## 代数幾何学セミナー

過去の記録 ～01/29｜次回の予定｜今後の予定 01/30～

開催情報 | 火曜日 10:30～11:30 or 12:00 数理科学研究科棟(駒場) ハイブリッド開催/002号室 |
---|---|

担当者 | 權業 善範・中村 勇哉・田中公 |

**過去の記録**

### 2015年04月13日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

An orbifold version of Miyaoka's semi-positivity theorem and applications (English)

**Frédéric Campana 氏**(Université de Lorraine)An orbifold version of Miyaoka's semi-positivity theorem and applications (English)

[ 講演概要 ]

This `orbifold' version of Miyaoka's theorem says that if (X,D)

is a projective log-canonical pair with K_X+D pseudo-effective,

then its 'cotangent' sheaf $¥Omega^1(X,D)$ is generically semi-positive.

The definitions will be given. The original proof of Miyaoka, which

mixes

char 0 and char p>0 arguments could not be adapted. Our proof is in char

0 only.

A first consequence is when (X,D) is log-smooth with reduced boudary D,

in which case the cotangent sheaf is the classical Log-cotangent sheaf:

if some tensor power of $¥omega^1_X(log(D))$ contains a 'big' line

bundle, then K_X+D is 'big' too. This implies, together with work of

Viehweg-Zuo,

the `hyperbolicity conjecture' of Shafarevich-Viehweg.

The preceding is joint work with Mihai Paun.

A second application (joint work with E. Amerik) shows that if D is a

non-uniruled smooth divisor in aprojective hyperkaehler manifold with

symplectic form s,

then its characteristic foliation is algebraic only if X is a K3 surface.

This was shown previously bt Hwang-Viehweg assuming D to be of general

type. This result has some further consequences.

This `orbifold' version of Miyaoka's theorem says that if (X,D)

is a projective log-canonical pair with K_X+D pseudo-effective,

then its 'cotangent' sheaf $¥Omega^1(X,D)$ is generically semi-positive.

The definitions will be given. The original proof of Miyaoka, which

mixes

char 0 and char p>0 arguments could not be adapted. Our proof is in char

0 only.

A first consequence is when (X,D) is log-smooth with reduced boudary D,

in which case the cotangent sheaf is the classical Log-cotangent sheaf:

if some tensor power of $¥omega^1_X(log(D))$ contains a 'big' line

bundle, then K_X+D is 'big' too. This implies, together with work of

Viehweg-Zuo,

the `hyperbolicity conjecture' of Shafarevich-Viehweg.

The preceding is joint work with Mihai Paun.

A second application (joint work with E. Amerik) shows that if D is a

non-uniruled smooth divisor in aprojective hyperkaehler manifold with

symplectic form s,

then its characteristic foliation is algebraic only if X is a K3 surface.

This was shown previously bt Hwang-Viehweg assuming D to be of general

type. This result has some further consequences.

### 2015年01月26日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Positivity in varieties of maximal Albanese dimension (ENGLISH)

**Jungkai Chen 氏**(National Taiwan University)Positivity in varieties of maximal Albanese dimension (ENGLISH)

[ 講演概要 ]

Given a variety of maximal Albanese dimension, it is known that the holomorphic Euler characteristic is non-negative. It is an interesting question to characterize varieties with vanishing Euler characteristic.

In our previous work (jointly with Debarre and Jiang), we prove that Ein-Lazarsgfeld's example is essentially the only variety of maximal Albanese and Kodaira dimension with vanishing Euler characteristic in dimension three. In the recent joint work with Jiang, we prove a decomposition theorem for the push-forward of canonical sheaf. As a consequence, we are able to generalized our previous characterization. The purpose of this talk is give a survey of these two works.

Given a variety of maximal Albanese dimension, it is known that the holomorphic Euler characteristic is non-negative. It is an interesting question to characterize varieties with vanishing Euler characteristic.

In our previous work (jointly with Debarre and Jiang), we prove that Ein-Lazarsgfeld's example is essentially the only variety of maximal Albanese and Kodaira dimension with vanishing Euler characteristic in dimension three. In the recent joint work with Jiang, we prove a decomposition theorem for the push-forward of canonical sheaf. As a consequence, we are able to generalized our previous characterization. The purpose of this talk is give a survey of these two works.

