## Algebraic Geometry Seminar

Seminar information archive ～11/28｜Next seminar｜Future seminars 11/29～

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**

### 2010/06/21

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

Pseudo-index and minimal length of extremal rays for Fano manifolds (JAPANESE)

**Toru Tsukioka**(Osaka Prefecture University)Pseudo-index and minimal length of extremal rays for Fano manifolds (JAPANESE)

[ Abstract ]

The minimum of intersection numbers of the anticanonical

divisor with rational curves on a Fano manifold is called pseudo-index.

In view of the fact that the geometry of Fano manifolds is governed by

its extremal rays, it is important to consider the extremal rational

curves. In this talk, we show that for Fano 4-folds having birational

contractions, the minimal length of extremal rays coincides with the

pseudo-index.

The minimum of intersection numbers of the anticanonical

divisor with rational curves on a Fano manifold is called pseudo-index.

In view of the fact that the geometry of Fano manifolds is governed by

its extremal rays, it is important to consider the extremal rational

curves. In this talk, we show that for Fano 4-folds having birational

contractions, the minimal length of extremal rays coincides with the

pseudo-index.

### 2010/06/14

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

Slope of smooth rational curves in an anticanonically polarized Fano manifold (ENGLISH)

**Yongnam Lee**(Sogang University)Slope of smooth rational curves in an anticanonically polarized Fano manifold (ENGLISH)

[ Abstract ]

Ross and Thomas introduce the concept of slope stability to study K-stability, which has conjectural relation with the existence of constant scalar curvature metric. Since K-stability implies slope stability, slope stability gives an algebraic obstruction to theexistence of constant scalar curvature. This talk presents a systematic study of slope stability of anticanonically polarized Fano manifolds with respect to smooth rational curves. Especially, we prove that an anticanonically polarized Fano maniold is slope semistable with respect to any free smooth rational curves, and that an anticanonically polarized Fano threefold X with Picard number 1 is slope stable with respect to any smooth rational curves unless X is the project space. It is a joint work with Jun-Muk Hwang and Hosung Kim.

Ross and Thomas introduce the concept of slope stability to study K-stability, which has conjectural relation with the existence of constant scalar curvature metric. Since K-stability implies slope stability, slope stability gives an algebraic obstruction to theexistence of constant scalar curvature. This talk presents a systematic study of slope stability of anticanonically polarized Fano manifolds with respect to smooth rational curves. Especially, we prove that an anticanonically polarized Fano maniold is slope semistable with respect to any free smooth rational curves, and that an anticanonically polarized Fano threefold X with Picard number 1 is slope stable with respect to any smooth rational curves unless X is the project space. It is a joint work with Jun-Muk Hwang and Hosung Kim.

### 2010/06/07

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

Genus 2 curve configurations on Fano surfaces (ENGLISH)

**Xavier Roulleau**(The University of Tokyo)Genus 2 curve configurations on Fano surfaces (ENGLISH)

### 2010/05/31

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

On Pfaffian Calabi-Yau Varieties and Mirror Symmetry (JAPANESE)

**Atsushi Kanazawa**(The University of Tokyo)On Pfaffian Calabi-Yau Varieties and Mirror Symmetry (JAPANESE)

[ Abstract ]

We construct new smooth CY 3-folds with 1-dimensional Kaehler moduli and

determine their fundamental topological invariants. The existence of CY

3-folds with the computed invariants was previously conjectured. We then

report mirror symmetry for these non-complete intersection CY 3-folds.

We explicitly build their mirror partners, some of which have 2 LCSLs,

and carry out instanton computations for g=0,1.

We construct new smooth CY 3-folds with 1-dimensional Kaehler moduli and

determine their fundamental topological invariants. The existence of CY

3-folds with the computed invariants was previously conjectured. We then

report mirror symmetry for these non-complete intersection CY 3-folds.

We explicitly build their mirror partners, some of which have 2 LCSLs,

and carry out instanton computations for g=0,1.

