## 代数幾何学セミナー

過去の記録 ～03/27｜次回の予定｜今後の予定 03/28～

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**

### 2016年07月25日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

今週は月曜日にセミナーがあります。また13:30--15:00と15:30--17:00に二つの講演があります。This week's seminar will be held on Monday, and we have two seminars from 13:30--15:00 and from 15:30--17:00.

On the geometry of thin exceptional sets in Manin’s conjecture

今週は月曜日にセミナーがあります。また13:30--15:00と15:30--17:00に二つの講演があります。This week's seminar will be held on Monday, and we have two seminars from 13:30--15:00 and from 15:30--17:00.

**谷本 祥 氏**(University of Copenhagen)On the geometry of thin exceptional sets in Manin’s conjecture

[ 講演概要 ]

Manin’s conjecture predicts the asymptotic formula for the counting function of rational points on a Fano variety X after removing the exceptional sets. The original conjecture, which removes a proper closed subset, is wrong due to covering families of subvarieties violating the compatibility of Manin’s conjecture, and its refinement, suggested by Emmanuel Peyre, removes a thin set instead of a closed set. In this talk, first I would like to explain that subvarieties which conjecturally have more points than X only form a thin set using the minimal model program and the boundedness of log Fano varieties. After that, I would like to discuss our conjecture on the birational boundedness of covers violating the compatibility of Manin’s conjecture, and present some results in dimension 2 and 3. This is joint work with Brian Lehmann.

Manin’s conjecture predicts the asymptotic formula for the counting function of rational points on a Fano variety X after removing the exceptional sets. The original conjecture, which removes a proper closed subset, is wrong due to covering families of subvarieties violating the compatibility of Manin’s conjecture, and its refinement, suggested by Emmanuel Peyre, removes a thin set instead of a closed set. In this talk, first I would like to explain that subvarieties which conjecturally have more points than X only form a thin set using the minimal model program and the boundedness of log Fano varieties. After that, I would like to discuss our conjecture on the birational boundedness of covers violating the compatibility of Manin’s conjecture, and present some results in dimension 2 and 3. This is joint work with Brian Lehmann.

### 2016年07月12日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

同じ日の13:30--15:00, 126室で、同講師による標数0の特異点解消の講義があります. We have a complimentary lecture by Matsuki-sensei on the resolution in characteristic 0 (from 13:30-15:00 at room#126).

Hypersurfaces of maximal contact and jumping phenomenon in the problem of resolution of singularities in positive characteristic (English)

https://www.math.purdue.edu/people/bio/kmatsuki/home

同じ日の13:30--15:00, 126室で、同講師による標数0の特異点解消の講義があります. We have a complimentary lecture by Matsuki-sensei on the resolution in characteristic 0 (from 13:30-15:00 at room#126).

**Kenji Matsuki 氏**(Purdue/RIMS)Hypersurfaces of maximal contact and jumping phenomenon in the problem of resolution of singularities in positive characteristic (English)

[ 講演概要 ]

According to our approach for resolution of singularities in positive characteristic (called the Idealistic Filtration Program, alias the I.F.P.) the algorithm is divided into the following two steps:

Step 1. Reduction of the general case to the monomial case.

Step 2. Solution in the monomial case.

While we have established Step 1 in abritrary dimension, Step 2 becomes very subtle and difficult in positive characteristic. This is in clear contrast to the classical setting in characteristic zero, where the solution in the monomial case is quite easy.

The talk consists of the two parts.

・Part I [13:30--15:00]: This part is mainly for the students, who are not familiar with the classical results in characteristic zero. Through Hironaka's reformulation of the problem of resolution of singularities, we will see how the notion of a hypersurface of maximal contact provides an inductive structure on dimension to the problem, and hence leading to a solution. Since our I.F.P. is closely modelled upon the classical algorithm in characteristic zero, this part should also give some background material and motivation for our approach in positive characteristic.

in

・Part II [15:30--17:00]: This is the main body of my talk. I will proceed according to the following menu.

{\bf Framewrok of the I.F.P.}: First I will explain the framewrok of the I.F.P., which further extends Hironaka's refomulation. The biggest obstacle to establish Step 1 is the fact that, in positive characteristic, a smooth hypersurface of maximal contact does not exist in general. In order to overcome this obstacle, we introduce the notion of the Leading Generator System, which is the collection of multiple singular hypersurfaces of maximal contcat.

{\bf Monomial Case}: As metioned above, then the problem is reduced to the one in the monomial case.

・ {\bf Inductive scheme on the invariant \boldmath$\tau$}: We firstly observe that, by the inductive scheme on the invariant $\tau$, we have only to consider the case with $\tau = 1$, i.e., the case where there is only one single singular hypersurface of maximal contact.

・ {\bf Tight Monomail Case}: We secondly observe that, if we reach the so-called Tight Monomial Case, then we can easily solve the problem.

