumann algebras (JAPANESE)

**Yusuke Isono**(Univ. Tokyo)Introduction to rigidity theory of von Neumann algebras (JAPANESE)

### 2011/06/15

#### Number Theory Seminar

17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Product formula for $p$-adic epsilon factors (ENGLISH)

**Tomoyuki Abe**(IPMU)Product formula for $p$-adic epsilon factors (ENGLISH)

[ Abstract ]

I would like to talk about my recent work jointly with A. Marmora on a product formula for $p$-adic epsilon factors. In 80's Deligne conjectured that a constant appearing in the functional equation of $L$-function of $\\ell$-adic lisse sheaf can be written by means of local contributions, and proved some particular cases. This conjecture was proven later by Laumon, and was used in the Lafforgue's proof of the Langlands' program for functional filed case. In my talk, I would like to prove a $p$-adic analog of this product formula.

I would like to talk about my recent work jointly with A. Marmora on a product formula for $p$-adic epsilon factors. In 80's Deligne conjectured that a constant appearing in the functional equation of $L$-function of $\\ell$-adic lisse sheaf can be written by means of local contributions, and proved some particular cases. This conjecture was proven later by Laumon, and was used in the Lafforgue's proof of the Langlands' program for functional filed case. In my talk, I would like to prove a $p$-adic analog of this product formula.

### 2011/06/14

#### Tuesday Seminar on Topology

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Donaldson-Tian-Yau's Conjecture (JAPANESE)

**Toshiki Mabuchi**(Osaka University)Donaldson-Tian-Yau's Conjecture (JAPANESE)

[ Abstract ]

For polarized algebraic manifolds, the concept of K-stability

introduced by Tian and Donaldson is conjecturally strongly correlated

to the existence of constant scalar curvature metrics (or more

generally extremal K\\"ahler metrics) in the polarization class. This is

known as Donaldson-Tian-Yau's conjecture. Recently, a remarkable

progress has been made by many authors toward its solution. In this

talk, I'll discuss the topic mainly with emphasis on the existence

part of the conjecture.

For polarized algebraic manifolds, the concept of K-stability

introduced by Tian and Donaldson is conjecturally strongly correlated

to the existence of constant scalar curvature metrics (or more

generally extremal K\\"ahler metrics) in the polarization class. This is

known as Donaldson-Tian-Yau's conjecture. Recently, a remarkable

progress has been made by many authors toward its solution. In this

talk, I'll discuss the topic mainly with emphasis on the existence

part of the conjecture.

### 2011/06/13

#### Seminar on Geometric Complex Analysis

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)

#### Lectures

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Regularity of Solutions for a Class of Degenerate Equations (ENGLISH)

**CHEN Hua**(Wuhan University)Regularity of Solutions for a Class of Degenerate Equations (ENGLISH)

[ Abstract ]

In this talk, I would report some recent joint results on the Gevrey (or analytic) regularities of solutions for some degenerate partial differential equations, which including

(1) generalized Kolmogorov equations,

(2) Fokker-Planck equations,

(3) Landau equations and

(4) sub-elliptic Monge-Ampere equations.

In this talk, I would report some recent joint results on the Gevrey (or analytic) regularities of solutions for some degenerate partial differential equations, which including

(1) generalized Kolmogorov equations,

(2) Fokker-Planck equations,

(3) Landau equations and

(4) sub-elliptic Monge-Ampere equations.

### 2011/06/09

#### Operator Algebra Seminars

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

On the free Fisher information distance and the free logarithmic Sobolev inequality (JAPANESE)

**Hiroaki Yoshida**(Ochanomizu Univ.)On the free Fisher information distance and the free logarithmic Sobolev inequality (JAPANESE)

#### Applied Analysis

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

Spectral representations and scattering for Schr\\"odinger operators on star graphs (JAPANESE)

**Kiyoshi Mochizuki**(Tokyo Metropolitan University, Emeritus Professor)Spectral representations and scattering for Schr\\"odinger operators on star graphs (JAPANESE)

[ Abstract ]

We consider Schr\\"odinger operators defined on star graphs with Kirchhoff boundary conditions. Under suitable decay conditions on the potential, we construct a complete set of eigenfunctions to obtain spectral representations of the operator. The results are applied to give a time dependent formulation of the scattering theory. Also we use the spectral representation to determine an integral equation of Marchenko which is fundamental to enter into the inverse scattering problems.

