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代数幾何学セミナー
過去の記録 ～07/26｜次回の予定｜今後の予定 07/27～
| 開催情報 | 金曜日 13:30～15:00 数理科学研究科棟(駒場) ハイブリッド開催/117号室 |
|---|---|
| 担当者 | 權業 善範、中村 勇哉、田中 公 |
過去の記録
2023年02月06日(月)
13:00-14:30 数理科学研究科棟(駒場) 123号室
レクチャーシリーズ第２回目
Chenyang Xu 氏 (プリンストン大学)
K-stability of Fano varieties. (English)
レクチャーシリーズ第２回目
Chenyang Xu 氏 (プリンストン大学)
K-stability of Fano varieties. (English)
[ 講演概要 ]
The notion of K-stability of Fano varieties was first introduced to characterize the existence of Kahler-Einstein metric. Recently, a purely algebro-geometric theory has been developed and it has yielded many striking results, such as the solution of the Yau-Tian-Donaldson Conjecture for all Fano varieties, as well as the construction of a projective moduli scheme, called K-moduli, parametrizing K-polystable Fano varieties.
In this lecture series, I will survey the recent progress. The first two lectures will be devoted to explain the evolution of algebraic geometer’s understanding of various aspects of the notion of K-stability. The Lecture 3 and 4 will be devoted to discuss the construction of the K-moduli space.
The notion of K-stability of Fano varieties was first introduced to characterize the existence of Kahler-Einstein metric. Recently, a purely algebro-geometric theory has been developed and it has yielded many striking results, such as the solution of the Yau-Tian-Donaldson Conjecture for all Fano varieties, as well as the construction of a projective moduli scheme, called K-moduli, parametrizing K-polystable Fano varieties.
In this lecture series, I will survey the recent progress. The first two lectures will be devoted to explain the evolution of algebraic geometer’s understanding of various aspects of the notion of K-stability. The Lecture 3 and 4 will be devoted to discuss the construction of the K-moduli space.
2023年01月31日(火)
14:30-16:00 数理科学研究科棟(駒場) ハイブリッド開催/002号室
いつもと時間が異なります.
Shiji Lyu 氏 (プリンストン大学)
Some properties of splinters via ultrapower (English)
いつもと時間が異なります.
Shiji Lyu 氏 (プリンストン大学)
Some properties of splinters via ultrapower (English)
[ 講演概要 ]
A Noetherian (reduced) ring is called a splinter if it is a direct summand of every finite ring extension of it. This notion is related to various interesting notions of singularities, but far less properties are known about splinters.
In this talk, we will discuss the question of "regular ascent"; in the simplest (but already essential) form, we ask, for a Noetherian splinter R, is the polynomial ring R[X] always a splinter. We will see how ultrapower, a construction mainly belonging to model theory, is involved.
A Noetherian (reduced) ring is called a splinter if it is a direct summand of every finite ring extension of it. This notion is related to various interesting notions of singularities, but far less properties are known about splinters.
In this talk, we will discuss the question of "regular ascent"; in the simplest (but already essential) form, we ask, for a Noetherian splinter R, is the polynomial ring R[X] always a splinter. We will see how ultrapower, a construction mainly belonging to model theory, is involved.
2023年01月27日(金)
13:00-14:30 数理科学研究科棟(駒場) 056号室
全4回:1/27 (金) 13:00―14:30, 数理科学研究科056室 2/6 (月) 13:00―14:30, 数理科学研究科123室, 2/17 (金) 10:00―11:30, 数理科学研究科123室, 2/20 (月) 10:00ー11:30, 数理科学研究科056室
Chenyang Xu 氏 (プリンストン大学)
K-stability of Fano varieties (English)
全4回:1/27 (金) 13:00―14:30, 数理科学研究科056室 2/6 (月) 13:00―14:30, 数理科学研究科123室, 2/17 (金) 10:00―11:30, 数理科学研究科123室, 2/20 (月) 10:00ー11:30, 数理科学研究科056室
Chenyang Xu 氏 (プリンストン大学)
K-stability of Fano varieties (English)
[ 講演概要 ]
The notion of K-stability of Fano varieties was first introduced to characterize the existence of Kahler-Einstein metric. Recently, a purely algebro-geometric theory has been developed and it has yielded many striking results, such as the solution of the Yau-Tian-Donaldson Conjecture for all Fano varieties, as well as the construction of a projective moduli scheme, called K-moduli, parametrizing K-polystable Fano varieties.
In this lecture series, I will survey the recent progress. The first two lectures will be devoted to explain the evolution of algebraic geometer’s understanding of various aspects of the notion of K-stability. The Lecture 3 and 4 will be devoted to discuss the construction of the K-moduli space.
