## Algebraic Geometry Seminar

Seminar information archive ～10/10｜Next seminar｜Future seminars 10/11～

Date, time & place | Friday 13:30 - 15:00 ハイブリッド開催/117Room #ハイブリッド開催/117 (Graduate School of Math. Sci. Bldg.) |
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Organizer(s) | GONGYO Yoshinori, NAKAMURA Yusuke, TANAKA Hiromu |

**Seminar information archive**

### 2024/10/04

13:30-15:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Arithmetic finiteness of Mukai varieties of genus 7 (日本語)

**Teppei Takamatsu**(Kyoto University)Arithmetic finiteness of Mukai varieties of genus 7 (日本語)

[ Abstract ]

Fano threefolds over C with Picard number and index equal to 1 are known to be classified by their genus g (where 2≤g≤12 and g≠11). In particular, Mukai has shown that those with genus 7 can be described as hyperplane sections of a connected component of the 10-dimensional orthogonal Grassmannian.

In this talk, we discuss the arithmetic properties of these genus 7 threefolds and their higher-dimensional generalizations (called Mukai varieties of genus 7). More precisely, we consider the finiteness problem of varieties over a ring of S-integers (so called the Shafarevich conjecture), and the existence problem of varieties over the rational integer ring Z.

This talk is based on a joint work with Tetsushi Ito, Akihiro Kanemitsu, and Yuuji Tanaka.

Fano threefolds over C with Picard number and index equal to 1 are known to be classified by their genus g (where 2≤g≤12 and g≠11). In particular, Mukai has shown that those with genus 7 can be described as hyperplane sections of a connected component of the 10-dimensional orthogonal Grassmannian.

In this talk, we discuss the arithmetic properties of these genus 7 threefolds and their higher-dimensional generalizations (called Mukai varieties of genus 7). More precisely, we consider the finiteness problem of varieties over a ring of S-integers (so called the Shafarevich conjecture), and the existence problem of varieties over the rational integer ring Z.

This talk is based on a joint work with Tetsushi Ito, Akihiro Kanemitsu, and Yuuji Tanaka.

### 2024/07/04

13:00-14:30 Room #ハイブリッド開催/118 (Graduate School of Math. Sci. Bldg.)

On a principle of Ogus: the Hasse invariant's order of vanishing and "Frobenius and the Hodge filtration'' (English)

**Stefan Reppen**(University of Tokyo)On a principle of Ogus: the Hasse invariant's order of vanishing and "Frobenius and the Hodge filtration'' (English)

[ Abstract ]

In joint work with W. Goldring we generalize a result of Ogus that, under certain technical conditions, the vanishing order of the Hasse invariant of a family $Y/X$ of $n$-dimensional Calabi-Yau varieties in characteristic $p$ at a point $x$ of $X$ equals the "conjugate line position" of $H^n_{\dR}(Y/X)$ at $x$, i.e. the largest $i$ such that the line of the conjugate filtration is contained in $\text{Fil}^i$ of the Hodge filtration. For every triple $(G,\mu,r)$ consisting of a connected, reductive $\mathbb{F}_p$-group $G$, a cocharacter $\mu \in X_*(G)$ and an $\mathbb{F}_p$-representation $r$ of $G$, we state a generalized Ogus Principle. If $\zeta:X \to \GZip^{\mu}$ is a smooth morphism, then the group theoretic Ogus Principle implies an Ogus Principle on $X$. We deduce an Ogus Principle for several Hodge and abelian-type Shimura varieties and the moduli space of K3 surfaces. In the talk I will present this work.