We consider Schr\\"odinger operators defined on star graphs with Kirchhoff boundary conditions. Under suitable decay conditions on the potential, we construct a complete set of eigenfunctions to obtain spectral representations of the operator. The results are applied to give a time dependent formulation of the scattering theory. Also we use the spectral representation to determine an integral equation of Marchenko which is fundamental to enter into the inverse scattering problems.

### 2011/06/08

#### Number Theory Seminar

16:30-17:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Congruences of modular forms and the Iwasawa λ-invariants (JAPANESE)

**Yuichi Hirano**(University of Tokyo)Congruences of modular forms and the Iwasawa λ-invariants (JAPANESE)

### 2011/06/07

#### Algebraic Geometry Seminar

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Log canonical closure (ENGLISH)

**Chenyang Xu**(MIT)Log canonical closure (ENGLISH)

[ Abstract ]

(joint with Christopher Hacon) In this talk, we will address the problem on given a log canonical variety, how we compactify it. Our approach is via MMP. The result has a few applications. Especially I will explain the one on the moduli of stable schemes.

If time permits, I will also talk about how a similar approach can be applied to give a proof of the existence of log canonical flips and a conjecture due to Kollár on the geometry of log centers.

(joint with Christopher Hacon) In this talk, we will address the problem on given a log canonical variety, how we compactify it. Our approach is via MMP. The result has a few applications. Especially I will explain the one on the moduli of stable schemes.

If time permits, I will also talk about how a similar approach can be applied to give a proof of the existence of log canonical flips and a conjecture due to Kollár on the geometry of log centers.

#### Numerical Analysis Seminar

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Linear stability analyses of flow fields driven by propellers on

the water surface for water quality improvement

(JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

**Takashi Nakazawa**(Okayama University)Linear stability analyses of flow fields driven by propellers on

the water surface for water quality improvement

(JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

#### Lie Groups and Representation Theory

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Rigidity of group actions via invariant geometric structures

(JAPANESE)

**Masahiko Kanai**(the University of Tokyo)Rigidity of group actions via invariant geometric structures

(JAPANESE)

[ Abstract ]

It is a homomorphism into a FINITE dimensional Lie group that is concerned with in the classical RIGIDITY theorems such as those of Mostow and Margulis. In the meantime, differentiable GROUP ACTIONS for which we ask rigidity problems is a homomorphism into a diffeomorphism group, which is a typical example of INFINITE dimensional Lie groups. The purpose of the present talk is exhibiting several rigidity theorems for group actions in which I have been involved for years. Although quite a few fields of mathematics, such as ergodic theory, the theory of smooth dynamical systems, representation theory and so on, have made remarkable contributions to rigidity problems, I would rather emphasis geometric aspects: I would focus on those rigidity phenomenon for group actions that are observed by showing that the actions have INVARIANT GEOMETRIC STRUCTURES.

It is a homomorphism into a FINITE dimensional Lie group that is concerned with in the classical RIGIDITY theorems such as those of Mostow and Margulis. In the meantime, differentiable GROUP ACTIONS for which we ask rigidity problems is a homomorphism into a diffeomorphism group, which is a typical example of INFINITE dimensional Lie groups. The purpose of the present talk is exhibiting several rigidity theorems for group actions in which I have been involved for years. Although quite a few fields of mathematics, such as ergodic theory, the theory of smooth dynamical systems, representation theory and so on, have made remarkable contributions to rigidity problems, I would rather emphasis geometric aspects: I would focus on those rigidity phenomenon for group actions that are observed by showing that the actions have INVARIANT GEOMETRIC STRUCTURES.

#### Tuesday Seminar on Topology

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Rigidity of group actions via invariant geometric structures (JAPANESE)

**Masahiko Kanai**(The University of Tokyo)Rigidity of group actions via invariant geometric structures (JAPANESE)

[ Abstract ]

It is a homomorphism into a FINITE dimensional Lie group that is concerned with in the classical RIGIDITY theorems such as those of Mostow and Margulis. In the meantime, differentiable GROUP ACTIONS for which we ask rigidity problems is a homomorphism into a diffeomorphism group, which is a typical example of INFINITE dimensional Lie groups. The purpose of the present talk is exhibiting several rigidity theorems for group actions in which I have been involved for years. Although quite a few fields of mathematics, such as ergodic theory, the theory of smooth dynamical systems, representation theory and so on, have made remarkable contributions to rigidity problems, I would rather emphasis geometric aspects: I would focus on those rigidity phenomenon for group actions that are observed by showing that the actions have INVARIANT GEOMETRIC STRUCTURES.