The notion of K-stability of Fano varieties was first introduced to characterize the existence of Kahler-Einstein metric. Recently, a purely algebro-geometric theory has been developed and it has yielded many striking results, such as the solution of the Yau-Tian-Donaldson Conjecture for all Fano varieties, as well as the construction of a projective moduli scheme, called K-moduli, parametrizing K-polystable Fano varieties.
In this lecture series, I will survey the recent progress. The first two lectures will be devoted to explain the evolution of algebraic geometer’s understanding of various aspects of the notion of K-stability. The Lecture 3 and 4 will be devoted to discuss the construction of the K-moduli space.
2023年01月10日(火)
10:30-12:00 数理科学研究科棟(駒場) ハイブリッド開催/002号室
講演は対面で行い、Zoomで中継します。
東谷 章弘 氏 (大阪大情報)
Toric Fano varieties arising from posets and their combinatorial mutation equivalence (日本語)
講演は対面で行い、Zoomで中継します。
東谷 章弘 氏 (大阪大情報)
Toric Fano varieties arising from posets and their combinatorial mutation equivalence (日本語)
[ 講演概要 ]
In 1986, Stanley introduced two polytopes arising from posets, called order polytopes and chain polytopes. Since then, those polytopes have been studied from viewpoints of combinatorics. Projective toric varieties arising from order polytopes are called Hibi toric varieties in these days. On the other hand, combinatorial mutations were introduced by Akhtar-Coates-Galkin-Kasprzyk in 2012 in the context of the classification problem of Fano varieties using mirror symmetry.
In this talk, after surveying two poset polytopes and combinatorial mutations, we discuss the combinatorial mutation equivalence of two poset polytopes. Those equivalence implies qG-deformation equivalence of projective toric varieties arising from two poset polytopes.
Moreover, it turns out that order polytopes, chain polytopes and their intermediate polytopes correspond to some toric Fano varieties.
In 1986, Stanley introduced two polytopes arising from posets, called order polytopes and chain polytopes. Since then, those polytopes have been studied from viewpoints of combinatorics. Projective toric varieties arising from order polytopes are called Hibi toric varieties in these days. On the other hand, combinatorial mutations were introduced by Akhtar-Coates-Galkin-Kasprzyk in 2012 in the context of the classification problem of Fano varieties using mirror symmetry.
In this talk, after surveying two poset polytopes and combinatorial mutations, we discuss the combinatorial mutation equivalence of two poset polytopes. Those equivalence implies qG-deformation equivalence of projective toric varieties arising from two poset polytopes.
Moreover, it turns out that order polytopes, chain polytopes and their intermediate polytopes correspond to some toric Fano varieties.
2022年12月21日(水)
13:00-14:00 or 14:30 数理科学研究科棟(駒場) ハイブリッド開催/056号室
いつもと部屋が異なります. 京大代数幾何セミナーと共催です.
Hsueh-Yung Lin 氏 (NTU)
Towards a geometric origin of the dynamical filtrations (English)
いつもと部屋が異なります. 京大代数幾何セミナーと共催です.
Hsueh-Yung Lin 氏 (NTU)
Towards a geometric origin of the dynamical filtrations (English)
[ 講演概要 ]
Let X be a smooth projective variety with an automorphism f. When X is a threefold, Serge Cantat asked whether X has a non-trivial equivariant rational fibration, if the action of f on the Néron-Severi space is non-trivial and unipotent. We will propose a precise conjecture related to Cantat's question for minimal varieties in arbitrary dimension, in light of the "dynamical filtrations" arising in the study of zero entropy group actions. This conjecture also suggests a geometric origin of dynamical filtrations, whose definition is purely cohomological. We will provide some heuristic evidence from the relative abundance conjecture.
If time permits, we will also explain how the study of dynamical filtrations leads to new results about solvable group actions, which are not necessarily of zero entropy.
Let X be a smooth projective variety with an automorphism f. When X is a threefold, Serge Cantat asked whether X has a non-trivial equivariant rational fibration, if the action of f on the Néron-Severi space is non-trivial and unipotent. We will propose a precise conjecture related to Cantat's question for minimal varieties in arbitrary dimension, in light of the "dynamical filtrations" arising in the study of zero entropy group actions. This conjecture also suggests a geometric origin of dynamical filtrations, whose definition is purely cohomological. We will provide some heuristic evidence from the relative abundance conjecture.
If time permits, we will also explain how the study of dynamical filtrations leads to new results about solvable group actions, which are not necessarily of zero entropy.
2022年12月20日(火)
9:30-10:30 数理科学研究科棟(駒場) オンラインZoom号室
Takumi Murayama 氏 (パーデュー大学)
The relative minimal model program for excellent algebraic spaces and analytic spaces in equal characteristic zero (English)
Takumi Murayama 氏 (パーデュー大学)
The relative minimal model program for excellent algebraic spaces and analytic spaces in equal characteristic zero (English)
[ 講演概要 ]
In 2010, Birkar, Cascini, Hacon, and McKernan proved a relative version of the minimal model program for projective morphisms of complex quasi-projective varieties, called the relative minimal model program with scaling. Their result is now fundamental to our understanding of the birational classification of quasi-projective varieties and has numerous applications.