・ {\bf Introduction of the invariant `` \boldmath$\mathrm{inv}_{\mathrm{MON},real}$''}: Thus our final task is, after arriving at the monimial case with $\tau = 1$, to reach the Tight Monomial Case, which is characterized by $\mathrm{inv}_{\mathrm{MON},real} = 0$.

・ {\bf Moh-Hauser Jumping phenomenon}: The invariant $\mathrm{inv}_{\mathrm{MON},real}$ usually behaves well, i.e., decreases after each blow up. But under some circustances, it strictly increases. I will explain this well-known Moh-Jumping phenomenon by giving a simple example.

・ {\bf Eventual decrease of the jumping peaks}: At last, the problem boils down to analyzing and overcoming the Moh-Hauser Jumping phenomenon. For this purpose, we will present the conjecture of ``Eventual decrease of the jumping peaks'', which is affirmatively solved in dimension 3, and is the current focus of our research in dimension 4.

[ 参考URL ]According to our approach for resolution of singularities in positive characteristic (called the Idealistic Filtration Program, alias the I.F.P.) the algorithm is divided into the following two steps:

Step 1. Reduction of the general case to the monomial case.

Step 2. Solution in the monomial case.

While we have established Step 1 in abritrary dimension, Step 2 becomes very subtle and difficult in positive characteristic. This is in clear contrast to the classical setting in characteristic zero, where the solution in the monomial case is quite easy.

The talk consists of the two parts.

・Part I [13:30--15:00]: This part is mainly for the students, who are not familiar with the classical results in characteristic zero. Through Hironaka's reformulation of the problem of resolution of singularities, we will see how the notion of a hypersurface of maximal contact provides an inductive structure on dimension to the problem, and hence leading to a solution. Since our I.F.P. is closely modelled upon the classical algorithm in characteristic zero, this part should also give some background material and motivation for our approach in positive characteristic.

in

・Part II [15:30--17:00]: This is the main body of my talk. I will proceed according to the following menu.

{\bf Framewrok of the I.F.P.}: First I will explain the framewrok of the I.F.P., which further extends Hironaka's refomulation. The biggest obstacle to establish Step 1 is the fact that, in positive characteristic, a smooth hypersurface of maximal contact does not exist in general. In order to overcome this obstacle, we introduce the notion of the Leading Generator System, which is the collection of multiple singular hypersurfaces of maximal contcat.

{\bf Monomial Case}: As metioned above, then the problem is reduced to the one in the monomial case.

・ {\bf Inductive scheme on the invariant \boldmath$\tau$}: We firstly observe that, by the inductive scheme on the invariant $\tau$, we have only to consider the case with $\tau = 1$, i.e., the case where there is only one single singular hypersurface of maximal contact.

・ {\bf Tight Monomail Case}: We secondly observe that, if we reach the so-called Tight Monomial Case, then we can easily solve the problem.

・ {\bf Introduction of the invariant `` \boldmath$\mathrm{inv}_{\mathrm{MON},real}$''}: Thus our final task is, after arriving at the monimial case with $\tau = 1$, to reach the Tight Monomial Case, which is characterized by $\mathrm{inv}_{\mathrm{MON},real} = 0$.

・ {\bf Moh-Hauser Jumping phenomenon}: The invariant $\mathrm{inv}_{\mathrm{MON},real}$ usually behaves well, i.e., decreases after each blow up. But under some circustances, it strictly increases. I will explain this well-known Moh-Jumping phenomenon by giving a simple example.

・ {\bf Eventual decrease of the jumping peaks}: At last, the problem boils down to analyzing and overcoming the Moh-Hauser Jumping phenomenon. For this purpose, we will present the conjecture of ``Eventual decrease of the jumping peaks'', which is affirmatively solved in dimension 3, and is the current focus of our research in dimension 4.

https://www.math.purdue.edu/people/bio/kmatsuki/home

### 2016年07月05日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Generalized Bogomolov-Gieseker type inequality for Fano 3-folds (English)

**Dulip Piyaratne 氏**(IPMU)Generalized Bogomolov-Gieseker type inequality for Fano 3-folds (English)

[ 講演概要 ]

Construction of Bridgeland stability conditions on a given smooth projective 3-fold is an important problem. A conjectural construction for any 3-fold was introduced by Bayer, Macri and Toda, and the problem is reduced to proving so-called Bogomolov-Gieseker type inequality holds for certain stable objects in the derived category. It has been shown to hold for Fano 3-folds of Picard rank one due to the works of Macri, Schmidt and Li. However, Schmidt gave a counter-example for a Fano 3-fold of higher Picard rank. In this talk, I will explain how to modify the original conjectural inequality for general Fano 3-folds and why it holds.

Construction of Bridgeland stability conditions on a given smooth projective 3-fold is an important problem. A conjectural construction for any 3-fold was introduced by Bayer, Macri and Toda, and the problem is reduced to proving so-called Bogomolov-Gieseker type inequality holds for certain stable objects in the derived category. It has been shown to hold for Fano 3-folds of Picard rank one due to the works of Macri, Schmidt and Li. However, Schmidt gave a counter-example for a Fano 3-fold of higher Picard rank. In this talk, I will explain how to modify the original conjectural inequality for general Fano 3-folds and why it holds.