### 2015年01月19日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Crepant resolutions of Slodowy slice in nilpotent orbit closure in sl_N(C) (JAPANESE)

**山岸 亮 氏**(京都大学理学部)Crepant resolutions of Slodowy slice in nilpotent orbit closure in sl_N(C) (JAPANESE)

[ 講演概要 ]

Nilpotent orbit closures and their intersections with Slodowy slices are typical examples of symplectic varieties. It is known that every crepant resolution of a nilpotent orbit closure is obtained as a Springer resolution. In this talk, we show that every crepant resolution of a Slodowy slice in nilpotent orbit closure in sl_N(C) is obtained as the restriction of a Springer resolution and explain how to count the number of crepant resolutions. The proof of the main results is based on the fact that Slodowy slices can be described as quiver varieties.

Nilpotent orbit closures and their intersections with Slodowy slices are typical examples of symplectic varieties. It is known that every crepant resolution of a nilpotent orbit closure is obtained as a Springer resolution. In this talk, we show that every crepant resolution of a Slodowy slice in nilpotent orbit closure in sl_N(C) is obtained as the restriction of a Springer resolution and explain how to count the number of crepant resolutions. The proof of the main results is based on the fact that Slodowy slices can be described as quiver varieties.

### 2014年12月15日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

A characterization of ordinary abelian varieties in positive characteristic (JAPANESE)

**三内 顕義 氏**(東京大学数理科学研究科)A characterization of ordinary abelian varieties in positive characteristic (JAPANESE)

[ 講演概要 ]

This is joint work with Hiromu Tanaka. In this talk, we study F^e_*O_X on a projective variety over the algebraic closed field of positive characteristic. For an ordinary abelian variety X, F^e_*O_X is decomposed into line bundles for every positive integer e. Conversely, if a smooth projective variety X satisfies this property and its Kodaira dimension is non-negative, then X is an ordinary abelian variety.

This is joint work with Hiromu Tanaka. In this talk, we study F^e_*O_X on a projective variety over the algebraic closed field of positive characteristic. For an ordinary abelian variety X, F^e_*O_X is decomposed into line bundles for every positive integer e. Conversely, if a smooth projective variety X satisfies this property and its Kodaira dimension is non-negative, then X is an ordinary abelian variety.

### 2014年12月01日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Induced Automorphisms on Hyperkaehler Manifolds (ENGLISH)

**Malte Wandel 氏**(RIMS)Induced Automorphisms on Hyperkaehler Manifolds (ENGLISH)

[ 講演概要 ]

in this talk I want to report on a joint project with Giovanni Mongardi (Milano). We study automorphisms of hyperkaehler manifolds. All known deformation classes of these manifolds contain moduli spaces of stable sheaves on surfaces. If the underlying surface admits a non-trivial automorphism, it is often possible to transfer this automorphism to a moduli space of sheaves. In this way we obtain a big class of interesting examples of automorphisms of hyperkaehler manifolds. I will present a criterion to 'detect' automorphisms in this class and discuss several applications for the classification of automorphisms of manifolds of K3^[n]- and kummer n-type. If time permits I will try to talk about generalisations to O'Grady's sporadic examples.

in this talk I want to report on a joint project with Giovanni Mongardi (Milano). We study automorphisms of hyperkaehler manifolds. All known deformation classes of these manifolds contain moduli spaces of stable sheaves on surfaces. If the underlying surface admits a non-trivial automorphism, it is often possible to transfer this automorphism to a moduli space of sheaves. In this way we obtain a big class of interesting examples of automorphisms of hyperkaehler manifolds. I will present a criterion to 'detect' automorphisms in this class and discuss several applications for the classification of automorphisms of manifolds of K3^[n]- and kummer n-type. If time permits I will try to talk about generalisations to O'Grady's sporadic examples.

### 2014年10月27日(月)

14:50-16:20 数理科学研究科棟(駒場) 122号室

いつもと開始時間が異なります。

On projective varieties with very large canonical volume (ENGLISH)

いつもと開始時間が異なります。

**Meng Chen 氏**(Fudan University)On projective varieties with very large canonical volume (ENGLISH)

[ 講演概要 ]

For any positive integer n>0, a theorem of Hacon-McKernan, Takayama and Tsuji says that there is a constant c(n) so that the m-canonical map is birational onto its image for all smooth projective n-folds and all m>=c(n). We are interested in the following problem "P(n)": is there a constant M(n) so that, for all smooth projective n-fold X with Vol(X)>M(n), the m-canonical map of X is birational for all m>=c(n-1). The answer to “P_n" is positive due to Bombieri when $n=2$ and to Todorov when $n=3$. The aim of this talk is to introduce my joint work with Zhi Jiang from Universite Paris-Sud. We give a positive answer in dimensions 4 and 5.