### 2010/05/24

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

A counterexample of the birational Torelli problem via Fourier--Mukai transforms (JAPANESE)

**Hokuto Uehara**(Tokyo Metropolitan University)A counterexample of the birational Torelli problem via Fourier--Mukai transforms (JAPANESE)

[ Abstract ]

We study the Fourier--Mukai numbers of rational elliptic surfaces. As

its application, we give an example of a pair of minimal 3-folds $X$

with Kodaira dimensions 1, $h^1(O_X)=h^2(O_X)=0$ such that they are

mutually derived equivalent, deformation equivalent, but not

birationally equivalent. It also supplies a counterexample of the

birational Torelli problem.

We study the Fourier--Mukai numbers of rational elliptic surfaces. As

its application, we give an example of a pair of minimal 3-folds $X$

with Kodaira dimensions 1, $h^1(O_X)=h^2(O_X)=0$ such that they are

mutually derived equivalent, deformation equivalent, but not

birationally equivalent. It also supplies a counterexample of the

birational Torelli problem.

### 2010/05/17

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

On the GIT stability of Polarized Varieties (JAPANESE)

**Yuji Odaka**(Research Institute for Mathematical Sciences)On the GIT stability of Polarized Varieties (JAPANESE)

[ Abstract ]

Background:

Original GIT-stability notion for polarized variety is

"asymptotic stability", studied by Mumford, Gieseker etc around 1970s.

Recently a version appeared, so-called "K-stability", introduced by

Tian(1997) and reformulated by Donaldson(2002), by the way of seeking

the analogue of Kobayashi-Hitchin correspondence, which gives

"differential geometric" interpretation of "stability". These two have

subtle but interesting differences in dimension higher than 1.

Contents:

(1*) Any semistable (in any sense) polarized variety should have only

"semi-log-canonical" singularities. (Partly observed around 1970s)

(2) On the other hand, we proved some stabilities, which corresponds to

"Calabi conjecture", also with admitting mild singularities.

As applications these yield

(3*) Compact moduli spaces with GIT interpretations.

(4) Many counterexamples (as orbifolds) to folklore conjecture:

"K-stability implies asymptotic stability".

(*: Some technical points are yet to be settled.

Some parts for (1)(2) are available on arXiv:0910.1794.)

Background:

Original GIT-stability notion for polarized variety is

"asymptotic stability", studied by Mumford, Gieseker etc around 1970s.

Recently a version appeared, so-called "K-stability", introduced by

Tian(1997) and reformulated by Donaldson(2002), by the way of seeking

the analogue of Kobayashi-Hitchin correspondence, which gives

"differential geometric" interpretation of "stability". These two have

subtle but interesting differences in dimension higher than 1.

Contents:

(1*) Any semistable (in any sense) polarized variety should have only

"semi-log-canonical" singularities. (Partly observed around 1970s)

(2) On the other hand, we proved some stabilities, which corresponds to

"Calabi conjecture", also with admitting mild singularities.

As applications these yield

(3*) Compact moduli spaces with GIT interpretations.

(4) Many counterexamples (as orbifolds) to folklore conjecture:

"K-stability implies asymptotic stability".

(*: Some technical points are yet to be settled.

Some parts for (1)(2) are available on arXiv:0910.1794.)

### 2010/05/10

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

Toric degenerations of Grassmannians and mirror symmetry (JAPANESE)

**Makoto Miura**

(The University of Tokyo)Toric degenerations of Grassmannians and mirror symmetry (JAPANESE)

[ Abstract ]

I will talk about toric degenerarions of Grassmannians and

an application to the mirror constructions for complete intersection

Calabi-Yau manifolds in Grassmannians.

In particular, if we focus on toric degenerations by term orderings on

polynomial rings,

we have to choose a term ordering for which the coordinate ring has an

uniformly homogeneous sagbi basis.

We discuss this condition for some examples of ordinary Grassmannians

and a spinor variety.