In joint work with W. Goldring we generalize a result of Ogus that, under certain technical conditions, the vanishing order of the Hasse invariant of a family $Y/X$ of $n$-dimensional Calabi-Yau varieties in characteristic $p$ at a point $x$ of $X$ equals the "conjugate line position" of $H^n_{\dR}(Y/X)$ at $x$, i.e. the largest $i$ such that the line of the conjugate filtration is contained in $\text{Fil}^i$ of the Hodge filtration. For every triple $(G,\mu,r)$ consisting of a connected, reductive $\mathbb{F}_p$-group $G$, a cocharacter $\mu \in X_*(G)$ and an $\mathbb{F}_p$-representation $r$ of $G$, we state a generalized Ogus Principle. If $\zeta:X \to \GZip^{\mu}$ is a smooth morphism, then the group theoretic Ogus Principle implies an Ogus Principle on $X$. We deduce an Ogus Principle for several Hodge and abelian-type Shimura varieties and the moduli space of K3 surfaces. In the talk I will present this work.

### 2024/06/28

13:30-15:00 Room #ハイブリッド開催/117 (Graduate School of Math. Sci. Bldg.)

Stable rationality of hypersurfaces in mock toric varieties (日本語)

**Taro Yoshino**(The University of Tokyo)Stable rationality of hypersurfaces in mock toric varieties (日本語)

[ Abstract ]

In recent years, there has been a development in approaching rationality problems through motivic methods. This approach requires the explicit construction of degeneration families over curves with favorable properties. However, the specific construction is generally difficult. Nicaise and Ottem combined combinatorial methods to construct degeneration families of hypersurfaces in toric varieties and mentioned the stable rationality of a very general hypersurface in projective spaces. In this talk, we mention the following two points: First, I introduce the notion of mock toric varieties, which are generalizations of toric varieties. Second, I combinatorially construct degeneration families of hypersurfaces in mock toric varieties, and I mention the irrationality of a very general hypersurface in the complex Grassmannian variety Gr(2, n).

In recent years, there has been a development in approaching rationality problems through motivic methods. This approach requires the explicit construction of degeneration families over curves with favorable properties. However, the specific construction is generally difficult. Nicaise and Ottem combined combinatorial methods to construct degeneration families of hypersurfaces in toric varieties and mentioned the stable rationality of a very general hypersurface in projective spaces. In this talk, we mention the following two points: First, I introduce the notion of mock toric varieties, which are generalizations of toric varieties. Second, I combinatorially construct degeneration families of hypersurfaces in mock toric varieties, and I mention the irrationality of a very general hypersurface in the complex Grassmannian variety Gr(2, n).

### 2024/06/21

13:30-15:00 Room #ハイブリッド開催/056 (Graduate School of Math. Sci. Bldg.)

ON THE POWER SERIES OF DENEF AND LOESER'S MOTIVIC VANISHING CYCLES OF JET POLYNOMIALS (English)

**Kien Nguyen Huu**(Normandie Université/KU Leuven)ON THE POWER SERIES OF DENEF AND LOESER'S MOTIVIC VANISHING CYCLES OF JET POLYNOMIALS (English)

[ Abstract ]

Let f be a non-constant polynomial in n variables over a field k of characteristic

0. Denef and Loeser introduced the notion of motivic vanishing cycles of f as an element in

the localization Mμˆ of the Grothendieck ring Kμˆ(Var ) of k-varieties with a good action of k0k

μˆ := lim μm by inverting the affne line equipped with the trivial action of μˆ, where μm

is the group scheme over k of mth roots of unity. In particular, if k is the field of complex

numbers then Denef and Loeser showed that their motivic vanishing cycles and the complex

φf [n − 1] has the same Hodge characteristic, where φf is the complex of vanishing cycles

in the usual sense. Motivated by the Igusa conjecture for exponential sums and the strong

monodromy conjecture, we introduce the notion of Poincaré series of Denef-Loeser's van-

ishing cycles of jet polynomials of f, where jet polynomials of f are polynomials appearing

naturally when we compute the jet schemes of f. By using Davison-Meinhardt's conjecture

which was proved by Nicaise and Payne in 2019, we can show that our Poincaré series is a

rational function over a quotient ring of Mμˆ by very natural relations. In particular, we can k

recovery Denef and Loeser's motivic vanishing cycles from our Poincaré series. Moreover, we can show that our Poincaré series owns a universal property in the sense that if k is a number field then the Igusa local zeta functions, the motivic Igusa zeta functions, the Poincaré series of exponential sums modulo pm of f can be obtained from our Poincaré se- ries by suitable specialization maps preserving the rationality. If time permits, I will present some initial consequences that have arisen during the study of our Poincaré series.