For any positive integer n>0, a theorem of Hacon-McKernan, Takayama and Tsuji says that there is a constant c(n) so that the m-canonical map is birational onto its image for all smooth projective n-folds and all m>=c(n). We are interested in the following problem "P(n)": is there a constant M(n) so that, for all smooth projective n-fold X with Vol(X)>M(n), the m-canonical map of X is birational for all m>=c(n-1). The answer to “P_n" is positive due to Bombieri when $n=2$ and to Todorov when $n=3$. The aim of this talk is to introduce my joint work with Zhi Jiang from Universite Paris-Sud. We give a positive answer in dimensions 4 and 5.

### 2014年07月07日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Balanced line bundles (JAPANESE)

**谷本翔 氏**(Rice University)Balanced line bundles (JAPANESE)

[ 講演概要 ]

A conjecture of Batyrev and Manin relates arithmetic properties of

varieties with big anticanonical class to geometric invariants; in

particular, counting functions defined by metrized ample line bundles

and the corresponding asymptotics of rational points of bounded height

are interpreted in terms of cones of effective divisors and certain

thresholds with respect to these cones. This framework leads to the

notion of balanced line bundles, whose counting functions, conjecturally,

capture generic distributions of rational points. We investigate

balanced line bundles in the context of the Minimal Model Program, with

special regard to the classification of Fano threefolds and Mori fiber

spaces.

This is joint work with Brian Lehmann and Yuri Tschinkel.

A conjecture of Batyrev and Manin relates arithmetic properties of

varieties with big anticanonical class to geometric invariants; in

particular, counting functions defined by metrized ample line bundles

and the corresponding asymptotics of rational points of bounded height

are interpreted in terms of cones of effective divisors and certain

thresholds with respect to these cones. This framework leads to the

notion of balanced line bundles, whose counting functions, conjecturally,

capture generic distributions of rational points. We investigate

balanced line bundles in the context of the Minimal Model Program, with

special regard to the classification of Fano threefolds and Mori fiber

spaces.

This is joint work with Brian Lehmann and Yuri Tschinkel.

### 2014年06月30日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Invariant subrings of the Cox rings of K3surfaces by automorphism groups (JAPANESE)

**三内顕義 氏**(東京大学数理科学研究科)Invariant subrings of the Cox rings of K3surfaces by automorphism groups (JAPANESE)

[ 講演概要 ]

Cox rings were introduced by D.Cox and are important rings which appeared in algebraic geometry. One of the main topic related with Cox rings is the finite generation of them. In this talk, we consider the Cox rings of K3 surfaces and answer the following question asked by D. Huybrechts; Are the invariant subrings of the Cox rings of K3 surfaces by automorphism groups finitely generated in general?

Cox rings were introduced by D.Cox and are important rings which appeared in algebraic geometry. One of the main topic related with Cox rings is the finite generation of them. In this talk, we consider the Cox rings of K3 surfaces and answer the following question asked by D. Huybrechts; Are the invariant subrings of the Cox rings of K3 surfaces by automorphism groups finitely generated in general?

### 2014年06月02日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

On base point free theorem for log canonical three folds over the algebraic closure of a finite field (JAPANESE)

**中村勇哉 氏**(東京大学数理科学研究科)On base point free theorem for log canonical three folds over the algebraic closure of a finite field (JAPANESE)

[ 講演概要 ]

We will discuss about the base point free theorem on three-dimensional

pairs defined over the algebraic closure of a finite field.

We know the base point free theorem on arbitrary-dimensional Kawamata

log terminal pairs in characteristic zero. By Birkar and Xu, the base

point free theorem in positive characteristic is known for big line

bundles on three-dimensional Kawamata log terminal pairs defined over

an algebraically closed field of characteristic larger than 5. Over the

algebraic closure of a finite field, a stronger result was proved by Keel.

