## Seminar information archive

Seminar information archive ～10/22｜Today's seminar 10/23 | Future seminars 10/24～

#### Lie Groups and Representation Theory

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

Geometry of multiplicity-free representations of SO(N) and visible actions (JAPANESE)

**Yuichiro Tanaka**(the University of Tokyo, Graduate School of Mathematical Sciences)Geometry of multiplicity-free representations of SO(N) and visible actions (JAPANESE)

[ Abstract ]

For a connected compact simple Lie group of type B or D,

we find pairs $(V_{1},V_{2})$ of irreducible representations of G such that the tensor product representation $V_{1}¥otimes V_{2}$ is multiplicity-free by a geometric consideration based on

a notion of visible actions on complex manifolds,

introduced by T. Kobayashi. The pairs we find exhaust

all the multiplicity-free pairs by an earlier

combinatorial classification due to Stembridge.

For a connected compact simple Lie group of type B or D,

we find pairs $(V_{1},V_{2})$ of irreducible representations of G such that the tensor product representation $V_{1}¥otimes V_{2}$ is multiplicity-free by a geometric consideration based on

a notion of visible actions on complex manifolds,

introduced by T. Kobayashi. The pairs we find exhaust

all the multiplicity-free pairs by an earlier

combinatorial classification due to Stembridge.

### 2013/10/28

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Minimal singular metrics of a line bundle admitting no Zariski decomposition (JAPANESE)

**Takayuki Koike**(The University of Tokyo)Minimal singular metrics of a line bundle admitting no Zariski decomposition (JAPANESE)

[ Abstract ]

We give a concrete expression of a minimal singular metric of a big line bundle on a compact Kähler manifold which is the total space of a toric bundle over a complex torus. In this class of manifolds, Nakayama constructed examples which have line bundles admitting no Zariski decomposition even after any proper modifications. As an application, we discuss the Zariski closedness of non-nef loci and the openness conjecture of Demailly and Kollar in this class.

We give a concrete expression of a minimal singular metric of a big line bundle on a compact Kähler manifold which is the total space of a toric bundle over a complex torus. In this class of manifolds, Nakayama constructed examples which have line bundles admitting no Zariski decomposition even after any proper modifications. As an application, we discuss the Zariski closedness of non-nef loci and the openness conjecture of Demailly and Kollar in this class.

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Weak Borisov-Alexeev-Borisov conjecture for 3-fold Mori Fiber spaces (ENGLISH)

**Chen Jiang**(University of Tokyo)Weak Borisov-Alexeev-Borisov conjecture for 3-fold Mori Fiber spaces (ENGLISH)

[ Abstract ]

We investigate $\\epsilon$-klt log Fano 3-folds with some Mori fiber space structure, more precisely, with a del Pezzo fibration structure, or a conic bundle structure over projective plane. We give a bound for the log anti-canonical volume of such pair. The method is constructing non-klt centers and using connectedness lemma. This result is related to birational boundedness of log Fano varieties.

We investigate $\\epsilon$-klt log Fano 3-folds with some Mori fiber space structure, more precisely, with a del Pezzo fibration structure, or a conic bundle structure over projective plane. We give a bound for the log anti-canonical volume of such pair. The method is constructing non-klt centers and using connectedness lemma. This result is related to birational boundedness of log Fano varieties.

### 2013/10/25

#### Seminar on Probability and Statistics

14:50-16:00 Room #006 (Graduate School of Math. Sci. Bldg.)

Sparse coding and structured dictionary learning (JAPANESE)

[ Reference URL ]

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/05.html

**MURATA, Noboru**(Waseda University)Sparse coding and structured dictionary learning (JAPANESE)

[ Reference URL ]

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/05.html

### 2013/10/24

#### Geometry Colloquium

10:00-11:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Some Uniqueness Theorems for Smoothing Singularities in Special Lagrangian Geometry (JAPANESE)

**Yohsuke Imagi**(Kyoto University)Some Uniqueness Theorems for Smoothing Singularities in Special Lagrangian Geometry (JAPANESE)

[ Abstract ]

Special Lagrangian submanifolds are area-minimizing Lagrangian submanifolds of Calabi--Yau manifolds. I'll talk mainly about the singularities of two special Lagrangian planes intersecting transversely. I'll determine a class of smoothing models for the singularities.