It is a homomorphism into a FINITE dimensional Lie group that is concerned with in the classical RIGIDITY theorems such as those of Mostow and Margulis. In the meantime, differentiable GROUP ACTIONS for which we ask rigidity problems is a homomorphism into a diffeomorphism group, which is a typical example of INFINITE dimensional Lie groups. The purpose of the present talk is exhibiting several rigidity theorems for group actions in which I have been involved for years. Although quite a few fields of mathematics, such as ergodic theory, the theory of smooth dynamical systems, representation theory and so on, have made remarkable contributions to rigidity problems, I would rather emphasis geometric aspects: I would focus on those rigidity phenomenon for group actions that are observed by showing that the actions have INVARIANT GEOMETRIC STRUCTURES.

### 2011/06/06

#### Seminar on Geometric Complex Analysis

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)

#### Algebraic Geometry Seminar

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Multiplier ideals via Mather discrepancies (JAPANESE)

**Shihoko Ishii**(University of Tokyo)Multiplier ideals via Mather discrepancies (JAPANESE)

[ Abstract ]

For an arbitrary variety we define a multiplier ideal by using Mather discrepancy.

This ideal coincides with the usual multiplier ideal if the variety is normal and complete intersection.

In the talk I will show a local vanishing theorem for this ideal and as corollaries we obtain restriction theorem, subadditivity theorem, Skoda type theorem, and Briancon-Skoda type theorem.

For an arbitrary variety we define a multiplier ideal by using Mather discrepancy.

This ideal coincides with the usual multiplier ideal if the variety is normal and complete intersection.

In the talk I will show a local vanishing theorem for this ideal and as corollaries we obtain restriction theorem, subadditivity theorem, Skoda type theorem, and Briancon-Skoda type theorem.

### 2011/06/02

#### Infinite Analysis Seminar Tokyo

16:30-17:30 Room #056 (Graduate School of Math. Sci. Bldg.)

On the module category of $¥overline{U}_q(¥mathfrak{sl}_2)$ (JAPANESE)

**Yoshihisa Saito**(Graduate School of Mathematical Sciences, Univ. of Tokyo)On the module category of $¥overline{U}_q(¥mathfrak{sl}_2)$ (JAPANESE)

[ Abstract ]

In the representation theory of quantum groups at roots of unity, it is

often assumed that the parameter $q$ is a primitive $n$-th root of unity

where $n$ is a odd prime number. However, there has recently been

increasing interest in the the cases where $n$ is an even integer ---

for example, in the study of logarithmic conformal field theories, or in

knot invariants. In this talk,

we work out a fairly detailed study on the category of finite

dimensional

modules of the restricted quantum $¥overline{U}_q(¥mathfrak{sl}_2)$

where

$q$ is a $2p$-th root of unity, $p¥ge2$.

In the representation theory of quantum groups at roots of unity, it is

often assumed that the parameter $q$ is a primitive $n$-th root of unity

where $n$ is a odd prime number. However, there has recently been

increasing interest in the the cases where $n$ is an even integer ---

for example, in the study of logarithmic conformal field theories, or in

knot invariants. In this talk,

we work out a fairly detailed study on the category of finite

dimensional

modules of the restricted quantum $¥overline{U}_q(¥mathfrak{sl}_2)$

where

$q$ is a $2p$-th root of unity, $p¥ge2$.

### 2011/05/31

#### Tuesday Seminar on Topology

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Measures with maximum total exponent and generic properties of $C^

{1}$ expanding maps (JAPANESE)

**Takehiko Morita**(Osaka University)Measures with maximum total exponent and generic properties of $C^

{1}$ expanding maps (JAPANESE)

[ Abstract ]

This is a joint work with Yusuke Tokunaga. Let $M$ be an $N$

dimensional compact connected smooth Riemannian manifold without

boundary and let $\\mathcal{E}^{r}(M,M)$ be the space of $C^{r}$

expandig maps endowed with $C^{r}$ topology. We show that

each of the following properties for element $T$ in $\\mathcal{E}

^{1}(M,M)$ is generic.

\\begin{itemize}

\\item[(1)] $T$ has a unique measure with maximum total exponent.

\\item[(2)] Any measure with maximum total exponent for $T$ has

zero entropy.

\\item[(3)] Any measure with maximum total exponent for $T$ is

fully supported.

\\end{itemize}

On the contrary, we show that for $r\\ge 2$, a generic element

in $\\mathcal{E}^{r}(M,M)$ has no fully supported measures with

maximum total exponent.