### 2016年06月27日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

いつもと曜日が異なります。This week's seminar will be held on Monday, not on Tuesday.

Generic vanishing and birational geometry in char p>0 (ENGLISH)

http://www.math.utah.edu/~hacon/

いつもと曜日が異なります。This week's seminar will be held on Monday, not on Tuesday.

**Christopher Hacon 氏**(University of Utah)Generic vanishing and birational geometry in char p>0 (ENGLISH)

[ 講演概要 ]

Many precise results on the birational geometry of irregular varieties have been obtained by combining the generic vanishing theorems of Green and Lazarsfeld with the Fourier-Mukai transform. In this talk we will discuss the failure of the generic vanishing theorems of Green and Lazarsfeld in positive characteristic. We will then explain a different approach to generic vanishing based on the theory of F-singularities that leads to concrete applications in birational geometry in positive characteristics

[ 参考URL ]Many precise results on the birational geometry of irregular varieties have been obtained by combining the generic vanishing theorems of Green and Lazarsfeld with the Fourier-Mukai transform. In this talk we will discuss the failure of the generic vanishing theorems of Green and Lazarsfeld in positive characteristic. We will then explain a different approach to generic vanishing based on the theory of F-singularities that leads to concrete applications in birational geometry in positive characteristics

http://www.math.utah.edu/~hacon/

### 2016年06月20日(月)

16:30-18:00 数理科学研究科棟(駒場) 123号室

普段と曜日、時間、部屋が異なります。The day of the week, the time and the room are different from usual.

BUILDING BLOCKS OF POLARIZED ENDOMORPHISMS OF NORMAL PROJECTIVE VARIETIES (English)

http://www.math.nus.edu.sg/~matzdq/

普段と曜日、時間、部屋が異なります。The day of the week, the time and the room are different from usual.

**De-Qi Zhang 氏**(National University of Singapore)BUILDING BLOCKS OF POLARIZED ENDOMORPHISMS OF NORMAL PROJECTIVE VARIETIES (English)

[ 講演概要 ]

An endomorphism f of a normal projective variety X is polarized if f∗H ∼ qH for some ample Cartier divisor H and integer q > 1.

We first assert that a suitable maximal rationally connected fibration (MRC) can be made f-equivariant using a construction of N. Nakayama, that f descends to a polarized endomorphism of the base Y of this MRC and that this Y is a Q-abelian variety (quasi- ́etale quotient of an abelian variety). Next we show that we can run the minimal model program (MMP) f-equivariantly for mildly singular X and reach either a Q-abelian variety or a Fano variety of Picard number one.

As a consequence, the building blocks of polarized endomorphisms are those of Q- abelian varieties and those of Fano varieties of Picard number one.

Along the way, we show that f always descends to a polarized endomorphism of the Albanese variety Alb(X) of X, and that a power of f acts as a scalar on the Neron-Severi group of X (modulo torsion) when X is smooth and rationally connected.

Partial answers about X being of Calabi-Yau type or Fano type are also given with an extra primitivity assumption on f which seems necessary by an example.

This is a joint work with S. Meng.

[ 参考URL ]An endomorphism f of a normal projective variety X is polarized if f∗H ∼ qH for some ample Cartier divisor H and integer q > 1.

We first assert that a suitable maximal rationally connected fibration (MRC) can be made f-equivariant using a construction of N. Nakayama, that f descends to a polarized endomorphism of the base Y of this MRC and that this Y is a Q-abelian variety (quasi- ́etale quotient of an abelian variety). Next we show that we can run the minimal model program (MMP) f-equivariantly for mildly singular X and reach either a Q-abelian variety or a Fano variety of Picard number one.

As a consequence, the building blocks of polarized endomorphisms are those of Q- abelian varieties and those of Fano varieties of Picard number one.

Along the way, we show that f always descends to a polarized endomorphism of the Albanese variety Alb(X) of X, and that a power of f acts as a scalar on the Neron-Severi group of X (modulo torsion) when X is smooth and rationally connected.

Partial answers about X being of Calabi-Yau type or Fano type are also given with an extra primitivity assumption on f which seems necessary by an example.

This is a joint work with S. Meng.

http://www.math.nus.edu.sg/~matzdq/

### 2016年06月20日(月)

14:45-16:15 数理科学研究科棟(駒場) 123号室

普段と曜日、時間、部屋が異なります。The day of the week, the time and the room are different from usual.

Fujita's freeness conjecture for 5-fold (English)

普段と曜日、時間、部屋が異なります。The day of the week, the time and the room are different from usual.

**Zhixian Zhu 氏**(KIAS)Fujita's freeness conjecture for 5-fold (English)

[ 講演概要 ]

Let X be a smooth projective variety of dimension n and L any ample line bundle. Fujita conjectured that the adjoint line bundle O(K_X+mL) is globally generated for any m greater or equal to dimX+1. By studying the singularity of pairs, we prove Fujita's freeness conjecture for smooth 5-folds.