Let f be a non-constant polynomial in n variables over a field k of characteristic

0. Denef and Loeser introduced the notion of motivic vanishing cycles of f as an element in

the localization Mμˆ of the Grothendieck ring Kμˆ(Var ) of k-varieties with a good action of k0k

μˆ := lim μm by inverting the affne line equipped with the trivial action of μˆ, where μm

is the group scheme over k of mth roots of unity. In particular, if k is the field of complex

numbers then Denef and Loeser showed that their motivic vanishing cycles and the complex

φf [n − 1] has the same Hodge characteristic, where φf is the complex of vanishing cycles

in the usual sense. Motivated by the Igusa conjecture for exponential sums and the strong

monodromy conjecture, we introduce the notion of Poincaré series of Denef-Loeser's van-

ishing cycles of jet polynomials of f, where jet polynomials of f are polynomials appearing

naturally when we compute the jet schemes of f. By using Davison-Meinhardt's conjecture

which was proved by Nicaise and Payne in 2019, we can show that our Poincaré series is a

rational function over a quotient ring of Mμˆ by very natural relations. In particular, we can k

recovery Denef and Loeser's motivic vanishing cycles from our Poincaré series. Moreover, we can show that our Poincaré series owns a universal property in the sense that if k is a number field then the Igusa local zeta functions, the motivic Igusa zeta functions, the Poincaré series of exponential sums modulo pm of f can be obtained from our Poincaré se- ries by suitable specialization maps preserving the rationality. If time permits, I will present some initial consequences that have arisen during the study of our Poincaré series.

### 2024/06/07

13:30-15:00 Room #ハイブリッド開催/056 (Graduate School of Math. Sci. Bldg.)

K-stability of pointless Fano 3-folds (English)

**Ivan Cheltsov**(University of Edinburgh)K-stability of pointless Fano 3-folds (English)

[ Abstract ]

In this talk we will show how to prove that all pointless smooth Fano 3-folds defined over a subfield of the field of complex numbers are Kahler-Einstein unless they belong to 8 exceptional deformation families. This is a joint work in progress with Hamid Abban (Nottingham) and Frederic Mangolte (Marseille).

In this talk we will show how to prove that all pointless smooth Fano 3-folds defined over a subfield of the field of complex numbers are Kahler-Einstein unless they belong to 8 exceptional deformation families. This is a joint work in progress with Hamid Abban (Nottingham) and Frederic Mangolte (Marseille).

### 2024/05/24

13:30-15:00 Room #ハイブリッド開催/056 (Graduate School of Math. Sci. Bldg.)

Boundedness of weak Fano threefolds with fixed Gorenstein index in positive characteristic

**Kenta Sato**(Kyusyu University)Boundedness of weak Fano threefolds with fixed Gorenstein index in positive characteristic

[ Abstract ]

In this talk, we give a partial affirmative answer to the BAB conjecture for 3-folds in characteristic p>5. Specifically, we prove that a set of weak Fano 3-folds over an uncountable algebraically closed field is bounded, if each element X satisfies certain conditions regarding the Gorenstein index, a complement and Kodaira type vanishing. In the course of the proof, we also study a uniform lower bound for Seshadri constants of nef and big invertible sheaves on projective 3-folds.

In this talk, we give a partial affirmative answer to the BAB conjecture for 3-folds in characteristic p>5. Specifically, we prove that a set of weak Fano 3-folds over an uncountable algebraically closed field is bounded, if each element X satisfies certain conditions regarding the Gorenstein index, a complement and Kodaira type vanishing. In the course of the proof, we also study a uniform lower bound for Seshadri constants of nef and big invertible sheaves on projective 3-folds.