I will talk about toric degenerarions of Grassmannians and

an application to the mirror constructions for complete intersection

Calabi-Yau manifolds in Grassmannians.

In particular, if we focus on toric degenerations by term orderings on

polynomial rings,

we have to choose a term ordering for which the coordinate ring has an

uniformly homogeneous sagbi basis.

We discuss this condition for some examples of ordinary Grassmannians

and a spinor variety.

### 2010/04/26

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

The unirationality of the moduli spaces of 2-elementary K3

surfaces (JAPANESE)

**Shouhei Ma**(The University of Tokyo)The unirationality of the moduli spaces of 2-elementary K3

surfaces (JAPANESE)

[ Abstract ]

We prove the unirationality of the moduli spaces of K3 surfaces

with non-symplectic involution. As a by-product, we describe the

configuration spaces of 5, 6, 7, 8 points in the projective plane as

arithmetic quotients of type IV.

We prove the unirationality of the moduli spaces of K3 surfaces

with non-symplectic involution. As a by-product, we describe the

configuration spaces of 5, 6, 7, 8 points in the projective plane as

arithmetic quotients of type IV.

### 2010/04/19

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

制限型体積と因子的ザリスキー分解

**松村 慎一**(東大数理)制限型体積と因子的ザリスキー分解

[ Abstract ]

豊富な因子の部分多様体に沿った自己交点数は, 基本的かつ重要である.

(部分多様体に沿った)自己交点数の巨大な因子への一般化である制限型体積は,

多くの状況で出現する重要な概念である.

様々な部分多様体に沿った制限型体積の振る舞いと

巨大な因子のザリスキー分解可能性の関係について考察したい.

また, 時間が許せば, 元々の問題意識であった制限型体積の複素解析的な側面に

ついても触れたい.

豊富な因子の部分多様体に沿った自己交点数は, 基本的かつ重要である.

(部分多様体に沿った)自己交点数の巨大な因子への一般化である制限型体積は,

多くの状況で出現する重要な概念である.

様々な部分多様体に沿った制限型体積の振る舞いと

巨大な因子のザリスキー分解可能性の関係について考察したい.

また, 時間が許せば, 元々の問題意識であった制限型体積の複素解析的な側面に

ついても触れたい.

### 2010/04/05

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

From Lang's Conjecture to finiteness properties of Torelli groups

**Alexandru Dimca**(Université Nice-Sophia Antipolis)From Lang's Conjecture to finiteness properties of Torelli groups

[ Abstract ]

First we recall one of Lang's conjectures in diophantine geometry

on the interplay between subvarieties and translated subgroups in a

commutative algebraic group

(proved by M. Laurent in the case of affine tori in 1984).

Then we present the technique of resonance and characteristic varieties,

a powerful tool in the study of fundamental groups of algebraic varieties.

Finally, using the two ingredients above, we show that the Torelli

groups $T_g$

have some surprising finiteness properties for $g>3$.

In particular, we show that for any subgroup $N$ in $T_g$ containing

the Johnson kernel $K_g$, the complex vector space $N_{ab} \\otimes C$

is finite dimensional.

All the details are available in our joint preprint with S. Papadima

arXiv:1002.0673.

First we recall one of Lang's conjectures in diophantine geometry

on the interplay between subvarieties and translated subgroups in a

commutative algebraic group

(proved by M. Laurent in the case of affine tori in 1984).

Then we present the technique of resonance and characteristic varieties,

a powerful tool in the study of fundamental groups of algebraic varieties.

Finally, using the two ingredients above, we show that the Torelli

groups $T_g$

have some surprising finiteness properties for $g>3$.

In particular, we show that for any subgroup $N$ in $T_g$ containing

the Johnson kernel $K_g$, the complex vector space $N_{ab} \\otimes C$

is finite dimensional.

All the details are available in our joint preprint with S. Papadima

arXiv:1002.0673.