By some results of Abouzaid and Smith one can determine the smoothing models up to quasi-isomorphism in a Fukaya category. I'll combine it with a technique of Thomas and Yau.

Special Lagrangian submanifolds are area-minimizing Lagrangian submanifolds of Calabi--Yau manifolds. I'll talk mainly about the singularities of two special Lagrangian planes intersecting transversely. I'll determine a class of smoothing models for the singularities.

By some results of Abouzaid and Smith one can determine the smoothing models up to quasi-isomorphism in a Fukaya category. I'll combine it with a technique of Thomas and Yau.

### 2013/10/23

#### Operator Algebra Seminars

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Classification of quantum homogeneous spaces (ENGLISH)

**Makoto Yamashita**(Ochanomizu Univ.)Classification of quantum homogeneous spaces (ENGLISH)

#### Seminar on Probability and Statistics

13:00-15:30 Room #006 (Graduate School of Math. Sci. Bldg.)

Limit theorems for ambit processes (ENGLISH)

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/04.html

**Mark Podolskij**(Universität Heidelberg)Limit theorems for ambit processes (ENGLISH)

[ Abstract ]

We present some recent limit theorems for high frequency observations of ambit processes. Ambit processes constitute a flexible class of models, which are usually used to describe turbulent motion in physics. Mathematically speaking, they have a continuous moving average structure with additional random component called intermittency. In the first part of the lecture we will demonstrate the asymptotic theory for ambit processes driven by Brownian motion. The second part will deal with Levy driven ambit processes. We will see that these two cases deliver completely different limiting results.

本講演は数物フロンティア・リーディング大学院のレクチャーとして行います.

[ Reference URL ]We present some recent limit theorems for high frequency observations of ambit processes. Ambit processes constitute a flexible class of models, which are usually used to describe turbulent motion in physics. Mathematically speaking, they have a continuous moving average structure with additional random component called intermittency. In the first part of the lecture we will demonstrate the asymptotic theory for ambit processes driven by Brownian motion. The second part will deal with Levy driven ambit processes. We will see that these two cases deliver completely different limiting results.

本講演は数物フロンティア・リーディング大学院のレクチャーとして行います.

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/04.html

### 2013/10/22

#### PDE Real Analysis Seminar

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Fractional harmonic maps and applications (ENGLISH)

**Armin Schikorra**(MPI for Mathematics in the Sciences, Leipzig)Fractional harmonic maps and applications (ENGLISH)

[ Abstract ]

Fractional harmonic mappings are critical points of a generalized Dirichlet Energy where the gradient is replaced with a (non-local) differential operator.

I will present aspects of the regularity theory of (non-local) fractional harmonic maps into manifolds, which extends (and contains) the theory of (poly-)harmonic mappings.

I also will mention, how one can show regularity for critical points of the Moebius (Knot-) Energy, applying the techniques developed in this theory.

Fractional harmonic mappings are critical points of a generalized Dirichlet Energy where the gradient is replaced with a (non-local) differential operator.

I will present aspects of the regularity theory of (non-local) fractional harmonic maps into manifolds, which extends (and contains) the theory of (poly-)harmonic mappings.

I also will mention, how one can show regularity for critical points of the Moebius (Knot-) Energy, applying the techniques developed in this theory.

#### Tuesday Seminar on Topology

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

Cluster algebra and complex volume of knots (JAPANESE)

**Rei Inoue**(Chiba University)Cluster algebra and complex volume of knots (JAPANESE)

[ Abstract ]

The cluster algebra was introduced by Fomin and Zelevinsky around

2000. The characteristic operation in the algebra called `mutation' is

related to various notions in mathematics and mathematical physics. In

this talk I review a basics of the cluster algebra, and introduce its

application to study the complex volume of knot complements in S^3.

Here a mutation corresponds to an ideal tetrahedron.

This talk is based on joint work with Kazuhiro Hikami (Kyushu University).