The purpose of this talk is to generalize the Keel's result. We will

prove the base point free theorem for big line bundles on

three-dimensional log canonical pairs defined over the algebraic closure

of a finite field. This theorem is not valid for another field.

This is joint work with Diletta Martinelli and Jakub Witaszek.

We will discuss about the base point free theorem on three-dimensional

pairs defined over the algebraic closure of a finite field.

We know the base point free theorem on arbitrary-dimensional Kawamata

log terminal pairs in characteristic zero. By Birkar and Xu, the base

point free theorem in positive characteristic is known for big line

bundles on three-dimensional Kawamata log terminal pairs defined over

an algebraically closed field of characteristic larger than 5. Over the

algebraic closure of a finite field, a stronger result was proved by Keel.

The purpose of this talk is to generalize the Keel's result. We will

prove the base point free theorem for big line bundles on

three-dimensional log canonical pairs defined over the algebraic closure

of a finite field. This theorem is not valid for another field.

This is joint work with Diletta Martinelli and Jakub Witaszek.

### 2014年05月12日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Higher Nash blowup on normal toric varieties and a higher order version of Nobile's theorem (ENGLISH)

**Andrés Daniel Duarte 氏**(Institut de Mathématiques de Toulouse)Higher Nash blowup on normal toric varieties and a higher order version of Nobile's theorem (ENGLISH)

[ 講演概要 ]

The higher Nash blowup of an algebraic variety replaces singular points with limits of certain vector spaces carrying first or higher order data associated to the variety at non-singular points. In the case of normal toric varieties, the higher Nash blowup has a combinatorial description in terms of the Gröbner fan. This description will allows us to prove a higher version of Nobile's theorem in this context: for a normal toric variety, the higher Nash blowup is an isomorphism if and only if the variety is non-singular. We will also present some further observations coming from computational experiments.

The higher Nash blowup of an algebraic variety replaces singular points with limits of certain vector spaces carrying first or higher order data associated to the variety at non-singular points. In the case of normal toric varieties, the higher Nash blowup has a combinatorial description in terms of the Gröbner fan. This description will allows us to prove a higher version of Nobile's theorem in this context: for a normal toric variety, the higher Nash blowup is an isomorphism if and only if the variety is non-singular. We will also present some further observations coming from computational experiments.

### 2014年04月28日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Syzygies of jacobian ideals and Torelli properties (ENGLISH)

**Alexandru Dimca 氏**(Institut Universitaire de France )Syzygies of jacobian ideals and Torelli properties (ENGLISH)

[ 講演概要 ]

Let $C$ be a reduced complex projective plane curve defined by a homogeneous equation $f(x,y,z)=0$. We consider syzygies of the type $af_x+bf_y+cf_z=0$, where $a,b,c$ are homogeneous polynomials and $f_x,f_y,f_z$ stand for the partial derivatives of $f$. In our talk we relate such syzygies with stable or splittable rank two vector bundles on the projective plane, and to Torelli properties of plane curves in the sense of Dolgachev-Kapranov.

Let $C$ be a reduced complex projective plane curve defined by a homogeneous equation $f(x,y,z)=0$. We consider syzygies of the type $af_x+bf_y+cf_z=0$, where $a,b,c$ are homogeneous polynomials and $f_x,f_y,f_z$ stand for the partial derivatives of $f$. In our talk we relate such syzygies with stable or splittable rank two vector bundles on the projective plane, and to Torelli properties of plane curves in the sense of Dolgachev-Kapranov.

### 2014年02月12日(水)

14:00-17:30 数理科学研究科棟(駒場) 122号室

いつもと曜日・時間が異なります

An injectivity theorem with multiplier ideal sheaves of singular metrics with transcendental singularities (ENGLISH)

Ohsawa-Takegoshi extension theorem for K\\"ahler manifolds (ENGLISH)

いつもと曜日・時間が異なります

**松村 慎一 氏**(鹿児島大学) 14:00-15:30An injectivity theorem with multiplier ideal sheaves of singular metrics with transcendental singularities (ENGLISH)

[ 講演概要 ]

In this talk, I give an injectivity theorem with multiplier ideal sheaves of singular metrics.

This theorem is a powerful generalization of various injectivity and vanishing theorems.