### 2010/02/01

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

Extensions of two Chow stability criteria to positive characteristics

**大川 新之介**(東大数理)Extensions of two Chow stability criteria to positive characteristics

[ Abstract ]

I will talk about two results on Chow (semi-)stability of cycles in positive characteristics, which were originally known in characteristic 0. One is on the stability of non-singular projective hypersurfaces of degree greater than 2, and the other is the criterion by Y. Lee in terms of the log canonical threshold of Chow divisor. A couple of examples will be discussed in detail.

I will talk about two results on Chow (semi-)stability of cycles in positive characteristics, which were originally known in characteristic 0. One is on the stability of non-singular projective hypersurfaces of degree greater than 2, and the other is the criterion by Y. Lee in terms of the log canonical threshold of Chow divisor. A couple of examples will be discussed in detail.

### 2010/01/25

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

On weak Fano varieties with log canonical singularities

**權業 善範**(東大数理)On weak Fano varieties with log canonical singularities

[ Abstract ]

We prove that the anti-canonical divisors of weak Fano

3-folds with log canonical singularities are semiample. Moreover, we consider

semiampleness of the anti-log canonical divisor of any weak log Fano pair

with log canonical singularities. We show semiampleness dose not hold in

general by constructing several examples. Based on those examples, we propose

sufficient conditions which seem to be the best possible and we prove

semiampleness under such conditions. In particular we derive semiampleness of the

anti-canonical divisors of log canonical weak Fano 4-folds whose lc centers

are at most 1-dimensional. We also investigate the Kleiman-Mori cones of

weak log Fano pairs with log canonical singularities.

We prove that the anti-canonical divisors of weak Fano

3-folds with log canonical singularities are semiample. Moreover, we consider

semiampleness of the anti-log canonical divisor of any weak log Fano pair

with log canonical singularities. We show semiampleness dose not hold in

general by constructing several examples. Based on those examples, we propose

sufficient conditions which seem to be the best possible and we prove

semiampleness under such conditions. In particular we derive semiampleness of the

anti-canonical divisors of log canonical weak Fano 4-folds whose lc centers

are at most 1-dimensional. We also investigate the Kleiman-Mori cones of

weak log Fano pairs with log canonical singularities.

### 2010/01/18

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

The divisor class group of terminal Gorenstein Fano 3-folds and rationality questions

**Anne-Sophie Kaloghiros**(RIMS)The divisor class group of terminal Gorenstein Fano 3-folds and rationality questions

[ Abstract ]

Let Y be a quartic hypersurface in CP^4 with mild singularities, e.g. no worse than ordinary double points.

If Y contains a surface that is not a hyperplane section, Y is not Q-factorial and the divisor class group of Y, Cl Y, contains divisors that are not Cartier. However, the rank of Cl Y is bounded.

In this talk, I will show that in most cases, it is possible to describe explicitly the divisor class group Cl Y by running a Minimal Model Program (MMP) on X, a small Q-factorialisation of Y. In this case, the generators of Cl Y/ Pic Y are ``topological traces " of K-negative extremal contractions on X.

This has surprising consequences: it is possible to conclude that a number of families of non-factorial quartic 3-folds are rational.

In particular, I give some examples of rational quartic hypersurfaces Y_4\\subset CP^4 with rk Cl Y=2 and show that when the divisor class group of Y has sufficiently high rank, Y is always rational.

Let Y be a quartic hypersurface in CP^4 with mild singularities, e.g. no worse than ordinary double points.

If Y contains a surface that is not a hyperplane section, Y is not Q-factorial and the divisor class group of Y, Cl Y, contains divisors that are not Cartier. However, the rank of Cl Y is bounded.

In this talk, I will show that in most cases, it is possible to describe explicitly the divisor class group Cl Y by running a Minimal Model Program (MMP) on X, a small Q-factorialisation of Y. In this case, the generators of Cl Y/ Pic Y are ``topological traces " of K-negative extremal contractions on X.