Let X be a smooth projective variety of dimension n and L any ample line bundle. Fujita conjectured that the adjoint line bundle O(K_X+mL) is globally generated for any m greater or equal to dimX+1. By studying the singularity of pairs, we prove Fujita's freeness conjecture for smooth 5-folds.

### 2016年05月31日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

A Characterization of Symplectic Grassmannians (JAPANESE)

**渡邉 究 氏**(埼玉大理)A Characterization of Symplectic Grassmannians (JAPANESE)

[ 講演概要 ]

In the series of their works, J. M. Hwang and N. Mok have been developing the theory of Varieties of Minimal Rational Tangents (VMRT for short). In this direction, the results of Mok and J. Hong-Hwang allow us to recognize a homogeneous Fano manifold X of Picard number one by looking at its VMRT at a general point. This characterization works for all rational homogeneous manifolds of Picard number one whenever the VMRT is rational homogeneous, which is always the case except for the short root cases; namely for symplectic Grassmannians, and for two varieties of type F*4*.

In this talk we show that, if we impose that the VMRT is the expected one at every point of the variety, we may still characterize symplectic Grassmannians. This is a joint work with G. Occhetta and L. E. Sola Conde (arXiv:1604.06867).

In the series of their works, J. M. Hwang and N. Mok have been developing the theory of Varieties of Minimal Rational Tangents (VMRT for short). In this direction, the results of Mok and J. Hong-Hwang allow us to recognize a homogeneous Fano manifold X of Picard number one by looking at its VMRT at a general point. This characterization works for all rational homogeneous manifolds of Picard number one whenever the VMRT is rational homogeneous, which is always the case except for the short root cases; namely for symplectic Grassmannians, and for two varieties of type F*4*.

In this talk we show that, if we impose that the VMRT is the expected one at every point of the variety, we may still characterize symplectic Grassmannians. This is a joint work with G. Occhetta and L. E. Sola Conde (arXiv:1604.06867).

### 2016年05月24日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

ON PSEUDO KOBAYASHI HYPERBOLICITY OF SUBVARIETIES OF ABELIAN VARIETIES

(tba)

**山ノ井克俊 氏**(大阪大学)ON PSEUDO KOBAYASHI HYPERBOLICITY OF SUBVARIETIES OF ABELIAN VARIETIES

(tba)

[ 講演概要 ]

We prove that the Kobayashi pseudo distance of a closed subvariety X of an abelian variety A is a true distance outside the special set Sp(X) of X, where Sp(X) is the union of all positive dimensional translated abelian subvarieties of A which are contained in X. More strongly, we prove that a closed subvariety X of an abelian variety is taut modulo Sp(X); Every sequence fn : D → X of holomorphic mappings from the unit disc D admits a subsequence which converges locally uniformly, unless the image fn(K) of a fixed compact set K of D eventually gets arbitrarily close to Sp(X) as n gets larger. These generalize a classical theorem on algebraic degeneracy of entire holomorphic curves in irregular varieties.

We prove that the Kobayashi pseudo distance of a closed subvariety X of an abelian variety A is a true distance outside the special set Sp(X) of X, where Sp(X) is the union of all positive dimensional translated abelian subvarieties of A which are contained in X. More strongly, we prove that a closed subvariety X of an abelian variety is taut modulo Sp(X); Every sequence fn : D → X of holomorphic mappings from the unit disc D admits a subsequence which converges locally uniformly, unless the image fn(K) of a fixed compact set K of D eventually gets arbitrarily close to Sp(X) as n gets larger. These generalize a classical theorem on algebraic degeneracy of entire holomorphic curves in irregular varieties.

### 2016年05月17日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

On dual defects of toric varieties (TBA)

https://sites.google.com/site/atsushiito221/

**伊藤敦 氏**(京大数学教室)On dual defects of toric varieties (TBA)

[ 講演概要 ]

For a projective variety embedded in a projective space,

we can define the dual variety in the dual projective space.

By dimension count, the codimension of the dual variety is expected to be one,

but it can be greater than one for some varieties.

For a smooth toric variety, it is known that the codimension of the dual variety is greater than one

if and only if the toric variety is a suitable projective bundle over some toric variety.

In this talk, I will explain a generalization of this result to toric varieties without the assumption of singularities.

This is a joint work with Katsuhisa Furukawa.

[ 参考URL ]For a projective variety embedded in a projective space,

we can define the dual variety in the dual projective space.

By dimension count, the codimension of the dual variety is expected to be one,

but it can be greater than one for some varieties.

For a smooth toric variety, it is known that the codimension of the dual variety is greater than one

if and only if the toric variety is a suitable projective bundle over some toric variety.

In this talk, I will explain a generalization of this result to toric varieties without the assumption of singularities.

This is a joint work with Katsuhisa Furukawa.

https://sites.google.com/site/atsushiito221/

### 2016年04月26日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

This talk is given in Japanese due to the speaker's intension.

