## 代数幾何学セミナー

過去の記録 ～05/28｜次回の予定｜今後の予定 05/29～

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05月26日(金)

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

Varieties in positive characteristic with numerically flat tangent bundle

**吉川 翔 氏**(東京工業大学, 理研)Varieties in positive characteristic with numerically flat tangent bundle

[ 講演概要 ]

The positivity condition imposed on the tangent bundle of a smooth projective variety is known to restrict the geometric structure of the variety. Demailly, Peternell and Schneider established a decomposition theorem for a smooth projective complex variety with nef tangent bundle. The theorem states that, up to an etale cover, such a variety has a smooth fibration admitting a smooth algebraic fiber space over an abelian variety whose fibers are Fano varieties, so one can say that such a variety decomposes into the "positive” part and the "flat” part. A positive characteristic analog of the above decomposition theorem was proved by Kanemitsu and Watanabe. The "flat” part of their theorem is a smooth projective variety with numerically flat tangent bundle. In this talk, I will introduce the result that every ordinary variety with numerically flat tangent bundle is an etale quotient of an ordinary Abelian variety. In particular, we obtain the decomposition theorem for Frobenius splitting varieties with nef tangent bundle. This talk is based on joint work with Sho Ejiri.

The positivity condition imposed on the tangent bundle of a smooth projective variety is known to restrict the geometric structure of the variety. Demailly, Peternell and Schneider established a decomposition theorem for a smooth projective complex variety with nef tangent bundle. The theorem states that, up to an etale cover, such a variety has a smooth fibration admitting a smooth algebraic fiber space over an abelian variety whose fibers are Fano varieties, so one can say that such a variety decomposes into the "positive” part and the "flat” part. A positive characteristic analog of the above decomposition theorem was proved by Kanemitsu and Watanabe. The "flat” part of their theorem is a smooth projective variety with numerically flat tangent bundle. In this talk, I will introduce the result that every ordinary variety with numerically flat tangent bundle is an etale quotient of an ordinary Abelian variety. In particular, we obtain the decomposition theorem for Frobenius splitting varieties with nef tangent bundle. This talk is based on joint work with Sho Ejiri.

### 2023年05月10日(水)

13:30-15:00 数理科学研究科棟(駒場) ハイブリッド開催/056号室

Singularities in mixed characteristic via the Riemann-Hilbert correspondence (English)

**Jakub Witaszek 氏**(Princeton University)Singularities in mixed characteristic via the Riemann-Hilbert correspondence (English)

[ 講演概要 ]

In my talk, I will start by reviewing how various properties of characteristic zero singularities can be understood topologically by ways of the Riemann-Hilbert correspondence. After that, I will explain how similar ideas can be applied in the study of mixed characteristic singularities. This is based on a joint work (in progress) with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker, and Joe Waldron.

In my talk, I will start by reviewing how various properties of characteristic zero singularities can be understood topologically by ways of the Riemann-Hilbert correspondence. After that, I will explain how similar ideas can be applied in the study of mixed characteristic singularities. This is based on a joint work (in progress) with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker, and Joe Waldron.

### 2023年04月28日(金)

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

On the degree of irrationality of complete intersections (Japanese )

**吉野太郎 氏**(東大数理)On the degree of irrationality of complete intersections (Japanese )

[ 講演概要 ]

The degree of irrationality of a variety X is the minimum degree of a dominant, generically finite rational map from X to a rational variety. This invariant gives a measure of how far X is from being rational. There were some varieties whose degree of irrationality was computed. For example, in 2017, Bastianelli, De Poi, Ein, Lazarsfeld, and Ullery computed the degree of irrationality of very general hypersurfaces of general type by using the positivity of the canonical line bundle. On the other hand, in 2020, Chen and Stapleton obtained the lower bound of the degree of irrationality of very general Fano hypersurfaces by using the reduction of modulo p.

In this talk, we will show that we can obtain the lower bound of the degree of irrationality of very general Fano complete intersections. For obtaining the bound, we make a minor adjustment to Chen--Stapleton's method using the trace map of differential modules.

This talk is based on joint work with Lucas Braune.

The degree of irrationality of a variety X is the minimum degree of a dominant, generically finite rational map from X to a rational variety. This invariant gives a measure of how far X is from being rational. There were some varieties whose degree of irrationality was computed. For example, in 2017, Bastianelli, De Poi, Ein, Lazarsfeld, and Ullery computed the degree of irrationality of very general hypersurfaces of general type by using the positivity of the canonical line bundle. On the other hand, in 2020, Chen and Stapleton obtained the lower bound of the degree of irrationality of very general Fano hypersurfaces by using the reduction of modulo p.