This has surprising consequences: it is possible to conclude that a number of families of non-factorial quartic 3-folds are rational.

In particular, I give some examples of rational quartic hypersurfaces Y_4\\subset CP^4 with rk Cl Y=2 and show that when the divisor class group of Y has sufficiently high rank, Y is always rational.

### 2009/12/21

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

Ampleness of two-sided tilting complexes

**源 泰幸**(京都大学理学部数学教室)Ampleness of two-sided tilting complexes

[ Abstract ]

From the view point of noncommutative algebraic geometry (NCAG),

a two-sided tilting complex is an analog of a line bundle.

In this talk we introduce the notion of ampleness for two-sided

tilting complexes over finite dimensional algebras.

From the view point of NCAG, the Serre functors are considered to be

shifted canonical bundles. We show by examples that the property

of shifted canonical bundle captures some representation theoretic

property of algebras.

From the view point of noncommutative algebraic geometry (NCAG),

a two-sided tilting complex is an analog of a line bundle.

In this talk we introduce the notion of ampleness for two-sided

tilting complexes over finite dimensional algebras.

From the view point of NCAG, the Serre functors are considered to be

shifted canonical bundles. We show by examples that the property

of shifted canonical bundle captures some representation theoretic

property of algebras.

### 2009/12/14

14:40-16:10 Room #126 (Graduate School of Math. Sci. Bldg.)

Invariants of Fano varieties via quantum D-module

**Sergey Galkin**(IPMU)Invariants of Fano varieties via quantum D-module

[ Abstract ]

We will introduce and compute Apery characteristic

class and Frobenius genera - invariants of Fano variety derived from

it's Gromov-Witten invariants. Then we will show how to compute them

and relate with other invariants.

We will introduce and compute Apery characteristic

class and Frobenius genera - invariants of Fano variety derived from

it's Gromov-Witten invariants. Then we will show how to compute them

and relate with other invariants.

### 2009/11/16

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

Rationality of the Brauer-Severi Varieties of Skylanin algebras

**Colin Ingalls**(University of New Brunswick and RIMS)Rationality of the Brauer-Severi Varieties of Skylanin algebras

[ Abstract ]

Iskovskih's conjecture states that a conic bundle over

a surface is rational if and only if the surface has a pencil of

rational curves which meet the discriminant in 3 or fewer points,

(with one exceptional case). We generalize Iskovskih's proof that

such conic bundles are rational, to the case of projective space

bundles of higher dimension. The proof involves maximal orders

and toric geometry. As a corollary we show that the Brauer-Severi

variety of a Sklyanin algebra is rational.

Iskovskih's conjecture states that a conic bundle over

a surface is rational if and only if the surface has a pencil of

rational curves which meet the discriminant in 3 or fewer points,

(with one exceptional case). We generalize Iskovskih's proof that

such conic bundles are rational, to the case of projective space

bundles of higher dimension. The proof involves maximal orders

and toric geometry. As a corollary we show that the Brauer-Severi

variety of a Sklyanin algebra is rational.

### 2009/11/02

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

Cohomology of moduli spaces of curves and modular forms

**Gerard van der Geer**(Universiteit van Amsterdam)Cohomology of moduli spaces of curves and modular forms

[ Abstract ]

The Eichler-Shimura theorem expresses cohomology of local systems

on the moduli of elliptic curves in terms of modular forms. The

cohomology of local systems can be succesfully explored by counting

points over finite fields. We show how this can be applied to

obtain a lot of information about the cohomology of other moduli spaces

of low genera and also about Siegel modular forms of genus 2 and 3.

This is joint work with Jonas Bergstroem and Carel Faber.

The Eichler-Shimura theorem expresses cohomology of local systems

on the moduli of elliptic curves in terms of modular forms. The

cohomology of local systems can be succesfully explored by counting

points over finite fields. We show how this can be applied to

obtain a lot of information about the cohomology of other moduli spaces

of low genera and also about Siegel modular forms of genus 2 and 3.