In this talk, I will discuss recent joint work with Shiji Lyu that establishes the relative minimal model program with scaling for excellent schemes, excellent algebraic spaces, and analytic spaces simultaneously in equal characteristic zero. This not only recovers previous results for complex varieties, complex algebraic spaces, and complex analytic spaces, but also greatly extends the scope of the relative minimal model program with scaling to a broader class of geometric spaces, including formal schemes, rigid analytic spaces, and Berkovich spaces, all in equal characteristic zero. Our results for (non-algebraic) schemes and rigid analytic spaces were previously only known in dimensions ≤3 and ≤2, respectively, and our results for formal schemes and Berkovich spaces are completely new.
In 2010, Birkar, Cascini, Hacon, and McKernan proved a relative version of the minimal model program for projective morphisms of complex quasi-projective varieties, called the relative minimal model program with scaling. Their result is now fundamental to our understanding of the birational classification of quasi-projective varieties and has numerous applications.
In this talk, I will discuss recent joint work with Shiji Lyu that establishes the relative minimal model program with scaling for excellent schemes, excellent algebraic spaces, and analytic spaces simultaneously in equal characteristic zero. This not only recovers previous results for complex varieties, complex algebraic spaces, and complex analytic spaces, but also greatly extends the scope of the relative minimal model program with scaling to a broader class of geometric spaces, including formal schemes, rigid analytic spaces, and Berkovich spaces, all in equal characteristic zero. Our results for (non-algebraic) schemes and rigid analytic spaces were previously only known in dimensions ≤3 and ≤2, respectively, and our results for formal schemes and Berkovich spaces are completely new.
2022年12月13日(火)
10:30-11:30 数理科学研究科棟(駒場) ハイブリッド開催/002号室
講演者はZoomにて遠隔講演 002でも遠隔で流そうと思います。
山岸亮 氏 (NTU)
Moduli of G-constellations and crepant resolutions (日本語)
講演者はZoomにて遠隔講演 002でも遠隔で流そうと思います。
山岸亮 氏 (NTU)
Moduli of G-constellations and crepant resolutions (日本語)
[ 講演概要 ]
For a finite subgroup G of SL_n(C), a moduli space of G-constellations is a generalization of the G-Hilbert scheme and is important from the viewpoint of McKay correspondence. In this talk I will explain its basic properties and show that every projective crepant resolution of C^3/G is isomorphic to such a moduli space.
For a finite subgroup G of SL_n(C), a moduli space of G-constellations is a generalization of the G-Hilbert scheme and is important from the viewpoint of McKay correspondence. In this talk I will explain its basic properties and show that every projective crepant resolution of C^3/G is isomorphic to such a moduli space.
2022年11月29日(火)
10:30-11:30 数理科学研究科棟(駒場) ハイブリッド開催/002号室
Thomas Hall 氏 (University of Nottingham)
The behaviour of Kahler-Einstein polygons under combinatorial mutation
(English)
Thomas Hall 氏 (University of Nottingham)
The behaviour of Kahler-Einstein polygons under combinatorial mutation
(English)
[ 講演概要 ]
Combinatorial mutations play an important role in the mirror symmetry approach to the classification of Fano varieties. Another important notion for Fano varieties is that of K-polystability, which turns out to have a nice combinatorial characterisation in the toric case. In this talk, I will give an overview of how mutations work and sketch the key ideas used to explore its interaction with Kahler-Einstein polygons (i.e. the Fano polygons whose associated toric variety is K-polystable).
Combinatorial mutations play an important role in the mirror symmetry approach to the classification of Fano varieties. Another important notion for Fano varieties is that of K-polystability, which turns out to have a nice combinatorial characterisation in the toric case. In this talk, I will give an overview of how mutations work and sketch the key ideas used to explore its interaction with Kahler-Einstein polygons (i.e. the Fano polygons whose associated toric variety is K-polystable).
2022年11月22日(火)
10:30-12:00 数理科学研究科棟(駒場) 002号室
90分ハイブリッド開催です。
谷本祥 氏 (名古屋多元)
Non-free sections of Fano fibrations (日本語)
90分ハイブリッド開催です。
谷本祥 氏 (名古屋多元)
Non-free sections of Fano fibrations (日本語)
[ 講演概要 ]
Manin’s Conjecture predicts the asymptotic formula for the counting function of rational points over number fields or global function fields. In the late 80’s, Batyrev developed a heuristic argument for Manin’s Conjecture over global function fields, and the assumptions underlying Batyrev’s heuristics are refined and formulated as Geometric Manin’s Conjecture. Geometric Manin’s Conjecture is a set of conjectures regarding properties of the space of sections of Fano fibrations, and it consists of three conjectures: (i) Pathological components are controlled by Fujita invariants; (ii) For each nef algebraic class, a non-pathological component which should be counted in Manin’s Conjecture is unique (This component is called as Manin components); (iii) Manin components exhibit homological or motivic stability. In this talk we discuss our proofs of GMC (i) over complex numbers using theory of foliations and the minimal model program. Using this result, we prove that these pathological components are coming from a bounded family of accumulating maps. This is joint work in progress with Brian Lehmann and Eric Riedl.
