## 代数幾何学セミナー

過去の記録 ～04/18｜次回の予定｜今後の予定 04/19～

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12月15日(金)

13:30-15:00 数理科学研究科棟(駒場) 118号室

On a pair of a smooth variety and a multi-ideal with a real exponent in positive characteristic (日本語)

**石井 志保子 氏**(東京大学)On a pair of a smooth variety and a multi-ideal with a real exponent in positive characteristic (日本語)

[ 講演概要 ]

In birational geometry, the behaviors of the invariants, mld (minimal log discrepancy) and lct (log canonical threshold), play important roles. These invariants are studied well in case the base field is characteristic zero, but not so in positive characteristic case. In this talk, I work on a pair consisting of smooth variety and a multi-ideal with a real exponent over an algebraically closed field of positive characteristic. We reduce some behaviors of the invariants for such pairs in positive characteristic case into characteristic zero.

In birational geometry, the behaviors of the invariants, mld (minimal log discrepancy) and lct (log canonical threshold), play important roles. These invariants are studied well in case the base field is characteristic zero, but not so in positive characteristic case. In this talk, I work on a pair consisting of smooth variety and a multi-ideal with a real exponent over an algebraically closed field of positive characteristic. We reduce some behaviors of the invariants for such pairs in positive characteristic case into characteristic zero.

### 2023年11月24日(金)

14:00-15:30 数理科学研究科棟(駒場) ハイブリッド開催/056号室

On Kawamata-Miyaoka type inequality

**Haidong Liu 氏**(Sun Yat-sen University)On Kawamata-Miyaoka type inequality

[ 講演概要 ]

For klt projective varieties with nef and big canonical divisors, there exists a Miyaoka-Yau type inequality concerning the first and the second Chern classes. In this talk, I will present a Kawamata-Miyaoka type inequality for terminal Q-Fano varieties, which is a mirror version of the Miyaoka-Yau type inequality. This is a joint work with Jie Liu.

For klt projective varieties with nef and big canonical divisors, there exists a Miyaoka-Yau type inequality concerning the first and the second Chern classes. In this talk, I will present a Kawamata-Miyaoka type inequality for terminal Q-Fano varieties, which is a mirror version of the Miyaoka-Yau type inequality. This is a joint work with Jie Liu.

### 2023年10月16日(月)

14:00-15:30 数理科学研究科棟(駒場) 002号室

Symmetries of Fano varieties

**Lena Ji 氏**(University of Michigan)Symmetries of Fano varieties

[ 講演概要 ]

Prokhorov and Shramov proved that the BAB conjecture (which Birkar later proved) implies the uniform Jordan property for automorphism groups of complex Fano varieties of fixed dimension. This property in particular gives an upper bound on the size of semi-simple groups (meaning those with no non-trivial normal abelian subgroups) acting faithfully on n-dimensional complex Fano varieties, and this bound only depends on n. In this talk, we investigate the consequences of a large action by a particular semi-simple group: the symmetric group. This work is joint with Louis Esser and Joaquín Moraga.

Prokhorov and Shramov proved that the BAB conjecture (which Birkar later proved) implies the uniform Jordan property for automorphism groups of complex Fano varieties of fixed dimension. This property in particular gives an upper bound on the size of semi-simple groups (meaning those with no non-trivial normal abelian subgroups) acting faithfully on n-dimensional complex Fano varieties, and this bound only depends on n. In this talk, we investigate the consequences of a large action by a particular semi-simple group: the symmetric group. This work is joint with Louis Esser and Joaquín Moraga.

### 2023年07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/002号室

(7/11更新) 中止となりました。来学期への延期を検討しています。

TBA

(7/11更新) 中止となりました。来学期への延期を検討しています。

**石井 志保子 氏**(東京大学)TBA

### 2023年07月21日(金)

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

普段と部屋が異なります。

The Demailly--Peternell--Schneider conjecture is true in positive characteristic

普段と部屋が異なります。

**江尻 祥 氏**(大阪公立大学)The Demailly--Peternell--Schneider conjecture is true in positive characteristic

[ 講演概要 ]

In 1993, Demailly, Peternell and Schneider conjectured that the Albanese morphism of a compact K\"{a}hler manifold with nef anti-canonical divisor is surjective. For smooth projective varieties of characteristic zero, the conjecture was verified by Zhang in 1996. In positive characteristic, the conjecture was solved under the assumption that the geometric generic fiber F of the Albanese morphism has only mild singularities. However, F may have bad singularities even if we restrict ourselves to the case when the anti-canonical divisor is nef. In this talk, we prove the conjecture in positive characteristic without any extra assumption. We also discuss properties of the Albanese morphism, such as flatness or local isotriviality. This talk is based on joint work with Zsolt Patakfalvi.