The cluster algebra was introduced by Fomin and Zelevinsky around

2000. The characteristic operation in the algebra called `mutation' is

related to various notions in mathematics and mathematical physics. In

this talk I review a basics of the cluster algebra, and introduce its

application to study the complex volume of knot complements in S^3.

Here a mutation corresponds to an ideal tetrahedron.

This talk is based on joint work with Kazuhiro Hikami (Kyushu University).

#### Lie Groups and Representation Theory

17:00-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Representation Theory and Microlocal Analysis (ENGLISH)

**Benjamin Harris**(Louisiana State University (USA))Representation Theory and Microlocal Analysis (ENGLISH)

[ Abstract ]

Suppose $H\\subset K$ are compact, connected Lie groups, and suppose $\\tau$ is an irreducible, unitary representation of $H$. In 1979, Kashiwara and Vergne proved a simple asymptotic formula for the decomposition of $Ind_H^K\\tau$ by microlocally studying the regularity of vectors in this representation, thought of as vector valued functions on $K$. In 1998, Kobayashi proved a powerful criterion for the discrete decomposability of an irreducible, unitary representation $\\pi$ of a reductive Lie group $G$ when restricted to a reductive subgroup $H$. One of his key ideas was to restrict $\\pi$ to a representation of a maximal compact subgroup $K\\subset G$, view $\\pi$ as a subrepresentation of $L^2(K)$, and then use ideas similar to those developed by Kashiwara and Vergne.

In a recent preprint the speaker wrote with Hongyu He and Gestur Olafsson, the authors consider the possibility of studying induction and restriction to a reductive Lie group $G$ by microlocally studying the regularity of the matrix coefficients of (possibly reducible) unitary representations of $G$, viewed as continuous functions on the (possibly noncompact) Lie group $G$. In this talk, we will outline the main results of this paper and give additional conjectures.

Suppose $H\\subset K$ are compact, connected Lie groups, and suppose $\\tau$ is an irreducible, unitary representation of $H$. In 1979, Kashiwara and Vergne proved a simple asymptotic formula for the decomposition of $Ind_H^K\\tau$ by microlocally studying the regularity of vectors in this representation, thought of as vector valued functions on $K$. In 1998, Kobayashi proved a powerful criterion for the discrete decomposability of an irreducible, unitary representation $\\pi$ of a reductive Lie group $G$ when restricted to a reductive subgroup $H$. One of his key ideas was to restrict $\\pi$ to a representation of a maximal compact subgroup $K\\subset G$, view $\\pi$ as a subrepresentation of $L^2(K)$, and then use ideas similar to those developed by Kashiwara and Vergne.

In a recent preprint the speaker wrote with Hongyu He and Gestur Olafsson, the authors consider the possibility of studying induction and restriction to a reductive Lie group $G$ by microlocally studying the regularity of the matrix coefficients of (possibly reducible) unitary representations of $G$, viewed as continuous functions on the (possibly noncompact) Lie group $G$. In this talk, we will outline the main results of this paper and give additional conjectures.

### 2013/10/18

#### PDE Real Analysis Seminar

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Some applications of partial balayage (ENGLISH)

**Björn Gustavsson**(KTH Royal Institute of Technology)Some applications of partial balayage (ENGLISH)

[ Abstract ]

Partial balayage is a rather recent tool in potential theory. One of its origins is the construction of quadrature domains for subharmonic functions by Makoto Sakai in the 1970's. It also gives a convenient way of describing weak solutions to a moving boundary problem for Hele-Shaw flow (Laplacian growth), and recently Stephen Gardiner and Tomas Sjödin have used partial balayage to make progress on an inverse problem in potential theory. I plan to discuss some of these, and related, matters.

Partial balayage is a rather recent tool in potential theory. One of its origins is the construction of quadrature domains for subharmonic functions by Makoto Sakai in the 1970's. It also gives a convenient way of describing weak solutions to a moving boundary problem for Hele-Shaw flow (Laplacian growth), and recently Stephen Gardiner and Tomas Sjödin have used partial balayage to make progress on an inverse problem in potential theory. I plan to discuss some of these, and related, matters.