In this talk, we will show that we can obtain the lower bound of the degree of irrationality of very general Fano complete intersections. For obtaining the bound, we make a minor adjustment to Chen--Stapleton's method using the trace map of differential modules.

This talk is based on joint work with Lucas Braune.

### 2023年04月21日(金)

14:00-15:30 数理科学研究科棟(駒場) 117号室

ハイブリッド開催

Endomorphisms of varieties and Bott vanishing (Japanese)

ハイブリッド開催

**河上 龍郎 氏**(京都大学)Endomorphisms of varieties and Bott vanishing (Japanese)

[ 講演概要 ]

In this talk, we show that a projective variety with an int-amplified endomorphism of degree invertible in the base field satisfies Bott vanishing. This is a new way to analyze which varieties have nontrivial endomorphisms. In particular, we extend some classification results on varieties admitting endomorphisms (for Fano threefolds of Picard number one and several other cases) to any characteristic. This talk is based on joint work with Burt Totaro.

In this talk, we show that a projective variety with an int-amplified endomorphism of degree invertible in the base field satisfies Bott vanishing. This is a new way to analyze which varieties have nontrivial endomorphisms. In particular, we extend some classification results on varieties admitting endomorphisms (for Fano threefolds of Picard number one and several other cases) to any characteristic. This talk is based on joint work with Burt Totaro.

### 2023年04月21日(金)

12:45-13:45 数理科学研究科棟(駒場) ハイブリッド開催/117号室

ACC of plc thresholds (English)

**Sung Rak Choi 氏**(Yonsei University )ACC of plc thresholds (English)

[ 講演概要 ]

The notion of potential pairs was developed as a means to bound the singularities while running the anti-MMP. They behave similarly with the usual klt, lc pairs.

We introduce potential log canonical threshold and prove that the set of these thresholds also satisfies the ascending chain condition (ACC). We also study the relation with the complements. This is a joint work with Sungwook Jang.

The notion of potential pairs was developed as a means to bound the singularities while running the anti-MMP. They behave similarly with the usual klt, lc pairs.

We introduce potential log canonical threshold and prove that the set of these thresholds also satisfies the ascending chain condition (ACC). We also study the relation with the complements. This is a joint work with Sungwook Jang.

### 2023年03月28日(火)

10:00-11:30 数理科学研究科棟(駒場) ハイブリッド開催/056号室

On the canonical bundle formula in positive characteristic (English)

**Paolo Cascini 氏**(Imperial College London)On the canonical bundle formula in positive characteristic (English)

[ 講演概要 ]

In a previous work in collaboration with F. Ambro, V. Shokurov and C. Spicer, we show that algebraically integrable foliations can be used to study the canonical bundle formula for fibrations which are not necessarily lc trivial.

I will discuss a work in progress by M. Benozzo on a generalisation of these results in positive characteristic.

In a previous work in collaboration with F. Ambro, V. Shokurov and C. Spicer, we show that algebraically integrable foliations can be used to study the canonical bundle formula for fibrations which are not necessarily lc trivial.

I will discuss a work in progress by M. Benozzo on a generalisation of these results in positive characteristic.

### 2023年03月10日(金)

13:15-14:45 数理科学研究科棟(駒場) ハイブリッド開催/123号室

On existence of flips for algebraically integrable foliations. (English)

**Paolo Cascini 氏**(Imperical College London)On existence of flips for algebraically integrable foliations. (English)

[ 講演概要 ]

Assuming termination of (classical) flips in dimension r, we show that flips exist for any algebraically integrable foliation of rank r with log canonical singularities. Joint work with C. Spicer.

Assuming termination of (classical) flips in dimension r, we show that flips exist for any algebraically integrable foliation of rank r with log canonical singularities. Joint work with C. Spicer.

### 2023年02月20日(月)

10:00-11:30 数理科学研究科棟(駒場) 056号室

レクチャーシリーズ第４回目

K-stability of Fano varieties. (English)

レクチャーシリーズ第４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02月17日(金)

10:00-11:30 数理科学研究科棟(駒場) 123号室

レクチャーシリーズ第３回

K-stability of Fano varieties. ( English)

レクチャーシリーズ第３回

**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

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.

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

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.

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)

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。

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.

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

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。

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

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。

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

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

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

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。

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

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

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。

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.