The proof is based on a combination of the theory of harmonic integrals and the L^2-method for the \\dbar-equation.

To treat transcendental singularities, after regularizing a given singular metric, we study the asymptotic behavior of the harmonic forms with respect to a family of the regularized metrics.

Moreover we obtain L^2-estimates of solutions of the \\dbar-equation, by using the \\check{C}ech complex.

As an application, we obtain a Nadel type vanishing theorem.

In this talk, I give an injectivity theorem with multiplier ideal sheaves of singular metrics.

This theorem is a powerful generalization of various injectivity and vanishing theorems.

The proof is based on a combination of the theory of harmonic integrals and the L^2-method for the \\dbar-equation.

To treat transcendental singularities, after regularizing a given singular metric, we study the asymptotic behavior of the harmonic forms with respect to a family of the regularized metrics.

Moreover we obtain L^2-estimates of solutions of the \\dbar-equation, by using the \\check{C}ech complex.

As an application, we obtain a Nadel type vanishing theorem.

**Junyan Cao 氏**(KIAS) 16:00-17:30Ohsawa-Takegoshi extension theorem for K\\"ahler manifolds (ENGLISH)

[ 講演概要 ]

In this talk, we first prove a version of the Ohsawa-Takegoshi

extension theorem valid for on arbitrary K\\"ahler manifolds, and for

holomorphic line bundles equipped with possibly singular metrics. As an

application, we generalise Berndtsson and Paun 's result about the

pseudo-effectivity of the relative canonical bundles to arbitrary

compact K\\"ahler families.

In this talk, we first prove a version of the Ohsawa-Takegoshi

extension theorem valid for on arbitrary K\\"ahler manifolds, and for

holomorphic line bundles equipped with possibly singular metrics. As an

application, we generalise Berndtsson and Paun 's result about the

pseudo-effectivity of the relative canonical bundles to arbitrary

compact K\\"ahler families.

### 2014年02月03日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Classification of log del Pezzo surfaces of index three (JAPANESE)

**藤田 健人 氏**(京都大学数理解析研究所)Classification of log del Pezzo surfaces of index three (JAPANESE)

[ 講演概要 ]

Log del Pezzo surfaces constitute an interesting class of rational surfaces and naturally appear in the minimal model program. I will describe an algorithm to classify all the log del Pezzo surfaces of fixed (Q-Gorenstein) index $a$. Especially, I will focus on the case that $a$ is equal to three. This is joint work with Kazunori Yasutake.

Log del Pezzo surfaces constitute an interesting class of rational surfaces and naturally appear in the minimal model program. I will describe an algorithm to classify all the log del Pezzo surfaces of fixed (Q-Gorenstein) index $a$. Especially, I will focus on the case that $a$ is equal to three. This is joint work with Kazunori Yasutake.

### 2014年01月22日(水)

15:00-16:30 数理科学研究科棟(駒場) 122号室

いつもと曜日・時間が異なります

Divisorial Extractions from Singular Curves in Smooth 3-Folds (ENGLISH)

いつもと曜日・時間が異なります

**Thomas Ducat 氏**(University of Warwick)Divisorial Extractions from Singular Curves in Smooth 3-Folds (ENGLISH)

[ 講演概要 ]

Consider a singular curve C contained in a smooth 3-fold X.

Assuming the existence of a Du Val general elephant S containing C,

I give a normal form for the equations of C in X and an outline of how to

construct a divisorial extraction from this curve. If the general S is

Du Val of type D_{2k}, E_6 or E_7 then I can give some explicit

conditions for the existence of a terminal extraction. A treatment of

the D_{2k+1} case should be possible by similar means.

Consider a singular curve C contained in a smooth 3-fold X.

Assuming the existence of a Du Val general elephant S containing C,

I give a normal form for the equations of C in X and an outline of how to

construct a divisorial extraction from this curve. If the general S is

Du Val of type D_{2k}, E_6 or E_7 then I can give some explicit

conditions for the existence of a terminal extraction. A treatment of

the D_{2k+1} case should be possible by similar means.