### 2013/10/17

#### GCOE Seminars

16:00-17:00 Room #470 (Graduate School of Math. Sci. Bldg.)

Stability for Inverse problems for Ultrahyperbolic Equations (ENGLISH)

**Fikret Goelgeleyen**(Bulent Ecevit University)Stability for Inverse problems for Ultrahyperbolic Equations (ENGLISH)

[ Abstract ]

In this work, we consider inverse problems of determining a coefficient or a source term in an ultrahyperbolic equation by some lateral boundary data.

We prove Hoelder estimates which are global and local and the key is Carleman estimates.

In this work, we consider inverse problems of determining a coefficient or a source term in an ultrahyperbolic equation by some lateral boundary data.

We prove Hoelder estimates which are global and local and the key is Carleman estimates.

#### GCOE Seminars

17:00-18:00 Room #470 (Graduate School of Math. Sci. Bldg.)

Fluid-structure interaction model and Levelset method (ENGLISH)

**kazufumi Ito**(North Carolina State University)Fluid-structure interaction model and Levelset method (ENGLISH)

[ Abstract ]

We derive a weak form and weak solution of the level set formulation of Cottet and Maitre for fluid-structure interaction problems with immersed surfaces. The method in particular exhibits appealing mass and energy conservation properties and a variational formulation of Peskin’s Immersed Boundary methods.

We derive a weak form and weak solution of the level set formulation of Cottet and Maitre for fluid-structure interaction problems with immersed surfaces. The method in particular exhibits appealing mass and energy conservation properties and a variational formulation of Peskin’s Immersed Boundary methods.

### 2013/10/16

#### Number Theory Seminar

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

Shimura varieties with infinite level, and torsion in the cohomology of locally symmetric spaces (ENGLISH)

**Peter Scholze**(Universität Bonn)Shimura varieties with infinite level, and torsion in the cohomology of locally symmetric spaces (ENGLISH)

[ Abstract ]

We will discuss the p-adic geometry of Shimura varieties with infinite level at p: They are perfectoid spaces, and there is a new period map defined at infinite level. As an application, we will discuss some results on torsion in the cohomology of locally symmetric spaces, and in particular the existence of Galois representations in this setup.

We will discuss the p-adic geometry of Shimura varieties with infinite level at p: They are perfectoid spaces, and there is a new period map defined at infinite level. As an application, we will discuss some results on torsion in the cohomology of locally symmetric spaces, and in particular the existence of Galois representations in this setup.

#### Operator Algebra Seminars

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Some prime factorization results for free quantum group factors (JAPANESE)

**Yusuke Isono**(Univ. Tokyo)Some prime factorization results for free quantum group factors (JAPANESE)

#### Seminar on Probability and Statistics

13:30-14:40 Room #052 (Graduate School of Math. Sci. Bldg.)

統計解析環境Rにおける多変量GARCHモデルの推定とパッケージ化 (JAPANESE)

[ Reference URL ]

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/03.html

**NAKATANI, Tomoaki**(Hokkaido University)統計解析環境Rにおける多変量GARCHモデルの推定とパッケージ化 (JAPANESE)

[ Reference URL ]

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/03.html

### 2013/10/15

#### Tuesday Seminar on Topology

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

Desingularizing special generic maps (JAPANESE)

**Masamichi Takase**(Seikei University)Desingularizing special generic maps (JAPANESE)

[ Abstract ]

This is a joint work with Osamu Saeki (IMI, Kyushu University).

A special generic map is a generic map which has only definite

fold as its singularities.

We study the condition for a special generic map from a closed

n-manifold to the p-space (n+1>p), to factor through a codimension

one immersion (or an embedding). In particular, for the cases

where p = 1 and 2 we obtain complete results.

Our techniques are related to Smale-Hirsch theory,

topology of the space of immersions, relation between the space

of topological immersions and that of smooth immersions,

sphere eversions, differentiable structures of homotopy spheres,

diffeomorphism group of spheres, free group actions on the sphere, etc.

This is a joint work with Osamu Saeki (IMI, Kyushu University).