A gentle introduction to K-stability and its recent development (Japanese)

https://sites.google.com/site/yujiodaka2013/

This talk is given in Japanese due to the speaker's intension.

**尾高悠志 氏**(京大数学教室)A gentle introduction to K-stability and its recent development (Japanese)

[ 講演概要 ]

K安定性とは複素代数多様体上の「標準的な」ケーラー計量の存在問題に端を発する，代数幾何的な概念です．二木先生や満渕先生等の先駆的な仕事に感化されて導入され，特に近年ホットに研究され始めている一方，未だその大半はより微分幾何的な研究者の方々や背景の中でなされているように講演者には感じられます．

代数幾何的にもどのように面白いか，どういった意義があるかに私見で軽く触れた上で，その基礎付けをより拡張した枠組みで説明しつつ，最先端でどのようなことが問題になっているかをいくらか（私の力量と時間の許す限り）解説しつつ，文献をご紹介できればと思っています

[ 参考URL ]K安定性とは複素代数多様体上の「標準的な」ケーラー計量の存在問題に端を発する，代数幾何的な概念です．二木先生や満渕先生等の先駆的な仕事に感化されて導入され，特に近年ホットに研究され始めている一方，未だその大半はより微分幾何的な研究者の方々や背景の中でなされているように講演者には感じられます．

代数幾何的にもどのように面白いか，どういった意義があるかに私見で軽く触れた上で，その基礎付けをより拡張した枠組みで説明しつつ，最先端でどのようなことが問題になっているかをいくらか（私の力量と時間の許す限り）解説しつつ，文献をご紹介できればと思っています

https://sites.google.com/site/yujiodaka2013/

### 2016年04月19日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Isomorphic quartic K3 surfaces and Cremona transformations (JAPANESE)

**小木曽 啓示 氏**(東京大学大学院数理科学研究科)Isomorphic quartic K3 surfaces and Cremona transformations (JAPANESE)

[ 講演概要 ]

We show that

(i) there is a pair of smooth complex quartic K3 surfaces such that they are isomorphic as abstract varieties but not Cremona equivalent.

(ii) there is a pair of smooth complex quartic K3 surfaces such that they are Cemona equivalent but not projectively equivalent.

These two results are much inspired by e-mails from Professors Tuyen Truong and J\'anos Koll\'ar.

We show that

(i) there is a pair of smooth complex quartic K3 surfaces such that they are isomorphic as abstract varieties but not Cremona equivalent.

(ii) there is a pair of smooth complex quartic K3 surfaces such that they are Cemona equivalent but not projectively equivalent.

These two results are much inspired by e-mails from Professors Tuyen Truong and J\'anos Koll\'ar.

### 2016年04月11日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Gysin maps, duality and Schubert classes

(English)

https://www.impan.pl/~pragacz/main.htm

**Piotr Pragacz 氏**(Institute of Mathematics, Polish Academy of Sciences )Gysin maps, duality and Schubert classes

(English)

[ 講演概要 ]

We establish a Gysin formula for Schubert bundles

and a strong version of the duality theorem in Schubert calculus

on Grassmann bundles. We then combine them to compute the fundamental

classes of Schubert bundles in Grassmann bundles, which yields a new

proof of the Giambelli formula for vector bundles. This is a joint

work with Lionel Darondeau.

[ 参考URL ]We establish a Gysin formula for Schubert bundles

and a strong version of the duality theorem in Schubert calculus

on Grassmann bundles. We then combine them to compute the fundamental

classes of Schubert bundles in Grassmann bundles, which yields a new

proof of the Giambelli formula for vector bundles. This is a joint

work with Lionel Darondeau.

https://www.impan.pl/~pragacz/main.htm

### 2015年12月17日(木)

15:30-17:00 数理科学研究科棟(駒場) 122号室

このセミナーは講演者が急病のためキャンセルになりました。This seminer is canceled due to the speaker 's sick.

Polarization and stability on a derived equivalent abelian variety (English)

http://db.ipmu.jp/member/personal/3989en.html

このセミナーは講演者が急病のためキャンセルになりました。This seminer is canceled due to the speaker 's sick.

**Dulip Piyaratne 氏**(IPMU)Polarization and stability on a derived equivalent abelian variety (English)

[ 講演概要 ]

In this talk I will explain how one can define a polarization on a derived equivalent abelian variety by using Fourier-Mukai theory. Furthermore, we see how such a realisations is connected with stability conditions on their derived categories. Then I will discuss these ideas for abelian surfaces and abelian 3-folds in detail.