### 2014年01月20日(月)

15:30-17:00 数理科学研究科棟(駒場) 126号室

いつもと部屋が異なります

Deforming elephants of Q-Fano 3-folds (ENGLISH)

いつもと部屋が異なります

**佐野 太郎 氏**(University of Warwick)Deforming elephants of Q-Fano 3-folds (ENGLISH)

[ 講演概要 ]

Shokurov and Reid proved that a Fano 3-fold with canonical

Gorenstein singularities has a Du Val elephant, that is,

a member of the anticanonical linear system with only Du Val singularities.

The classification of Fano 3-folds is based on this fact.

However, for a Fano 3-fold with non-Gorenstein terminal singularities,

the anticanonical system does not contain such a member in general.

Alt{\\i}nok--Brown--Reid conjectured that, if the anticanonical system is non-empty,

a Q-Fano 3-fold can be deformed to that with a Du Val elephant.

In this talk, I will explain how to deform an elephant with isolated

singularities to a Du Val elephant.

Shokurov and Reid proved that a Fano 3-fold with canonical

Gorenstein singularities has a Du Val elephant, that is,

a member of the anticanonical linear system with only Du Val singularities.

The classification of Fano 3-folds is based on this fact.

However, for a Fano 3-fold with non-Gorenstein terminal singularities,

the anticanonical system does not contain such a member in general.

Alt{\\i}nok--Brown--Reid conjectured that, if the anticanonical system is non-empty,

a Q-Fano 3-fold can be deformed to that with a Du Val elephant.

In this talk, I will explain how to deform an elephant with isolated

singularities to a Du Val elephant.

### 2013年12月09日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

On birationally tririgid Q-Fano threefolds (JAPANESE)

**岡田 拓三 氏**(佐賀大学)On birationally tririgid Q-Fano threefolds (JAPANESE)

[ 講演概要 ]

I will talk about birational geometry of Q-Fano threefolds. A Mori

fiber space birational to a given Q-Fano threefold is called a birational Mori fiber structure of the threefold. The existence of Q-Fano threefolds with a unique birational Mori fiber structure (resp. with two birational Mori fiber structures) is known. In this talk I will give an example of Q-Fano threefolds with three birational Mori fiber structures and also discuss about the behavior of birational Mori fiber structures in a family.

I will talk about birational geometry of Q-Fano threefolds. A Mori

fiber space birational to a given Q-Fano threefold is called a birational Mori fiber structure of the threefold. The existence of Q-Fano threefolds with a unique birational Mori fiber structure (resp. with two birational Mori fiber structures) is known. In this talk I will give an example of Q-Fano threefolds with three birational Mori fiber structures and also discuss about the behavior of birational Mori fiber structures in a family.

### 2013年11月25日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Minimal singular metrics of some line bundles with infinitely generated section rings (JAPANESE)

**小池 貴之 氏**(東大数理)Minimal singular metrics of some line bundles with infinitely generated section rings (JAPANESE)

[ 講演概要 ]

We consider Hermitian metrics of pseudo-effective line bundles on smooth

projective varieties defined over $\\mathbb{C}$.

Especially we are interested in (possibly singular) Hermitian metrics

with semi-positive curvatures when the section rings are not finitely generated.

We study where and how minimal singular metrics, special Hermitian

metrics with semi-positive curvatures, diverges in the following two situations;

a line bundle admitting no Zariski decomposition even after any

modifications (Nakayama example)

and a nef line bundle $L$ on $X$ satisfying $D \\subset |mL|$ and $|mL-D|

= \\emptyset$ for some divisor $D \\subset X$ and for all $m \\geq 1$ (

Zariski example).

We consider Hermitian metrics of pseudo-effective line bundles on smooth

projective varieties defined over $\\mathbb{C}$.

Especially we are interested in (possibly singular) Hermitian metrics

with semi-positive curvatures when the section rings are not finitely generated.

We study where and how minimal singular metrics, special Hermitian

metrics with semi-positive curvatures, diverges in the following two situations;

a line bundle admitting no Zariski decomposition even after any

modifications (Nakayama example)

and a nef line bundle $L$ on $X$ satisfying $D \\subset |mL|$ and $|mL-D|

= \\emptyset$ for some divisor $D \\subset X$ and for all $m \\geq 1$ (

Zariski example).

### 2013年11月18日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

この講演はGCOEレクチャーズとの共同セミナーになります

An injectivity theorem (ENGLISH)

この講演はGCOEレクチャーズとの共同セミナーになります

**Florin Ambro 氏**(IMAR)An injectivity theorem (ENGLISH)

[ 講演概要 ]

I will discuss a generalization of the injectivity theorem of Esnault-Viehweg, and an

application to the problem of lifting sections from the non-log canonical locus of a log variety.