This is a joint work with Yusuke Tokunaga. Let $M$ be an $N$

dimensional compact connected smooth Riemannian manifold without

boundary and let $\\mathcal{E}^{r}(M,M)$ be the space of $C^{r}$

expandig maps endowed with $C^{r}$ topology. We show that

each of the following properties for element $T$ in $\\mathcal{E}

^{1}(M,M)$ is generic.

\\begin{itemize}

\\item[(1)] $T$ has a unique measure with maximum total exponent.

\\item[(2)] Any measure with maximum total exponent for $T$ has

zero entropy.

\\item[(3)] Any measure with maximum total exponent for $T$ is

fully supported.

\\end{itemize}

On the contrary, we show that for $r\\ge 2$, a generic element

in $\\mathcal{E}^{r}(M,M)$ has no fully supported measures with

maximum total exponent.

#### Lie Groups and Representation Theory

16:30-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

On character tables of association schemes based on attenuated

spaces (JAPANESE)

**Hirotake Kurihara**(Tohoku University)On character tables of association schemes based on attenuated

spaces (JAPANESE)

[ Abstract ]

An association scheme is a pair of a finite set $X$

and a set of relations $\\{R_i\\}_{0\\le i\\le d}$

on $X$ which satisfies several axioms of regularity.

The notion of association schemes is viewed as some axiomatized

properties of transitive permutation groups in terms of combinatorics, and also the notion of association schemes is regarded as a generalization of the subring of the group ring spanned by the conjugacy classes of finite groups.

Thus, the theory of association schemes had been developed in the

study of finite permutation groups and representation theory.

To determine the character tables of association schemes is an

important first step to a systematic study of association schemes, and is helpful toward the classification of those schemes.

In this talk, we determine the character tables of association schemes based on attenuated spaces.

These association schemes are obtained from subspaces of a given

dimension in attenuated spaces.

An association scheme is a pair of a finite set $X$

and a set of relations $\\{R_i\\}_{0\\le i\\le d}$

on $X$ which satisfies several axioms of regularity.

The notion of association schemes is viewed as some axiomatized

properties of transitive permutation groups in terms of combinatorics, and also the notion of association schemes is regarded as a generalization of the subring of the group ring spanned by the conjugacy classes of finite groups.

Thus, the theory of association schemes had been developed in the

study of finite permutation groups and representation theory.

To determine the character tables of association schemes is an

important first step to a systematic study of association schemes, and is helpful toward the classification of those schemes.

In this talk, we determine the character tables of association schemes based on attenuated spaces.

These association schemes are obtained from subspaces of a given

dimension in attenuated spaces.

### 2011/05/30

#### Algebraic Geometry Seminar

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Kodaira Dimension of Irregular Varieties (ENGLISH)

**Jungkai Alfred Chen**(National Taiwan University and RIMS)Kodaira Dimension of Irregular Varieties (ENGLISH)

[ Abstract ]

$f:X\\to Y$ be an algebraic fiber space with generic geometric fiber $F$, $\\dim X=n$ and $\\dim Y=m$. Then Iitaka's $C_{n,m}$ conjecture states $$\\kappa (X)\\geq \\kappa (Y)+\\kappa (F).$$ In particular, if $X$ is a variety with $\\kappa(X)=0$ and $f: X \\to Y$ is the Albanese map, then Ueno conjecture that $\\kappa(F)=0$. One can regard Ueno’s conjecture an important test case of Iitaka’s conjecture in general.

These conjectures are of fundamental importance in the classification of higher dimensional complex projective varieties. In a recent joint work with Hacon, we are able to prove Ueno’s conjecture and $C_{n,m}$ conjecture holds when $Y$ is of maximal Albanese dimension. In this talk, we will introduce some relative results and briefly sketch the proof.

$f:X\\to Y$ be an algebraic fiber space with generic geometric fiber $F$, $\\dim X=n$ and $\\dim Y=m$. Then Iitaka's $C_{n,m}$ conjecture states $$\\kappa (X)\\geq \\kappa (Y)+\\kappa (F).$$ In particular, if $X$ is a variety with $\\kappa(X)=0$ and $f: X \\to Y$ is the Albanese map, then Ueno conjecture that $\\kappa(F)=0$. One can regard Ueno’s conjecture an important test case of Iitaka’s conjecture in general.

These conjectures are of fundamental importance in the classification of higher dimensional complex projective varieties. In a recent joint work with Hacon, we are able to prove Ueno’s conjecture and $C_{n,m}$ conjecture holds when $Y$ is of maximal Albanese dimension. In this talk, we will introduce some relative results and briefly sketch the proof.

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