[ 参考URL ]In this talk I will explain how one can define a polarization on a derived equivalent abelian variety by using Fourier-Mukai theory. Furthermore, we see how such a realisations is connected with stability conditions on their derived categories. Then I will discuss these ideas for abelian surfaces and abelian 3-folds in detail.

http://db.ipmu.jp/member/personal/3989en.html

### 2015年12月14日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Extending Hori-Vafa toric mirror symmetry via SYZ and modular forms (English)

**金沢篤 氏**(ハーバード大学)Extending Hori-Vafa toric mirror symmetry via SYZ and modular forms (English)

[ 講演概要 ]

In this talk, I will introduce partial compactification for a class of toric Calabi-Yau manifolds. A fundamental question is how the Hori-Vafa toric mirror symmetry extends to this new class of Calabi-Yau manifolds. The answer leads us to a new connection between SYZ mirror symmetry and modular forms. If time permits, I will also discuss higher dimensional analogues of the Yau-Zaslow formula (for an elliptic K3 surface) in terms of Siegel modular forms. This talk is based on a joint work with Siu-Cheong Lau.

In this talk, I will introduce partial compactification for a class of toric Calabi-Yau manifolds. A fundamental question is how the Hori-Vafa toric mirror symmetry extends to this new class of Calabi-Yau manifolds. The answer leads us to a new connection between SYZ mirror symmetry and modular forms. If time permits, I will also discuss higher dimensional analogues of the Yau-Zaslow formula (for an elliptic K3 surface) in terms of Siegel modular forms. This talk is based on a joint work with Siu-Cheong Lau.

### 2015年12月07日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Flops and spherical functors (English)

**Alexey Bondal 氏**(IPMU)Flops and spherical functors (English)

[ 講演概要 ]

I will describe various functors on derived categories of coherent sheaves

related to flops and relations between these functors. A categorical

version of deformation theory of systems of objects in abelian categories

will be outlined and its relation to flop spherical functors will be

presented.

I will describe various functors on derived categories of coherent sheaves

related to flops and relations between these functors. A categorical

version of deformation theory of systems of objects in abelian categories

will be outlined and its relation to flop spherical functors will be

presented.

### 2015年11月30日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Interesting surfaces which are coverings of a rational surface branched over few lines (English)

**Fabrizio Catanese 氏**(Universität Bayreuth)Interesting surfaces which are coverings of a rational surface branched over few lines (English)

[ 講演概要 ]

Surfaces which are covers of the plane branched over 5 or 6 lines have provided answers to long standing questions, for instance the BCD surfaces for Fujita's question on semiampleness of VHS (Dettweiler-Cat); and examples of ball quotients (Hirzebruch), automorphisms acting trivially on integral cohomology (Cat-Gromadtzki), canonical maps with high degree or image-degree (Pardini, Bauer-Cat). I shall speak especially about the above Abelian coverings of the plane, the geometry of the del Pezzo surface of degree 5, the rigidity of BCD surfaces, and a criterion for a fibred surface to be a projective classifying space.

Surfaces which are covers of the plane branched over 5 or 6 lines have provided answers to long standing questions, for instance the BCD surfaces for Fujita's question on semiampleness of VHS (Dettweiler-Cat); and examples of ball quotients (Hirzebruch), automorphisms acting trivially on integral cohomology (Cat-Gromadtzki), canonical maps with high degree or image-degree (Pardini, Bauer-Cat). I shall speak especially about the above Abelian coverings of the plane, the geometry of the del Pezzo surface of degree 5, the rigidity of BCD surfaces, and a criterion for a fibred surface to be a projective classifying space.

### 2015年11月16日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Counting curves on surface in Calabi-Yau threefolds and the proof of S-duality modularity conjecture (English)

**Artan Sheshmani 氏**(IPMU/ Ohio State University)Counting curves on surface in Calabi-Yau threefolds and the proof of S-duality modularity conjecture (English)

[ 講演概要 ]

I will talk about recent joint works with Amin Gholampour, Richard Thomas and Yukinobu Toda, on an algebraic-geometric proof of the S-duality conjecture in superstring theory, made formerly by physicists Gaiotto, Strominger, Yin, regarding the modularity of DT invariants of sheaves supported on hyperplane sections of the quintic Calabi-Yau threefold. Our strategy is to first use degeneration and localization techniques to reduce the threefold theory to a certain intersection theory over the relative Hilbert scheme of points on surfaces and then prove modularity; More precisely, together with Gholampour we have proven that the generating series, associated to the top intersection numbers of the Hilbert scheme of points, relative to an effective divisor, on a smooth quasi-projective surface is a modular form. This is a generalization of the result of Okounkov-Carlsson, where they used representation theory and the machinery of vertex operators to prove this statement for absolute Hilbert schemes. These intersection numbers eventually, together with the generating series of Noether-Lefschetz numbers as I will explain, will provide the ingredients to achieve a complete algebraic-geometric proof of S-duality modularity conjecture.