I will discuss a generalization of the injectivity theorem of Esnault-Viehweg, and an

application to the problem of lifting sections from the non-log canonical locus of a log variety.

### 2013年11月11日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Geography via the base loci (ENGLISH)

**Sung Rak Choi 氏**(POSTECH)Geography via the base loci (ENGLISH)

[ 講演概要 ]

The geography of log model refers to the decomposition of the set of effective adjoint divisors into the cells defined by the resulting models that are obtained by the log minimal model program.

We will describe the geography in terms of the asymptotic base loci and Zariski decompositions of divisors.

As an application, we give a partial answer to a question of B. Totaro concerning the structure of partially ample cones.

The geography of log model refers to the decomposition of the set of effective adjoint divisors into the cells defined by the resulting models that are obtained by the log minimal model program.

We will describe the geography in terms of the asymptotic base loci and Zariski decompositions of divisors.

As an application, we give a partial answer to a question of B. Totaro concerning the structure of partially ample cones.

### 2013年10月28日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

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

**江 辰 氏**(東京大学数理科学研究科)Weak Borisov-Alexeev-Borisov conjecture for 3-fold Mori Fiber spaces (ENGLISH)

[ 講演概要 ]

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年07月22日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Equivariant degenerations of spherical modules (ENGLISH)

**Stavros Papadakis 氏**(RIMS)Equivariant degenerations of spherical modules (ENGLISH)

[ 講演概要 ]

Given a reductive algebraic group G and an invariant

Hilbert function h, Alexeev and Brion have defined

a moduli scheme M which parametrizes affine G-schemes X

with the property that the coordinate ring of X decomposes,

as G-module, according to the function h. The talk will

be about joint work with Bart Van Steirteghem (New York)

which studies the moduli scheme M under some additional

assumptions.

Given a reductive algebraic group G and an invariant

Hilbert function h, Alexeev and Brion have defined

a moduli scheme M which parametrizes affine G-schemes X

with the property that the coordinate ring of X decomposes,

as G-module, according to the function h. The talk will

be about joint work with Bart Van Steirteghem (New York)

which studies the moduli scheme M under some additional

assumptions.

### 2013年04月22日(月)

16:30-18:00 数理科学研究科棟(駒場) 118号室

Kodaira-Spencer classes, geometry of surfaces of general type and Torelli

theorem (ENGLISH)

**Professor Igor Reider 氏**(Universite d'Angers / RIMS)Kodaira-Spencer classes, geometry of surfaces of general type and Torelli

theorem (ENGLISH)

[ 講演概要 ]

In this talk I will explain a geometric interpretation of Kodaira-Spencer classes and apply

it to the study of the differential of the period map of weight 2 Hodge structures for surfaces

of general type.

My approach is based on interpreting Kodaira-Spencer classes as higher rank bundles and

then studing their stability. This naturally leads to two parts:

1) unstable case

2) stable case.

I will give a geometric characterization of the first case and show how to relate the second

case to a special family of vector bundles giving rise to a family of rational curves. This family

of rational curves is used to recover the surface in question.

In this talk I will explain a geometric interpretation of Kodaira-Spencer classes and apply

it to the study of the differential of the period map of weight 2 Hodge structures for surfaces

of general type.

My approach is based on interpreting Kodaira-Spencer classes as higher rank bundles and

then studing their stability. This naturally leads to two parts:

1) unstable case

2) stable case.

I will give a geometric characterization of the first case and show how to relate the second

case to a special family of vector bundles giving rise to a family of rational curves. This family

of rational curves is used to recover the surface in question.

### 2013年01月15日(火)

15:30-17:00 数理科学研究科棟(駒場) 128号室

いつもと曜日・場所が異なります

Three Dimensional Birational Geoemtry--updates and problems (ENGLISH)

いつもと曜日・場所が異なります

**Jungkai Alfred Chen 氏**(National Taiwan University)Three Dimensional Birational Geoemtry--updates and problems (ENGLISH)

[ 講演概要 ]

In this talk I will talk about some recent results on

biratioanl classification and biratioanl geoemtry of threefolds.

