## Algebraic Geometry Seminar

Seminar information archive ～12/08｜Next seminar｜Future seminars 12/09～

Date, time & place | Friday 13:30 - 15:00 ハイブリッド開催/117Room #ハイブリッド開催/117 (Graduate School of Math. Sci. Bldg.) |
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Organizer(s) | GONGYO Yoshinori, NAKAMURA Yusuke, TANAKA Hiromu |

**Seminar information archive**

### 2014/07/07

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

Balanced line bundles (JAPANESE)

**Sho Tanimoto**(Rice University)Balanced line bundles (JAPANESE)

[ Abstract ]

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.

### 2014/06/30

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

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

**Akiyoshi Sannai**(University of Tokyo)Invariant subrings of the Cox rings of K3surfaces by automorphism groups (JAPANESE)

[ Abstract ]

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?

### 2014/06/02

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

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

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

[ Abstract ]

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.

### 2014/05/12

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

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

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

[ Abstract ]

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.

### 2014/04/28

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

Syzygies of jacobian ideals and Torelli properties (ENGLISH)

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

[ Abstract ]

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.

### 2014/02/12

14:00-17:30 Room #122 (Graduate School of Math. Sci. Bldg.)

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

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

**Shin-ichi Matsumura**(Kagoshima University) 14:00-15:30An injectivity theorem with multiplier ideal sheaves of singular metrics with transcendental singularities (ENGLISH)

[ Abstract ]

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

[ Abstract ]

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.

### 2014/02/03

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

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

**Kento Fujita**(RIMS)Classification of log del Pezzo surfaces of index three (JAPANESE)

[ Abstract ]

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.

### 2014/01/22

15:00-16:30 Room #122 (Graduate School of Math. Sci. Bldg.)

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

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

[ Abstract ]

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.

### 2014/01/20

15:30-17:00 Room #126 (Graduate School of Math. Sci. Bldg.)

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

**Taro Sano**(University of Warwick)Deforming elephants of Q-Fano 3-folds (ENGLISH)

[ Abstract ]

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.

### 2013/12/09

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

On birationally tririgid Q-Fano threefolds (JAPANESE)

**Takuzo Okada**(Saga University)On birationally tririgid Q-Fano threefolds (JAPANESE)

[ Abstract ]

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.

### 2013/11/25

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

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

**Takayuki Koike**(The University of Tokyo)Minimal singular metrics of some line bundles with infinitely generated section rings (JAPANESE)

[ Abstract ]

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).

### 2013/11/18

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

An injectivity theorem (ENGLISH)

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

[ Abstract ]

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.

### 2013/11/11

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

Geography via the base loci (ENGLISH)

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

[ Abstract ]

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.

### 2013/10/28

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.

### 2013/07/22

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

Equivariant degenerations of spherical modules (ENGLISH)

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

[ Abstract ]

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.

### 2013/04/22

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

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

theorem (ENGLISH)

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

theorem (ENGLISH)

[ Abstract ]

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.

### 2013/01/15

15:30-17:00 Room #128 (Graduate School of Math. Sci. Bldg.)

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

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

[ Abstract ]

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

### 2012/12/13

10:40-12:10 Room #118 (Graduate School of Math. Sci. Bldg.)

The asymptotic variety of polynomial maps (ENGLISH)

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

[ Abstract ]

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.

### 2012/12/10

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

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

**Kotaro Kawatani**(Nagoya University)A hyperbolic metric and stability conditions on K3 surfaces with $¥rho=1$ (JAPANESE)

[ Abstract ]

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.

### 2012/11/26

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

A configuration of rational curves on the superspecial K3 surface (JAPANESE)

**Toshiyuki Katsura**(Hosei University)A configuration of rational curves on the superspecial K3 surface (JAPANESE)

### 2012/11/19

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

Stability conditions and birational geometry (JAPANESE)

**Yukinobu Toda**(IPMU)Stability conditions and birational geometry (JAPANESE)

[ Abstract ]

I propose a conjecture which claims that MMP for a smooth projective variety is realized as a variation of Bridgeland moduli spaces of semistable objects in the derived category of coherent sheaves. I will discuss the surface case and extremal contractions for 3-folds. In the former case, the conjecture is completely solved. In the latter case, I will construct the perverse t-structure associated to the extremal contraction, and construct a candidate of the desired stability condition as a double tilting of the perverse heart.

I propose a conjecture which claims that MMP for a smooth projective variety is realized as a variation of Bridgeland moduli spaces of semistable objects in the derived category of coherent sheaves. I will discuss the surface case and extremal contractions for 3-folds. In the former case, the conjecture is completely solved. In the latter case, I will construct the perverse t-structure associated to the extremal contraction, and construct a candidate of the desired stability condition as a double tilting of the perverse heart.

### 2012/11/12

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

On Fano fourfolds with nef vector bundles $Λ^2T_X$ (JAPANESE)

**Kazunori Yasutake**(Kyushu University)On Fano fourfolds with nef vector bundles $Λ^2T_X$ (JAPANESE)

[ Abstract ]

By using results about extremal contractions on smooth fourfolds, we give a classification of fano fourfolds whose the second exterior power of tangent bundles are numerically effective.

By using results about extremal contractions on smooth fourfolds, we give a classification of fano fourfolds whose the second exterior power of tangent bundles are numerically effective.

### 2012/11/05

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

The rationality of the moduli spaces of trigonal curves (JAPANESE)

**Shouhei Ma**(Nagoya University)The rationality of the moduli spaces of trigonal curves (JAPANESE)

### 2012/10/29

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

The Mukai conjecture for log Fano manifolds (JAPANESE)

**Kento Fujita**(RIMS)The Mukai conjecture for log Fano manifolds (JAPANESE)

[ Abstract ]

The concept of log Fano manifolds is one of the most natural generalization of the concept of Fano manifolds. We will give some structure theorems of log Fano manifolds. For example, we will show that the Mukai conjecture for Fano manifolds implies the `log Mukai conjecture' for log Fano manifolds.

The concept of log Fano manifolds is one of the most natural generalization of the concept of Fano manifolds. We will give some structure theorems of log Fano manifolds. For example, we will show that the Mukai conjecture for Fano manifolds implies the `log Mukai conjecture' for log Fano manifolds.

### 2012/10/15

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

On the moduli b-divisors of lc-trivial fibrations (JAPANESE)

**Yoshinori Gongyo**(University of Tokyo)On the moduli b-divisors of lc-trivial fibrations (JAPANESE)

[ Abstract ]

Roughly speaking, by using the semi-stable minimal model program, we prove that the moduli part of an lc-trivial fibration coincides with that of a klt-trivial fibration induced by adjunction after taking a suitable generically finite cover. As an application, we obtain that the moduli part of an lc-trivial fibration is b-nef and abundant by Ambro's result on klt-trivial fibrations. Moreover I may explain some applications of canonical bundle formulas. These are joint works with Osamu Fujino.

Roughly speaking, by using the semi-stable minimal model program, we prove that the moduli part of an lc-trivial fibration coincides with that of a klt-trivial fibration induced by adjunction after taking a suitable generically finite cover. As an application, we obtain that the moduli part of an lc-trivial fibration is b-nef and abundant by Ambro's result on klt-trivial fibrations. Moreover I may explain some applications of canonical bundle formulas. These are joint works with Osamu Fujino.