I will talk about recent joint works with Amin Gholampour, Richard Thomas and Yukinobu Toda, on an algebraic-geometric proof of the S-duality conjecture in superstring theory, made formerly by physicists Gaiotto, Strominger, Yin, regarding the modularity of DT invariants of sheaves supported on hyperplane sections of the quintic Calabi-Yau threefold. Our strategy is to first use degeneration and localization techniques to reduce the threefold theory to a certain intersection theory over the relative Hilbert scheme of points on surfaces and then prove modularity; More precisely, together with Gholampour we have proven that the generating series, associated to the top intersection numbers of the Hilbert scheme of points, relative to an effective divisor, on a smooth quasi-projective surface is a modular form. This is a generalization of the result of Okounkov-Carlsson, where they used representation theory and the machinery of vertex operators to prove this statement for absolute Hilbert schemes. These intersection numbers eventually, together with the generating series of Noether-Lefschetz numbers as I will explain, will provide the ingredients to achieve a complete algebraic-geometric proof of S-duality modularity conjecture.

### 2015年11月09日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

3-dimensional McKay correspondence (English)

**伊藤由佳理 氏**(名古屋大学)3-dimensional McKay correspondence (English)

[ 講演概要 ]

The original McKay correspondence is a relation between group theory of a finite subgroup G of SL(2,C) and geometry of the minimal resolution of the quotient singularity by G, and was generalized several ways. In particular, 3-dimensional generalization was extended to derived categorical eqivalence and the G-Hilbert scheme was useful to explain the correspondence. However, most results hold only for abelian subgroups. In this talk, I would like to introduce an iterated G-Hilbert scheme and show more geometrical McKay correspondence for non-abelian subgroups.

The original McKay correspondence is a relation between group theory of a finite subgroup G of SL(2,C) and geometry of the minimal resolution of the quotient singularity by G, and was generalized several ways. In particular, 3-dimensional generalization was extended to derived categorical eqivalence and the G-Hilbert scheme was useful to explain the correspondence. However, most results hold only for abelian subgroups. In this talk, I would like to introduce an iterated G-Hilbert scheme and show more geometrical McKay correspondence for non-abelian subgroups.

### 2015年11月05日(木)

15:30-17:00 数理科学研究科棟(駒場) 126号室

いつもと部屋と曜日が違います。The day of the week and room are different from usual.

Compact moduli of marked noncommutative del Pezzo surfaces via quivers (English)

いつもと部屋と曜日が違います。The day of the week and room are different from usual.

**大川新之介 氏**(阪大)Compact moduli of marked noncommutative del Pezzo surfaces via quivers (English)

[ 講演概要 ]

I will introduce certain GIT construction via quivers of compactified moduli spaces of marked noncommutative del Pezzo surfaces. For projective plane, quadric surface, and those of degree 3, 2, 1, we obtain projective toric varieties of dimension 2, 3, 8, 9, 10, respectively. Then I will discuss relations with deformation theory of abelian categories, blow-up of noncommutative projective planes, and three-block exceptional collections due to Karpov and Nogin. This talk is based on joint works in progress with Tarig Abdelgadir and Kazushi Ueda.

I will introduce certain GIT construction via quivers of compactified moduli spaces of marked noncommutative del Pezzo surfaces. For projective plane, quadric surface, and those of degree 3, 2, 1, we obtain projective toric varieties of dimension 2, 3, 8, 9, 10, respectively. Then I will discuss relations with deformation theory of abelian categories, blow-up of noncommutative projective planes, and three-block exceptional collections due to Karpov and Nogin. This talk is based on joint works in progress with Tarig Abdelgadir and Kazushi Ueda.

### 2015年10月26日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Asymptotic syzygies and the gonality conjecture (English)

**Lawrence Ein 氏**(University of Illinois at Chicago)Asymptotic syzygies and the gonality conjecture (English)

[ 講演概要 ]

We'll discuss my joint work with Lazarsfeld on the gonality conjecture about the syzygies of a smooth projective curve when it is embedded into the projective space by the complete linear system of a sufficiently very ample line bundles. We'll also discuss some results about the asymptotic syzygies f higher dimensional varieties.

We'll discuss my joint work with Lazarsfeld on the gonality conjecture about the syzygies of a smooth projective curve when it is embedded into the projective space by the complete linear system of a sufficiently very ample line bundles. We'll also discuss some results about the asymptotic syzygies f higher dimensional varieties.

### 2015年10月05日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Weighted Compactifications of Configuration Spaces (English)

**Evangelos Routis 氏**(IPMU)Weighted Compactifications of Configuration Spaces (English)

[ 講演概要 ]

In the early 90's, Fulton and MacPherson provided a natural and beautiful way of compactifying the configuration space F(X,n) of n distinct labeled points on an arbitrary nonsingular variety. In this talk, I will present an alternate compactification of F(X,n), which generalizes the work of Fulton and MacPherson and is parallel to Hassett's weighted generalization of the moduli space of n-pointed stable curves. After discussing its main properties, I will give a presentation of its intersection ring and as an application, I will describe the intersection ring of Hassett's spaces in genus 0. Finally, as time permits, I will discuss some additional moduli problems associated with weighted compactifications.

In the early 90's, Fulton and MacPherson provided a natural and beautiful way of compactifying the configuration space F(X,n) of n distinct labeled points on an arbitrary nonsingular variety. In this talk, I will present an alternate compactification of F(X,n), which generalizes the work of Fulton and MacPherson and is parallel to Hassett's weighted generalization of the moduli space of n-pointed stable curves. After discussing its main properties, I will give a presentation of its intersection ring and as an application, I will describe the intersection ring of Hassett's spaces in genus 0. Finally, as time permits, I will discuss some additional moduli problems associated with weighted compactifications.