Manin’s Conjecture predicts the asymptotic formula for the counting function of rational points over number fields or global function fields. In the late 80’s, Batyrev developed a heuristic argument for Manin’s Conjecture over global function fields, and the assumptions underlying Batyrev’s heuristics are refined and formulated as Geometric Manin’s Conjecture. Geometric Manin’s Conjecture is a set of conjectures regarding properties of the space of sections of Fano fibrations, and it consists of three conjectures: (i) Pathological components are controlled by Fujita invariants; (ii) For each nef algebraic class, a non-pathological component which should be counted in Manin’s Conjecture is unique (This component is called as Manin components); (iii) Manin components exhibit homological or motivic stability. In this talk we discuss our proofs of GMC (i) over complex numbers using theory of foliations and the minimal model program. Using this result, we prove that these pathological components are coming from a bounded family of accumulating maps. This is joint work in progress with Brian Lehmann and Eric Riedl.
2022年11月15日(火)
10:30-12:00 数理科学研究科棟(駒場) ハイブリッド開催/002号室
張 繼剛 氏 (NTU/東大数理)
Positivity of anticanonical divisors in algebraic fibre spaces (日本語)
張 繼剛 氏 (NTU/東大数理)
Positivity of anticanonical divisors in algebraic fibre spaces (日本語)
[ 講演概要 ]
It is known that the positivity of the anti-canonical divisor is an important property that is closely related to the geometric structure of a variety. Given an algebraic fibre space f : X → Y between normal projective varieties with mild singularities, and let F be a general fibre of f. In this talk, we will introduce results relating the positivity of −KX and −KY under some conditions on the asymptotic base loci of −KX. In particular, we will obtain an inequality between the anti-canonical Iitaka dimensions κ(X, −KX) ≤ κ(F, −KF ) + κ(Y, −KY ) under the assumption that the stable base locus B(−KX) does not dominant over Y .
It is known that the positivity of the anti-canonical divisor is an important property that is closely related to the geometric structure of a variety. Given an algebraic fibre space f : X → Y between normal projective varieties with mild singularities, and let F be a general fibre of f. In this talk, we will introduce results relating the positivity of −KX and −KY under some conditions on the asymptotic base loci of −KX. In particular, we will obtain an inequality between the anti-canonical Iitaka dimensions κ(X, −KX) ≤ κ(F, −KF ) + κ(Y, −KY ) under the assumption that the stable base locus B(−KX) does not dominant over Y .
2022年11月01日(火)
10:30-12:00 数理科学研究科棟(駒場) ハイブリッド開催/002号室
河上龍郎 氏 (京大数学教室)
Extendability of differential forms via Cartier operators (Japanese)
河上龍郎 氏 (京大数学教室)
Extendability of differential forms via Cartier operators (Japanese)
[ 講演概要 ]
For a normal variety X, we say X satisfies the extension theorem if, for every proper birational morphism from Y, every differential form on the regular locus of X extends to Y. This is a basic property relating differential forms and singularities, and many results are known over the field of complex numbers.
In this talk, we discuss the extension theorem in positive characteristic. Existing studies depend on geometric tools such as log resolutions, (mixed) Hodge theory, the minimal model program, and vanishing theorems, which are not expected to be true or are not known for higher-dimensional varieties in positive characteristic.
For this reason, I introduce a new algebraic approach to the extension theorem using Cartier operators. I also talk about an application of the theory of quasi-F-splitting, which is studied in joint work with Takamatsu-Tanaka-Witaszek-Yobuko-Yoshikawa, to the extension problem.
For a normal variety X, we say X satisfies the extension theorem if, for every proper birational morphism from Y, every differential form on the regular locus of X extends to Y. This is a basic property relating differential forms and singularities, and many results are known over the field of complex numbers.
In this talk, we discuss the extension theorem in positive characteristic. Existing studies depend on geometric tools such as log resolutions, (mixed) Hodge theory, the minimal model program, and vanishing theorems, which are not expected to be true or are not known for higher-dimensional varieties in positive characteristic.
For this reason, I introduce a new algebraic approach to the extension theorem using Cartier operators. I also talk about an application of the theory of quasi-F-splitting, which is studied in joint work with Takamatsu-Tanaka-Witaszek-Yobuko-Yoshikawa, to the extension problem.