A special generic map is a generic map which has only definite

fold as its singularities.

We study the condition for a special generic map from a closed

n-manifold to the p-space (n+1>p), to factor through a codimension

one immersion (or an embedding). In particular, for the cases

where p = 1 and 2 we obtain complete results.

Our techniques are related to Smale-Hirsch theory,

topology of the space of immersions, relation between the space

of topological immersions and that of smooth immersions,

sphere eversions, differentiable structures of homotopy spheres,

diffeomorphism group of spheres, free group actions on the sphere, etc.

### 2013/10/12

#### Harmonic Analysis Komaba Seminar

13:00-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

The global Cauchy problems for nonlinear dispersive equations on modulation spaces

(JAPANESE)

Path Integrals--Analysis on path space by time-slicing method (JAPANESE)

**Tomoya Kato**(Nagoya University) 13:30-15:00The global Cauchy problems for nonlinear dispersive equations on modulation spaces

(JAPANESE)

**Naoto Kumanogo**(Kogakuin University) 15:30-17:00Path Integrals--Analysis on path space by time-slicing method (JAPANESE)

### 2013/10/09

#### Operator Algebra Seminars

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Graphs of quantum groups and K-amenability (ENGLISH)

**Pierre Fima**(Univ. Paris VII)Graphs of quantum groups and K-amenability (ENGLISH)

#### GCOE Seminars

16:00-17:00 Room #370 (Graduate School of Math. Sci. Bldg.)

Determination of the first order terms for elliptic partial differential equations using the partial Cauchy data (ENGLISH)

**Oleg Emanouilov**(Colorado State Univ.)Determination of the first order terms for elliptic partial differential equations using the partial Cauchy data (ENGLISH)

[ Abstract ]

In the bounded domain we consider the variant of the Calderon's problem for the second order partial differential equation with unknown first order terms. Under some geometric condition on domain we prove that the coefficients of this equation can be determined from the partial Cauchy data up to the gauge equivalence.

In the bounded domain we consider the variant of the Calderon's problem for the second order partial differential equation with unknown first order terms. Under some geometric condition on domain we prove that the coefficients of this equation can be determined from the partial Cauchy data up to the gauge equivalence.

### 2013/10/08

#### Tuesday Seminar on Topology

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

An invariant of rational homology 3-spheres via vector fields. (JAPANESE)

**Tatsuro Shimizu**(The Univesity of Tokyo)An invariant of rational homology 3-spheres via vector fields. (JAPANESE)

[ Abstract ]

In this talk, we define an invariant of rational homology 3-spheres with

values in a space $\\mathcal A(\\emptyset)$ of Jacobi diagrams by using

vector fields.

The construction of our invariant is a generalization of both that of

the Kontsevich-Kuperberg-Thurston invariant $z^{KKT}$

and that of Fukaya and Watanabe's Morse homotopy invariant $z^{FW}$.

As an application of our invariant, we prove that $z^{KKT}=z^{FW}$ for

integral homology 3-spheres.

In this talk, we define an invariant of rational homology 3-spheres with

values in a space $\\mathcal A(\\emptyset)$ of Jacobi diagrams by using

vector fields.

The construction of our invariant is a generalization of both that of

the Kontsevich-Kuperberg-Thurston invariant $z^{KKT}$

and that of Fukaya and Watanabe's Morse homotopy invariant $z^{FW}$.

As an application of our invariant, we prove that $z^{KKT}=z^{FW}$ for

integral homology 3-spheres.

### 2013/10/07

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

The limits on boundary of orbifold Kähler-Einstein metrics and orbifold Kähler-Ricci flows over quasi-projective manifolds (JAPANESE)

**Shin Kikuta**(Sophia University)The limits on boundary of orbifold Kähler-Einstein metrics and orbifold Kähler-Ricci flows over quasi-projective manifolds (JAPANESE)

[ Abstract ]

In this talk, we consider a sequence of orbifold Kähler-Einstein metrics of negative Ricci curvature or corresponding orbifold normalized Kähler-Ricci flows on a quasi-projective manifold with ample log-canonical bundle for a simple normal crossing divisor. Tian-Yau, S. Bando and H. Tsuji established that the sequence of orbifold Kähler-Einstein metrics converged to the complete Käler-Einstein metric of negative Ricci curvature on the complement of the boundary divisor. The main purpose of this talk is to show that such a convergence is also true on the boundary for both of the orbifold Kähler-Einstein metrics and the orbifold normalized Kähler-Ricci flows.