### 2015年06月29日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Twisted cubics and cubic fourfolds (English)

**Manfred Lehn 氏**(Mainz/RIMS)Twisted cubics and cubic fourfolds (English)

[ 講演概要 ]

The moduli scheme of generalised twisted cubics on a smooth

cubic fourfold Y non containing a plane is smooth projective of

dimension 10 and admits a contraction to an 8-dimensional

holomorphic symplectic manifold Z(Y). The latter is shown to be

birational to the Hilbert scheme of four points on a K3 surface if

Y is of Pfaffian type. This is a report on joint work with C. Lehn,

C. Sorger and D. van Straten and with N. Addington.

The moduli scheme of generalised twisted cubics on a smooth

cubic fourfold Y non containing a plane is smooth projective of

dimension 10 and admits a contraction to an 8-dimensional

holomorphic symplectic manifold Z(Y). The latter is shown to be

birational to the Hilbert scheme of four points on a K3 surface if

Y is of Pfaffian type. This is a report on joint work with C. Lehn,

C. Sorger and D. van Straten and with N. Addington.

### 2015年06月22日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Rational cohomology tori

(English)

http://webusers.imj-prg.fr/~marti.lahoz/

**Martí Lahoz 氏**(Institut de Mathématiques de Jussieu )Rational cohomology tori

(English)

[ 講演概要 ]

Complex tori can be topologically characterised among compact Kähler

manifolds by their integral cohomology ring. I will discuss the

structure of compact Kähler manifolds whose rational cohomology ring is

isomorphic to the rational cohomology ring of a torus and give some

examples. This is joint work with Olivier Debarre and Zhi Jiang.

[ 参考URL ]Complex tori can be topologically characterised among compact Kähler

manifolds by their integral cohomology ring. I will discuss the

structure of compact Kähler manifolds whose rational cohomology ring is

isomorphic to the rational cohomology ring of a torus and give some

examples. This is joint work with Olivier Debarre and Zhi Jiang.

http://webusers.imj-prg.fr/~marti.lahoz/

### 2015年06月15日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Boundedness of the KSBA functor of

SLC models (English)

http://www.math.utah.edu/~hacon/

**Christopher Hacon 氏**(University of Utah/RIMS)Boundedness of the KSBA functor of

SLC models (English)

[ 講演概要 ]

Let $X$ be a canonically polarized smooth $n$-dimensional projective variety over $\mathbb C$ (so that $\omega _X$ is ample), then it is well-known that a fixed multiple of the canonical line bundle defines an embedding of $X$ in projective space. It then follows easily that if we fix certain invariants of $X$, then $X$ belongs to finitely many deformation types. Since canonical models are rarely smooth, it is important to generalize this result to canonically polarized $n$-dimensional projectivevarieties with canonical singularities. Moreover, since these varieties specialize to non-normal varieties it is also important to generalize this result to semi-log canonical pairs. In this talk we will explain a strong version of the above result that applies to semi-log canonical pairs.This is joint work with C. Xu and J. McKernan

[ 参考URL ]Let $X$ be a canonically polarized smooth $n$-dimensional projective variety over $\mathbb C$ (so that $\omega _X$ is ample), then it is well-known that a fixed multiple of the canonical line bundle defines an embedding of $X$ in projective space. It then follows easily that if we fix certain invariants of $X$, then $X$ belongs to finitely many deformation types. Since canonical models are rarely smooth, it is important to generalize this result to canonically polarized $n$-dimensional projectivevarieties with canonical singularities. Moreover, since these varieties specialize to non-normal varieties it is also important to generalize this result to semi-log canonical pairs. In this talk we will explain a strong version of the above result that applies to semi-log canonical pairs.This is joint work with C. Xu and J. McKernan

http://www.math.utah.edu/~hacon/

### 2015年06月01日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Rank 2 weak Fano bundles on cubic 3-folds (日本語)

**石川大蔵 氏**(早稲田)Rank 2 weak Fano bundles on cubic 3-folds (日本語)

[ 講演概要 ]

A vector bundle on a projective variety is called weak Fano if its

projectivization is a weak Fano manifold. This is a generalization of

Fano bundles.

In this talk, we will obtain a classification of rank 2 weak Fano

bundles on a nonsingular cubic hypersurface in a projective 4-space.

Specifically, we will show that there exist rank 2 indecomposable weak

Fano bundles on it.

A vector bundle on a projective variety is called weak Fano if its

projectivization is a weak Fano manifold. This is a generalization of

Fano bundles.

In this talk, we will obtain a classification of rank 2 weak Fano

bundles on a nonsingular cubic hypersurface in a projective 4-space.

Specifically, we will show that there exist rank 2 indecomposable weak

Fano bundles on it.