In 1993, Demailly, Peternell and Schneider conjectured that the Albanese morphism of a compact K\"{a}hler manifold with nef anti-canonical divisor is surjective. For smooth projective varieties of characteristic zero, the conjecture was verified by Zhang in 1996. In positive characteristic, the conjecture was solved under the assumption that the geometric generic fiber F of the Albanese morphism has only mild singularities. However, F may have bad singularities even if we restrict ourselves to the case when the anti-canonical divisor is nef. In this talk, we prove the conjecture in positive characteristic without any extra assumption. We also discuss properties of the Albanese morphism, such as flatness or local isotriviality. This talk is based on joint work with Zsolt Patakfalvi.

### 2023年06月28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

(6/27更新) 講演者の都合で中止となりました。

Preimages question and dynamical cancellation

(6/27更新) 講演者の都合で中止となりました。

**松澤 陽介 氏**(大阪公立大学)Preimages question and dynamical cancellation

[ 講演概要 ]

Pulling back an invariant subvariety by a self-morphism on projective variety, you will get a tower of increasing closed subsets. Working over a number field, we expect that the set of rational points contained in this increasing subsets eventually stabilizes. I am planning to discuss several results on this problem, such as the case of etale morphisms, morphisms on the product of two P^1. I will also present some counter examples that occur when we drop some of the assumptions. This work is based on a joint work with Matt Satriano and Jason Bell, and recent work in progress with Kaoru Sano.

Pulling back an invariant subvariety by a self-morphism on projective variety, you will get a tower of increasing closed subsets. Working over a number field, we expect that the set of rational points contained in this increasing subsets eventually stabilizes. I am planning to discuss several results on this problem, such as the case of etale morphisms, morphisms on the product of two P^1. I will also present some counter examples that occur when we drop some of the assumptions. This work is based on a joint work with Matt Satriano and Jason Bell, and recent work in progress with Kaoru Sano.

### 2023年06月23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

Minimal log discrepnacies for quotient singularities

**柴田 康介 氏**(東京電機大学)Minimal log discrepnacies for quotient singularities

[ 講演概要 ]

In this talk, I will discuss recent joint work with Yusuke Nakamura on minimal log discrepancies for quotient singularities. The minimal log discrepancy is an important invariant of singularities in birational geometry. The denominator of the minimal log discrepancy of a variety depends on the Gorenstein index. On the other hand, Shokurov conjectured that the Gorenstein index of a Q-Gorenstein germ can be bounded in terms of its dimension and minimal log discrepancy. In this talk, I will explain basic properties for quotient singularities and show Shokurov's index conjecture for quotient singularities.

In this talk, I will discuss recent joint work with Yusuke Nakamura on minimal log discrepancies for quotient singularities. The minimal log discrepancy is an important invariant of singularities in birational geometry. The denominator of the minimal log discrepancy of a variety depends on the Gorenstein index. On the other hand, Shokurov conjectured that the Gorenstein index of a Q-Gorenstein germ can be bounded in terms of its dimension and minimal log discrepancy. In this talk, I will explain basic properties for quotient singularities and show Shokurov's index conjecture for quotient singularities.

### 2023年06月14日(水)

14:00-15:30 数理科学研究科棟(駒場) ハイブリッド開催/056号室

普段と曜日・時間・場所が異なります。

Vanishing of local cohomology modules

普段と曜日・時間・場所が異なります。

**Wenliang Zhang 氏**(University of Illinois Chicago)Vanishing of local cohomology modules

[ 講演概要 ]

Studying the vanishing of local cohomology modules has a long and rich history, and is still an active research area. In this talk, we will discuss classic theorems (due to Grothendieck, Hartshorne, Peskine-Szpiro, and Ogus), recent developments, and some open problems.

Studying the vanishing of local cohomology modules has a long and rich history, and is still an active research area. In this talk, we will discuss classic theorems (due to Grothendieck, Hartshorne, Peskine-Szpiro, and Ogus), recent developments, and some open problems.

### 2023年06月07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

普段と曜日と部屋が異なります.

Quasi-F-splitting and Hodge-Witt

普段と曜日と部屋が異なります.

**呼子 笛太郎 氏**(名古屋大学)Quasi-F-splitting and Hodge-Witt

[ 講演概要 ]

Quasi-F-splitting is an extension of F-splitting, which is defined for schemes in positive characteristic. On the other hand, Hodge-Wittness is defined for smooth proper schemes over a perfect field using the de Rham-Witt complex and ordinarity implies Hodge-Wittness. In this talk, I will explain (unexpected) relations between F-split/quasi-F-split and ordinary/Hodge-Witt via examples and properties.

Quasi-F-splitting is an extension of F-splitting, which is defined for schemes in positive characteristic. On the other hand, Hodge-Wittness is defined for smooth proper schemes over a perfect field using the de Rham-Witt complex and ordinarity implies Hodge-Wittness. In this talk, I will explain (unexpected) relations between F-split/quasi-F-split and ordinary/Hodge-Witt via examples and properties.

### 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.