This is joint work with Jonas Bergstroem and Carel Faber.

### 2009/10/19

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

ファノ多様体上の有理曲線の鎖の長さについて

**渡辺 究**(早稲田大学基幹理工学研究科)ファノ多様体上の有理曲線の鎖の長さについて

[ Abstract ]

ピカール数1のファノ多様体に対し、一般の二点を結ぶために必要な

極小有理曲線の本数を「長さ」と呼び、それについて考える。特に、5次元以下の

ファノ多様体や余指数が3以下のファノ多様体などに対し、長さを求める。

ピカール数1のファノ多様体に対し、一般の二点を結ぶために必要な

極小有理曲線の本数を「長さ」と呼び、それについて考える。特に、5次元以下の

ファノ多様体や余指数が3以下のファノ多様体などに対し、長さを求める。

### 2009/10/05

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

代数曲面上の随伴束の基底点集合について

**伊藤 敦**(東大数理)代数曲面上の随伴束の基底点集合について

[ Abstract ]

偏極付き代数多様体上(X,L)は、Lに数値的な条件を付け加えると

その随伴束が自由になったり、基底点集合が具体的にかけることがある。しかし

、曲線の場合は簡単であるが高次元の場合は難しい。今回の講演では主に代数曲

面の場合について解説する。

偏極付き代数多様体上(X,L)は、Lに数値的な条件を付け加えると

その随伴束が自由になったり、基底点集合が具体的にかけることがある。しかし

、曲線の場合は簡単であるが高次元の場合は難しい。今回の講演では主に代数曲

面の場合について解説する。

### 2009/09/01

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

Arithmetic of K3 surfaces

**Matthias Schuett**(Leibniz University Hannover)Arithmetic of K3 surfaces

[ Abstract ]

This talk aims to review recent developments in the arithmetic of K3 surfaces, with emphasis on singular K3 surfaces.

We will consider in particular modularity, Galois action on Neron-Severi groups and behaviour in families.

This talk aims to review recent developments in the arithmetic of K3 surfaces, with emphasis on singular K3 surfaces.

We will consider in particular modularity, Galois action on Neron-Severi groups and behaviour in families.

### 2009/07/13

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

Seshadri constants on rational surfaces with anticanonical pencils

**佐野 太郎**(東大数理)Seshadri constants on rational surfaces with anticanonical pencils

[ Abstract ]

射影多様体上の豊富線束の$k$-jet ample性を測る不変量として

Seshadri定数と呼ばれる正の実数がある。

この不変量を調べることでしばしば幾何的な情報が得られる。

今回、1次元以上の反標準線形系をもつ有理曲面上のSeshadri定数を計算する公式

が得られた。

その公式を使うと、対数del Pezzo曲面の特異点の情報をSeshadri定数の値から

復元できる。

射影多様体上の豊富線束の$k$-jet ample性を測る不変量として

Seshadri定数と呼ばれる正の実数がある。

この不変量を調べることでしばしば幾何的な情報が得られる。

今回、1次元以上の反標準線形系をもつ有理曲面上のSeshadri定数を計算する公式

が得られた。

その公式を使うと、対数del Pezzo曲面の特異点の情報をSeshadri定数の値から

復元できる。

### 2009/07/06

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

アーベル曲面上の安定層とフーリエ向井変換について

**柳田 伸太郎**(神戸大学理学研究科)アーベル曲面上の安定層とフーリエ向井変換について

[ Abstract ]

今回の講演は吉岡康太との共同研究に基づくものである. 研究の発端は, 向井茂が1980年前後(フーリエ向井変換の発見前後)に考察し, 当時の講演記録に書き残した主張や予想の解読にある.

本研究は, 大まかに言うと, 半等質層とフーリエ向井変換を用いて, アーベル曲面上の安定層のモジュライ空間の構造を調べるというものである.