2022年10月25日(火)
10:30-11:45 数理科学研究科棟(駒場) ハイブリッド開催/002号室
伊藤敦 氏 (岡山大学)
Projective normality of general polarized abelian varieties (Japanese)
伊藤敦 氏 (岡山大学)
Projective normality of general polarized abelian varieties (Japanese)
[ 講演概要 ]
Projective normality is an important property of polarized varieties. Hwang and To prove that a general polarized abelian variety $(X,L)$ of dimension $g$ is projectively normal if $\chi(X,L) \geq (8g)^g/2g!$. In this talk, I will explain that their bound can be weaken as $\chi(X,L) > 2^{2g-1}$, which is sharp. A key tool in the proof is an invariant introduced by Jiang and Pareschi, which measures the basepoint freeness of $\mathbb{Q}$-divisors on abelian varieties.
Projective normality is an important property of polarized varieties. Hwang and To prove that a general polarized abelian variety $(X,L)$ of dimension $g$ is projectively normal if $\chi(X,L) \geq (8g)^g/2g!$. In this talk, I will explain that their bound can be weaken as $\chi(X,L) > 2^{2g-1}$, which is sharp. A key tool in the proof is an invariant introduced by Jiang and Pareschi, which measures the basepoint freeness of $\mathbb{Q}$-divisors on abelian varieties.
2022年10月05日(水)
13:00-14:00 数理科学研究科棟(駒場) 056号室
今学期より対面ハイブリッドでセミナーを再開します。本セミナーは京大と共催です。オンライン情報はメーリングリストで公開しています。
Yuri Tschinkel 氏 (Mathematics and Physical Sciences Division, Simons Foundation/ Courant Institute, New York University)
Equivariant birational geometry (joint with A. Kresch) (English)
今学期より対面ハイブリッドでセミナーを再開します。本セミナーは京大と共催です。オンライン情報はメーリングリストで公開しています。
Yuri Tschinkel 氏 (Mathematics and Physical Sciences Division, Simons Foundation/ Courant Institute, New York University)
Equivariant birational geometry (joint with A. Kresch) (English)
[ 講演概要 ]
Ideas from motivic integration led to the introduction of new invariants in equivariant birational geometry, the study of actions of finite groups on algebraic varieties, up to equivariant birational transformations.
These invariants allow us to distinguish actions in many new cases, shedding light on the structure of the Cremona group. The structure of the invariants themselves is also interesting: there are unexpected connections to modular curves and cohomology of arithmetic groups.
Ideas from motivic integration led to the introduction of new invariants in equivariant birational geometry, the study of actions of finite groups on algebraic varieties, up to equivariant birational transformations.
These invariants allow us to distinguish actions in many new cases, shedding light on the structure of the Cremona group. The structure of the invariants themselves is also interesting: there are unexpected connections to modular curves and cohomology of arithmetic groups.
2021年07月21日(水)
15:00-16:00 数理科学研究科棟(駒場) zoom号室
キャンセルになりました。京大と共催
宮本恵介 氏 (大阪大学)
TBA (日本語)
キャンセルになりました。京大と共催
宮本恵介 氏 (大阪大学)
TBA (日本語)
[ 講演概要 ]
TBA
TBA
2021年07月05日(月)
16:00-17:00 数理科学研究科棟(駒場) zoom号室
いつもと日時が異なります。京大と共催
Paolo Cascini 氏 (Imperial College London)
Birational geometry of foliations (English)
いつもと日時が異なります。京大と共催
Paolo Cascini 氏 (Imperial College London)
Birational geometry of foliations (English)
[ 講演概要 ]
I will survey about some recent progress towards the Minimal Model Program for foliations on complex varieties, focusing mainly on the case of threefolds and the case of algebraically integrable foliations.
I will survey about some recent progress towards the Minimal Model Program for foliations on complex varieties, focusing mainly on the case of threefolds and the case of algebraically integrable foliations.
2021年07月01日(木)
10:00-11:00 数理科学研究科棟(駒場) 号室
いつもと日時が違います。京大と共催です。
鈴木文顕 氏 (UCLA)
An O-acyclic variety of even index
いつもと日時が違います。京大と共催です。
鈴木文顕 氏 (UCLA)
An O-acyclic variety of even index
[ 講演概要 ]
I will construct a family of Enriques surfaces parametrized by P^1 such that any multi-section has even degree over the base P^1. Over the function field of a complex curve, this gives the first example of an O-acyclic variety (H^i(X,O)=0 for i>0) whose index is not equal to one, and an affirmative answer to a question of Colliot-Thélène and Voisin. I will also discuss applications to related problems, including the integral Hodge conjecture and Murre’s question on universality of the Abel-Jacobi maps. This is joint work with John Christian Ottem.
I will construct a family of Enriques surfaces parametrized by P^1 such that any multi-section has even degree over the base P^1. Over the function field of a complex curve, this gives the first example of an O-acyclic variety (H^i(X,O)=0 for i>0) whose index is not equal to one, and an affirmative answer to a question of Colliot-Thélène and Voisin. I will also discuss applications to related problems, including the integral Hodge conjecture and Murre’s question on universality of the Abel-Jacobi maps. This is joint work with John Christian Ottem.