In this talk, we consider a sequence of orbifold Kähler-Einstein metrics of negative Ricci curvature or corresponding orbifold normalized Kähler-Ricci flows on a quasi-projective manifold with ample log-canonical bundle for a simple normal crossing divisor. Tian-Yau, S. Bando and H. Tsuji established that the sequence of orbifold Kähler-Einstein metrics converged to the complete Käler-Einstein metric of negative Ricci curvature on the complement of the boundary divisor. The main purpose of this talk is to show that such a convergence is also true on the boundary for both of the orbifold Kähler-Einstein metrics and the orbifold normalized Kähler-Ricci flows.

### 2013/10/03

#### Geometry Colloquium

10:00-11:30 Room #122 (Graduate School of Math. Sci. Bldg.)

A rigidity lemma for cocycles over BS(1,k)-actions (JAPANESE)

**Masayuki ASAOKA**(Kyoto University)A rigidity lemma for cocycles over BS(1,k)-actions (JAPANESE)

[ Abstract ]

Existence of an invariant geometric structure is persistent for many known examples of group actions on homogeneous spaces. In this talk, I would like to report an attempt to explain such a rigidity from a unified point of view. We will see that some rigidity results are reduced to a rigidity lemma on Diff(R^n,0)-valued cocycles over BS(1,k)-actions, where BS(1,k) is the Baumslag-Solitar group .

Existence of an invariant geometric structure is persistent for many known examples of group actions on homogeneous spaces. In this talk, I would like to report an attempt to explain such a rigidity from a unified point of view. We will see that some rigidity results are reduced to a rigidity lemma on Diff(R^n,0)-valued cocycles over BS(1,k)-actions, where BS(1,k) is the Baumslag-Solitar group .

### 2013/10/02

#### Operator Algebra Seminars

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

A classification of flows on AFD factors with faithful Connes-Takesaki modules

(JAPANESE)

**Koichi Shimada**(Univ. Tokyo)A classification of flows on AFD factors with faithful Connes-Takesaki modules

(JAPANESE)

### 2013/10/01

#### Tuesday Seminar on Topology

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

The geography problem of Lefschetz fibrations (JAPANESE)

**Naoyuki Monden**(Tokyo University of Science)The geography problem of Lefschetz fibrations (JAPANESE)

[ Abstract ]

To consider holomorphic fibrations complex surfaces over complex curves

and Lefschetz fibrations over surfaces is one method for the study of

complex surfaces of general type and symplectic 4-manifods, respectively.

In this talk, by comparing the geography problem of relatively minimal

holomorphic fibrations with that of relatively minimal Lefschetz

fibrations (i.e., the characterization of pairs $(x,y)$ of certain

invariants $x$ and $y$ corresponding to relatively minimal holomorphic

fibrations and relatively minimal Lefschetz fibrations), we observe the

difference between complex surfaces of general type and symplectic

4-manifolds. In particular, we construct Lefschetz fibrations violating

the ``slope inequality" which holds for any relatively minimal holomorphic

fibrations.

To consider holomorphic fibrations complex surfaces over complex curves

and Lefschetz fibrations over surfaces is one method for the study of

complex surfaces of general type and symplectic 4-manifods, respectively.

In this talk, by comparing the geography problem of relatively minimal

holomorphic fibrations with that of relatively minimal Lefschetz

fibrations (i.e., the characterization of pairs $(x,y)$ of certain

invariants $x$ and $y$ corresponding to relatively minimal holomorphic

fibrations and relatively minimal Lefschetz fibrations), we observe the

difference between complex surfaces of general type and symplectic

4-manifolds. In particular, we construct Lefschetz fibrations violating

the ``slope inequality" which holds for any relatively minimal holomorphic

fibrations.

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