アーベル曲面上には半等質層と呼ばれる半安定層があり, その分類, 構成方法やコホモロジーが完全に知られている. アーベル曲面のフーリエ向井対は半等質層のモジュライ空間であることも知られている.

今回の研究はこの半等質層をbulding blockとして一般の安定層を構成することを考える. その際に"semi-homogeneous presentation"という概念が必要になる. これはアーベル曲面上の安定層の半等質層によるある種の分解のことである. 曲面のピカール数が1の時, この種の分解の存在が安定層のチャーン指標のみを用いて判定できる.

また安定層のフーリエ変換における振舞いの記述において, 算術群や整数係数2次形式が重要な役割を果たすことも分かる. この事と先に述べた表示の存在から, 安定層のモジュライとアーベル曲面上の点のヒルベルトスキームとの間の双有理変換が明示的に構成できる.

アーベル曲面のフーリエ向井変換のフォーマリズムはK3曲面の変換と共通する部分も少なくない. 講演ではそうした点にも触れつつ, 今回の結果とその証明の概要を解説したい.

今回の講演は吉岡康太との共同研究に基づくものである. 研究の発端は, 向井茂が1980年前後(フーリエ向井変換の発見前後)に考察し, 当時の講演記録に書き残した主張や予想の解読にある.

本研究は, 大まかに言うと, 半等質層とフーリエ向井変換を用いて, アーベル曲面上の安定層のモジュライ空間の構造を調べるというものである.

アーベル曲面上には半等質層と呼ばれる半安定層があり, その分類, 構成方法やコホモロジーが完全に知られている. アーベル曲面のフーリエ向井対は半等質層のモジュライ空間であることも知られている.

今回の研究はこの半等質層をbulding blockとして一般の安定層を構成することを考える. その際に"semi-homogeneous presentation"という概念が必要になる. これはアーベル曲面上の安定層の半等質層によるある種の分解のことである. 曲面のピカール数が1の時, この種の分解の存在が安定層のチャーン指標のみを用いて判定できる.

また安定層のフーリエ変換における振舞いの記述において, 算術群や整数係数2次形式が重要な役割を果たすことも分かる. この事と先に述べた表示の存在から, 安定層のモジュライとアーベル曲面上の点のヒルベルトスキームとの間の双有理変換が明示的に構成できる.

アーベル曲面のフーリエ向井変換のフォーマリズムはK3曲面の変換と共通する部分も少なくない. 講演ではそうした点にも触れつつ, 今回の結果とその証明の概要を解説したい.

### 2009/06/29

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

Moduli on the projective plane and the wall-crossing

**大川 領**(東京工業大学)Moduli on the projective plane and the wall-crossing

[ Abstract ]

射影平面上の半安定層のモジュライ空間を、Bridgeland 安定性条件

を用いることにより、ある有限次元代数の半安定表現のモジュライ空間

として構成する。階数が2以下の場合、表現の安定性条件を変化させること

により、壁越え現象としてのflip の記述を得る。

応用として、flip のBetti 数などが計算できる。

射影平面上の半安定層のモジュライ空間を、Bridgeland 安定性条件

を用いることにより、ある有限次元代数の半安定表現のモジュライ空間

として構成する。階数が2以下の場合、表現の安定性条件を変化させること

により、壁越え現象としてのflip の記述を得る。

応用として、flip のBetti 数などが計算できる。

### 2009/06/23

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

Group actions on affine cones

**岸本崇氏**(埼玉大学理工学研究科)Group actions on affine cones

[ Abstract ]

The action of the additive group scheme C_+ on normal affine varieties is one of main subjects in affine algebraic geometry for a long time. In this talk, we shall mainly consider the problem about the existence of C_+-actions on affine cones, more precisely, the question:

"Determine the affine cones over smooth projective varieties admitting a (non-trivial) C_+-action ".