### 2024/05/17

13:30-15:00 Room #ハイブリッド開催/056 (Graduate School of Math. Sci. Bldg.)

非分離Kummer曲面 (日本語)

**Yuya Matsumoto**(Tokyo University of Science)非分離Kummer曲面 (日本語)

[ Abstract ]

Kummer曲面Km(A)とは，+-1倍写像によるアーベル曲面Aの商の最小特異点解消として得られる曲面である．Aが標数≠2の場合（resp. 標数2で，超特異ではない場合）は，Km(A)はK3曲面であり，例外曲線は互いに交わらない（resp. 所定の交わり方をする）16本の有理曲線である．Aが標数2で超特異の場合はKm(A)はK3曲面にならない．また，Km(A)が標数2の超特異K3曲面になることはない．

本講演では，標数2の超特異K3曲面とその上の16本の有理曲線で所定の交わり方をするものに対し，非分離2重被覆Aを構成することができること，Aは非特異部分に群構造が入り「アーベル曲面もどき」になることを示す．Aの分類のために，RDP K3曲面のRDPの補集合から最小特異点解消への B_n \Omega^1（Cartier作用素を何回か適用すると消える1次微分形式の層）の延長に関する結果を用いるので，これにも言及したい．

プレプリントは https://arxiv.org/abs/2403.02770 でご覧いただけます．

Kummer曲面Km(A)とは，+-1倍写像によるアーベル曲面Aの商の最小特異点解消として得られる曲面である．Aが標数≠2の場合（resp. 標数2で，超特異ではない場合）は，Km(A)はK3曲面であり，例外曲線は互いに交わらない（resp. 所定の交わり方をする）16本の有理曲線である．Aが標数2で超特異の場合はKm(A)はK3曲面にならない．また，Km(A)が標数2の超特異K3曲面になることはない．

本講演では，標数2の超特異K3曲面とその上の16本の有理曲線で所定の交わり方をするものに対し，非分離2重被覆Aを構成することができること，Aは非特異部分に群構造が入り「アーベル曲面もどき」になることを示す．Aの分類のために，RDP K3曲面のRDPの補集合から最小特異点解消への B_n \Omega^1（Cartier作用素を何回か適用すると消える1次微分形式の層）の延長に関する結果を用いるので，これにも言及したい．

プレプリントは https://arxiv.org/abs/2403.02770 でご覧いただけます．

### 2024/04/26

14:00-15:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Frobenius stable Grauert-Riemenschneider vanishing fails (日本語)

**Tatsuro Kawakami**(Kyoto University)Frobenius stable Grauert-Riemenschneider vanishing fails (日本語)

[ Abstract ]

We show that the Frobenius stable version of Grauert-Riemenschneider vanishing fails for a terminal 3-fold in characteristic 2. To prove this, we introduce the notion of $F_p$-rationality for singularities in positive characteristic, and prove that 3-dimensional klt singularities are $F_p$-rational. I will also talk about the vanishing of $F_p$-cohomologies of log Fano threefolds. This is joint work with Jefferson Baudin and Fabio Bernasconi.

We show that the Frobenius stable version of Grauert-Riemenschneider vanishing fails for a terminal 3-fold in characteristic 2. To prove this, we introduce the notion of $F_p$-rationality for singularities in positive characteristic, and prove that 3-dimensional klt singularities are $F_p$-rational. I will also talk about the vanishing of $F_p$-cohomologies of log Fano threefolds. This is joint work with Jefferson Baudin and Fabio Bernasconi.

### 2023/12/15

13:30-15:00 Room #118 (Graduate School of Math. Sci. Bldg.)

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

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

[ Abstract ]

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.

### 2023/11/24

14:00-15:30 Room #ハイブリッド開催/056 (Graduate School of Math. Sci. Bldg.)