2021年06月14日(月)
17:00-18:00 数理科学研究科棟(駒場) 号室
京大と共催です。いつもと日時が異なります。
原和平 氏 (University of Glasgow)
Rank two weak Fano bundles on del Pezzo threefolds of degree 5 (日本語)
Zoom
京大と共催です。いつもと日時が異なります。
原和平 氏 (University of Glasgow)
Rank two weak Fano bundles on del Pezzo threefolds of degree 5 (日本語)
[ 講演概要 ]
射影化したとき反標準因子がネフかつ巨大になるようなベクトル束を弱Fanoベクトル束という．
本講演では，福岡氏，石川氏との共同研究で得られた，次数5の3次元del Pezzo多様体上の階数2のベクトル束の分類結果を紹介する．
今回は特に，導来圏の例外生成列を用いたベクトル束の分解を得る方法と，それを応用して得られるモジュライ空間についての諸結果に話題を絞って，証明を詳しく紹介したい．
[ 参考URL ]射影化したとき反標準因子がネフかつ巨大になるようなベクトル束を弱Fanoベクトル束という．
本講演では，福岡氏，石川氏との共同研究で得られた，次数5の3次元del Pezzo多様体上の階数2のベクトル束の分類結果を紹介する．
今回は特に，導来圏の例外生成列を用いたベクトル束の分解を得る方法と，それを応用して得られるモジュライ空間についての諸結果に話題を絞って，証明を詳しく紹介したい．
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2021年06月09日(水)
15:00-16:00 数理科学研究科棟(駒場) 122号室
京大と共催です。
Andrea Fanelli 氏 (Bordeaux)
Rational simple connectedness and Fano threefolds (English)
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京大と共催です。
Andrea Fanelli 氏 (Bordeaux)
Rational simple connectedness and Fano threefolds (English)
[ 講演概要 ]
The notion of rational simple connectedness can be seen as an algebro-geometric analogue of simple connectedness in topology. The work of de Jong, He and Starr has already produced several recent studies to understand this notion.
In this talk I will discuss the joint project with Laurent Gruson and Nicolas Perrin to study rational simple connectedness for Fano threefolds via explicit methods from birational geometry.
[ 参考URL ]The notion of rational simple connectedness can be seen as an algebro-geometric analogue of simple connectedness in topology. The work of de Jong, He and Starr has already produced several recent studies to understand this notion.
In this talk I will discuss the joint project with Laurent Gruson and Nicolas Perrin to study rational simple connectedness for Fano threefolds via explicit methods from birational geometry.
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2021年06月02日(水)
15:00-16:00 数理科学研究科棟(駒場) 号室
京大と共催です。
青木孔 氏 (東大数理)
Quasiexcellence implies strong generation (日本語)
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京大と共催です。
青木孔 氏 (東大数理)
Quasiexcellence implies strong generation (日本語)
[ 講演概要 ]
BondalとVan den Berghは（小さい）三角圏からの反変関手がいつ表現可能かという問題の考察の中で、対象が三角圏を強生成(strongly generate)することの定義を導入した。強生成する対象が存在するときは良い表現可能性定理が成立する。
どのような有限次元Noetherスキームに対してその連接層の導来圏が強生成であるかについてはBondal–Van den Bergh以降Rouquier, Keller–Van den Bergh, Aihara–Takahashi, Iyengar–Takahashiなどにより多くの結果が得られていたが、最近Neemanは別の手法を用いてそれをalterationが適用できる分離Noetherスキームに対して示した。それを講演者はGabberのweak local uniformizationを用いることでさらに分離的準優秀スキームにまで拡張した。講演ではこの結果およびその証明の手法を紹介する。
[ 参考URL ]BondalとVan den Berghは（小さい）三角圏からの反変関手がいつ表現可能かという問題の考察の中で、対象が三角圏を強生成(strongly generate)することの定義を導入した。強生成する対象が存在するときは良い表現可能性定理が成立する。
どのような有限次元Noetherスキームに対してその連接層の導来圏が強生成であるかについてはBondal–Van den Bergh以降Rouquier, Keller–Van den Bergh, Aihara–Takahashi, Iyengar–Takahashiなどにより多くの結果が得られていたが、最近Neemanは別の手法を用いてそれをalterationが適用できる分離Noetherスキームに対して示した。それを講演者はGabberのweak local uniformizationを用いることでさらに分離的準優秀スキームにまで拡張した。講演ではこの結果およびその証明の手法を紹介する。
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2021年05月26日(水)
15:00-16:00 数理科学研究科棟(駒場) zoom号室
京大と共催
山口樹 氏 (東大数理)
Multiplier ideals via ultraproducts (日本語)
京大と共催
山口樹 氏 (東大数理)
Multiplier ideals via ultraproducts (日本語)
[ 講演概要 ]
正標数の可換環と複素数体上の可換環の性質を比較する方法の一つにultraproductを用いた手法がある. このultraproductは超準解析において超実数の構成などに用いられているものである. これを可換環論へ応用する研究としてSchoutensによるnon-standard hullがある. この手法は等標数0の局所環に対するbig Cohen-Macaulay 代数の構成などにも応用がある. 彼の研究の一つに川又対数端末特異点のultraproductを用いた特徴付けがある. 本講演では, この結果の一般化として乗数イデアルがultraproductを用いて記述できることを説明する.