This question has an interest from a point of view of singularities. Indeed, a normal Cohen-Macaulay affine variety admitting an action by C_+ has at most rational singularities due to the result of H. Flenner and M. Zaidenberg. In the case of dimension 2, any affine cone over the projective line P^1 has a cyclic quotient singularity, and we can see that it admits, in fact, a C_+-action. Meanwhile, in case of dimension 3, i.e., affine cones over rational surfaces, the situation becomes more subtle.

One of the main results is concerned with a criterion for the existence of a C_+-action on affine cones (of any dimension) in terms of a cylinderlike open subset on the base variety. By making use of it, it is shown that, for any rational surface Y, we can take a suitable embedding of Y in such a way that the associated affine cone admits an action of C_+. Furthermore we are able to confirm that an affine cone over an anticanonically embedded del Pezzo surface of degree greater than or equal to 4 also admits such an action.

Nevertheless, our final purpose to decide whether or not there does exist a C_+-action on the fermat cubic: x^3+y^3+z^3+u^3 =0 in C^4, which is the affine cone over an anticanonically embedded cubic surface, say Y_3, is not yet accomplished. But, we can obtain certain informations about a linear pencil of rational curves on Y_3 arising from a C_+-action which seem to be useful in order to deny an existence of an action of C_+.

The action of the additive group scheme C_+ on normal affine varieties is one of main subjects in affine algebraic geometry for a long time. In this talk, we shall mainly consider the problem about the existence of C_+-actions on affine cones, more precisely, the question:

"Determine the affine cones over smooth projective varieties admitting a (non-trivial) C_+-action ".

This question has an interest from a point of view of singularities. Indeed, a normal Cohen-Macaulay affine variety admitting an action by C_+ has at most rational singularities due to the result of H. Flenner and M. Zaidenberg. In the case of dimension 2, any affine cone over the projective line P^1 has a cyclic quotient singularity, and we can see that it admits, in fact, a C_+-action. Meanwhile, in case of dimension 3, i.e., affine cones over rational surfaces, the situation becomes more subtle.

One of the main results is concerned with a criterion for the existence of a C_+-action on affine cones (of any dimension) in terms of a cylinderlike open subset on the base variety. By making use of it, it is shown that, for any rational surface Y, we can take a suitable embedding of Y in such a way that the associated affine cone admits an action of C_+. Furthermore we are able to confirm that an affine cone over an anticanonically embedded del Pezzo surface of degree greater than or equal to 4 also admits such an action.

Nevertheless, our final purpose to decide whether or not there does exist a C_+-action on the fermat cubic: x^3+y^3+z^3+u^3 =0 in C^4, which is the affine cone over an anticanonically embedded cubic surface, say Y_3, is not yet accomplished. But, we can obtain certain informations about a linear pencil of rational curves on Y_3 arising from a C_+-action which seem to be useful in order to deny an existence of an action of C_+.

### 2009/06/15

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

アーベル曲面の分解と2次形式

**馬 昭平氏**(東大数理)アーベル曲面の分解と2次形式

[ Abstract ]

複素Abel曲面が楕円曲線の積に分解可能である時、分解の仕方は一般に何通りも

ありうる。いくつかの場合に分解の個数公式が求められてきた(林田、塩田-三谷

)。本講演では、すべての分解可能な複素Abel曲面に対して、2次形式論の技法

を用いて分解数の公式を与える。関連して次のことも話す:合同モジュラー曲線

上のAtkin-Lehner対合の幾何学的意味;正定値2元2次形式の類数と判別式形式

の等長群の関係。

複素Abel曲面が楕円曲線の積に分解可能である時、分解の仕方は一般に何通りも

ありうる。いくつかの場合に分解の個数公式が求められてきた(林田、塩田-三谷

)。本講演では、すべての分解可能な複素Abel曲面に対して、2次形式論の技法

を用いて分解数の公式を与える。関連して次のことも話す:合同モジュラー曲線

上のAtkin-Lehner対合の幾何学的意味;正定値2元2次形式の類数と判別式形式

の等長群の関係。