On Kawamata-Miyaoka type inequality

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

[ Abstract ]

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.

### 2023/10/16

14:00-15:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Symmetries of Fano varieties

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

[ Abstract ]

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.

### 2023/07/28

13:30-15:00 Room #ハイブリッド開催/002 (Graduate School of Math. Sci. Bldg.)

TBA

**Shihoko Ishii**(The University of Tokyo)TBA

### 2023/07/21

13:30-15:00 Room #ハイブリッド開催/056 (Graduate School of Math. Sci. Bldg.)

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

**Sho Ejiri**(Osaka Metropolitan University)The Demailly--Peternell--Schneider conjecture is true in positive characteristic

[ Abstract ]

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.

### 2023/06/28

13:30-15:00 Room #ハイブリッド開催/056 (Graduate School of Math. Sci. Bldg.)

Preimages question and dynamical cancellation

**Yohsuke Matsuzawa**(Osaka Metropolitan University)Preimages question and dynamical cancellation

[ Abstract ]

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.

### 2023/06/23

13:30-15:00 Room #ハイブリッド開催/117 (Graduate School of Math. Sci. Bldg.)

Minimal log discrepnacies for quotient singularities

**Kohsuke Shibata**(Tokyo Denki University)Minimal log discrepnacies for quotient singularities

[ Abstract ]

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.

### 2023/06/14

14:00-15:30 Room #ハイブリッド開催/056 (Graduate School of Math. Sci. Bldg.)

Vanishing of local cohomology modules

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

[ Abstract ]

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.

### 2023/06/07

13:30-15:00 Room #ハイブリッド開催/056 (Graduate School of Math. Sci. Bldg.)

Quasi-F-splitting and Hodge-Witt

**Fuetaro Yobuko**(Nagoya University)Quasi-F-splitting and Hodge-Witt

[ Abstract ]

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.

### 2023/05/26

13:30-15:00 Room #ハイブリッド開催/117 (Graduate School of Math. Sci. Bldg.)

Varieties in positive characteristic with numerically flat tangent bundle

**Shou Yoshikawa**(Tokyo Institute of Technology, RIKEN)Varieties in positive characteristic with numerically flat tangent bundle

[ Abstract ]

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.

### 2023/05/10

13:30-15:00 Room #ハイブリッド開催/056 (Graduate School of Math. Sci. Bldg.)

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

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

[ Abstract ]

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.

### 2023/04/28

13:30-15:00 Room #ハイブリッド開催/117 (Graduate School of Math. Sci. Bldg.)

On the degree of irrationality of complete intersections (Japanese )

**Taro Yoshino**(Tokyo)On the degree of irrationality of complete intersections (Japanese )

[ Abstract ]

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.

### 2023/04/21

14:00-15:30 Room #117 (Graduate School of Math. Sci. Bldg.)

Endomorphisms of varieties and Bott vanishing (Japanese)

**Tatsuro Kawakami**(Kyoto University)Endomorphisms of varieties and Bott vanishing (Japanese)

[ Abstract ]

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.

### 2023/04/21

12:45-13:45 Room #ハイブリッド開催/117 (Graduate School of Math. Sci. Bldg.)

ACC of plc thresholds (English)

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

[ Abstract ]

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.

### 2023/03/28

10:00-11:30 Room #ハイブリッド開催/056 (Graduate School of Math. Sci. Bldg.)

On the canonical bundle formula in positive characteristic (English)

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

[ Abstract ]

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.

### 2023/03/10

13:15-14:45 Room #ハイブリッド開催/123 (Graduate School of Math. Sci. Bldg.)

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

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

[ Abstract ]

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.

### 2023/02/20

10:00-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

The 4th lecture of series talks

K-stability of Fano varieties. (English)

The 4th lecture of series talks

**Chenyang Xu**(Princeton University)K-stability of Fano varieties. (English)

[ Abstract ]

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.