Given a threefold of general type, we improved our previous result by

showing that $Vol \\ge 1/1680$ and $|mK_X|$ is biratioanl for $m \\ge

61$.

Compare with the worst known example that $X_{46} \\subset

\\mathbb{P}(4,5,6,7,23)$, one also knows that there are only finiteley

many singularities type

for threefolds of general type with $1/1680 \\le Vol \\le 1/420$. It is

then intereting to study threefolds of general type with given basket

of singularities and with given fiber structure.

Concerning threefolds with intermediate Kodaira dimension, we

considered the effective Iitaka fibration. For this purpose, it is

interesting to study threefolds with $\\kappa=1$ with given basket of

singularities and abelian fibration.

For explicit birational geoemtry, we will show our result that each

biratioanl map in minimal model program can be factored into a

sequence of following maps (or its inverse)

1. a divisorial contraction to a point of index r with discrepancy 1/r.

2. a blowup along a smooth curve

3. a flop

In this talk I will talk about some recent results on

biratioanl classification and biratioanl geoemtry of threefolds.

Given a threefold of general type, we improved our previous result by

showing that $Vol \\ge 1/1680$ and $|mK_X|$ is biratioanl for $m \\ge

61$.

Compare with the worst known example that $X_{46} \\subset

\\mathbb{P}(4,5,6,7,23)$, one also knows that there are only finiteley

many singularities type

for threefolds of general type with $1/1680 \\le Vol \\le 1/420$. It is

then intereting to study threefolds of general type with given basket

of singularities and with given fiber structure.

Concerning threefolds with intermediate Kodaira dimension, we

considered the effective Iitaka fibration. For this purpose, it is

interesting to study threefolds with $\\kappa=1$ with given basket of

singularities and abelian fibration.

For explicit birational geoemtry, we will show our result that each

biratioanl map in minimal model program can be factored into a

sequence of following maps (or its inverse)

1. a divisorial contraction to a point of index r with discrepancy 1/r.

2. a blowup along a smooth curve

3. a flop

### 2012年12月13日(木)

10:40-12:10 数理科学研究科棟(駒場) 118号室

いつもと曜日・時間・場所が異なります

The asymptotic variety of polynomial maps (ENGLISH)

いつもと曜日・時間・場所が異なります

**Jean-Paul Brasselet 氏**(CNRS (Luminy))The asymptotic variety of polynomial maps (ENGLISH)

[ 講演概要 ]

The asymptotic variety, or set of non-properness has been intensively studied by Zbigniew Jelonek. In a recent paper, Anna and Guillaume Valette associate to a polynomial map $F: {\\mathbb C}^n \\to {\\mathbb C}^n$ a singular variety $N_F$ and relate properness property of $F$ to the vanishing of some intersection homology groups of $N_F$. I will explain how stratifications of the asymptotic variety of $F$ play an important role in the story and how recently, one of my students, Nguyen Thi Bich Thuy, found a nice way to exhibit such a suitable stratification.

The asymptotic variety, or set of non-properness has been intensively studied by Zbigniew Jelonek. In a recent paper, Anna and Guillaume Valette associate to a polynomial map $F: {\\mathbb C}^n \\to {\\mathbb C}^n$ a singular variety $N_F$ and relate properness property of $F$ to the vanishing of some intersection homology groups of $N_F$. I will explain how stratifications of the asymptotic variety of $F$ play an important role in the story and how recently, one of my students, Nguyen Thi Bich Thuy, found a nice way to exhibit such a suitable stratification.

### 2012年12月10日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

A hyperbolic metric and stability conditions on K3 surfaces with $¥rho=1$ (JAPANESE)

**川谷 康太郎 氏**(名古屋大学多元数理科学研究科)A hyperbolic metric and stability conditions on K3 surfaces with $¥rho=1$ (JAPANESE)

[ 講演概要 ]

We introduce a hyperbolic metric on the (normalized) space of stability conditions on projective K3 surfaces $X$ with Picard rank 1. Furthermore we demonstrate how this hyperbolic metric is helpful for us by discussing two or three topics.

We introduce a hyperbolic metric on the (normalized) space of stability conditions on projective K3 surfaces $X$ with Picard rank 1. Furthermore we demonstrate how this hyperbolic metric is helpful for us by discussing two or three topics.