正標数の可換環と複素数体上の可換環の性質を比較する方法の一つにultraproductを用いた手法がある. このultraproductは超準解析において超実数の構成などに用いられているものである. これを可換環論へ応用する研究としてSchoutensによるnon-standard hullがある. この手法は等標数0の局所環に対するbig Cohen-Macaulay 代数の構成などにも応用がある. 彼の研究の一つに川又対数端末特異点のultraproductを用いた特徴付けがある. 本講演では, この結果の一般化として乗数イデアルがultraproductを用いて記述できることを説明する.
2021年05月17日(月)
17:00-18:00 数理科学研究科棟(駒場) zoom号室
いつもと日時が異なります。京大と共催
Ivan Cheltsov 氏 (エジンバラ)
Calabi problem for smooth Fano threefolds (English)
いつもと日時が異なります。京大と共催
Ivan Cheltsov 氏 (エジンバラ)
Calabi problem for smooth Fano threefolds (English)
[ 講演概要 ]
In this talk I will explain which three-dimensional complex Fano manifolds admit Kahler-Einstein metrics.
In this talk I will explain which three-dimensional complex Fano manifolds admit Kahler-Einstein metrics.
2021年05月13日(木)
9:00-10:00 数理科学研究科棟(駒場) zoom号室
村山匠 氏 (プリンストン大学)
Relative vanishing theorems for schemes of equal characteristic zero (Englishg)
村山匠 氏 (プリンストン大学)
Relative vanishing theorems for schemes of equal characteristic zero (Englishg)
[ 講演概要 ]
In 1953, Kodaira proved the Kodaira vanishing theorem, which states that if L is an ample divisor on a complex projective manifold X, then H^i(X,-L) = 0 for all i < dim(X). Since then, Kodaira's theorem and its generalizations have become indispensable tools in algebraic geometry over fields of characteristic zero. Even in this context, however, it is often necessary to work with schemes of finite type over power series rings, and a fundamental problem has been the lack of vanishing theorems in this setting.
We prove the analogue of the Kawamata-Viehweg vanishing theorem for proper morphisms of schemes of equal characteristic zero, which implies Kodaira's vanishing theorem in this context. This result resolves conjectures of Boutot and Kawakita, and is an important ingredient toward establishing the minimal model program for excellent schemes of equal characteristic zero.
In 1953, Kodaira proved the Kodaira vanishing theorem, which states that if L is an ample divisor on a complex projective manifold X, then H^i(X,-L) = 0 for all i < dim(X). Since then, Kodaira's theorem and its generalizations have become indispensable tools in algebraic geometry over fields of characteristic zero. Even in this context, however, it is often necessary to work with schemes of finite type over power series rings, and a fundamental problem has been the lack of vanishing theorems in this setting.
We prove the analogue of the Kawamata-Viehweg vanishing theorem for proper morphisms of schemes of equal characteristic zero, which implies Kodaira's vanishing theorem in this context. This result resolves conjectures of Boutot and Kawakita, and is an important ingredient toward establishing the minimal model program for excellent schemes of equal characteristic zero.
2021年04月28日(水)
15:00-16:00 数理科学研究科棟(駒場) Zoom号室
京大と共催です。
金城翼 氏 (東大数理/IPMU)
Dimensional reduction in cohomological Donaldson-Thomas theory (日本語)
京大と共催です。
金城翼 氏 (東大数理/IPMU)
Dimensional reduction in cohomological Donaldson-Thomas theory (日本語)
[ 講演概要 ]
三次元Calabi-Yau多様体のコホモロジー的Donaldson-Thomas不変量(CoDT不変量)とは、Joyceらによって導入されたDonaldson-Thomas不変量の圏化であり、
Kontsevich-Soibelmanによって導入されたポテンシャル付き箙のCoDT不変量の大域化とみなすことができるものである。
ポテンシャル付き箙のCoDT理論は表現論とのつながりなどの深い理論が知られているのに対し、
三次元Calabi-Yau多様体のCoDT理論は定義以外のことがほとんど知られていないのが現状である。
本講演では滑らかな代数曲面の標準束の全空間として与えられる三次元Calabi-Yau多様体のCoDT不変量と元の曲面の連接層のモジュライのBorel-Mooreホモロジーを関連付ける次元還元定理について説明を行う。
また、次元還元定理をトム同型の一般化とみなしオイラー類の構成を適用することで、仮想基本類の新しい構成が与えられることを説明する。
三次元Calabi-Yau多様体のコホモロジー的Donaldson-Thomas不変量(CoDT不変量)とは、Joyceらによって導入されたDonaldson-Thomas不変量の圏化であり、
Kontsevich-Soibelmanによって導入されたポテンシャル付き箙のCoDT不変量の大域化とみなすことができるものである。
ポテンシャル付き箙のCoDT理論は表現論とのつながりなどの深い理論が知られているのに対し、
三次元Calabi-Yau多様体のCoDT理論は定義以外のことがほとんど知られていないのが現状である。
本講演では滑らかな代数曲面の標準束の全空間として与えられる三次元Calabi-Yau多様体のCoDT不変量と元の曲面の連接層のモジュライのBorel-Mooreホモロジーを関連付ける次元還元定理について説明を行う。
また、次元還元定理をトム同型の一般化とみなしオイラー類の構成を適用することで、仮想基本類の新しい構成が与えられることを説明する。
2021年04月21日(水)
15:00-16:00 数理科学研究科棟(駒場) ZOOM号室
京大と共催です。
服部真史 氏 (京大数学教室)
A decomposition formula for J-stability and its applications (日本語)
京大と共催です。
服部真史 氏 (京大数学教室)
A decomposition formula for J-stability and its applications (日本語)
[ 講演概要 ]
J-stability is an analog of K-stability and plays an important role in K-stability for general polarized varieties (not only for Kahler-Einstein metrics). Strikingly, G.Chen proved uniform J-stability and slope uniform J-stability are equivalent, analogous to Ross-Thomas slope theory and Mumford-Takemoto slope theory for vector bundles, by differential geometric arguments recently. However, this fact has not been proved in algebro-geometric way before. In this talk, I would like to explain a decomposition formula of non-Archimedean J-functional, the (n+1)-dimensional intersection number into n-dimensional intersection numbers and its applications to prove the fact for surfaces and to construct a K-stable but not uniformly K-stable lc pair. Based on arXiv:2103.04603
J-stability is an analog of K-stability and plays an important role in K-stability for general polarized varieties (not only for Kahler-Einstein metrics). Strikingly, G.Chen proved uniform J-stability and slope uniform J-stability are equivalent, analogous to Ross-Thomas slope theory and Mumford-Takemoto slope theory for vector bundles, by differential geometric arguments recently. However, this fact has not been proved in algebro-geometric way before. In this talk, I would like to explain a decomposition formula of non-Archimedean J-functional, the (n+1)-dimensional intersection number into n-dimensional intersection numbers and its applications to prove the fact for surfaces and to construct a K-stable but not uniformly K-stable lc pair. Based on arXiv:2103.04603
2021年04月14日(水)
15:00-16:00 数理科学研究科棟(駒場) Zoom号室
京大と共催です。
佐藤謙太 氏 (九州大学)
Arithmetic deformation of F-singularities (日本語)
京大と共催です。
佐藤謙太 氏 (九州大学)
Arithmetic deformation of F-singularities (日本語)
[ 講演概要 ]
F正則特異点は，Frobenius写像の言葉で定義される正標数の特異点のクラスであるが，標数0のklt特異点と強い関係があることが知られている．例えば，標数0の特異点がkltであることと，無限個のpに関する正標数還元がF正則になることは同値である．近年Ma-Schwedeは，この関係の精密化として，total spaceがQ-Gorensteinという条件のもとで，一つのpでの正標数還元が強F正則ならば，もとの標数0の特異点はkltであることを証明した．
本講演では，total spaceがQ-Gorensteinでない場合にこの結果を一般化する．また，Liedtke-Martin-Matsumotoによる線形簡約商特異点に関する予想への本結果の応用や，F正則/klt特異点よりも少し広いクラスである，F純/lc特異点に関する同様の結果についても紹介する．これらの結果は，高木俊輔氏との共同研究である．
F正則特異点は，Frobenius写像の言葉で定義される正標数の特異点のクラスであるが，標数0のklt特異点と強い関係があることが知られている．例えば，標数0の特異点がkltであることと，無限個のpに関する正標数還元がF正則になることは同値である．近年Ma-Schwedeは，この関係の精密化として，total spaceがQ-Gorensteinという条件のもとで，一つのpでの正標数還元が強F正則ならば，もとの標数0の特異点はkltであることを証明した．
本講演では，total spaceがQ-Gorensteinでない場合にこの結果を一般化する．また，Liedtke-Martin-Matsumotoによる線形簡約商特異点に関する予想への本結果の応用や，F正則/klt特異点よりも少し広いクラスである，F純/lc特異点に関する同様の結果についても紹介する．これらの結果は，高木俊輔氏との共同研究である．
