## 代数幾何学セミナー

過去の記録 ～03/31｜次回の予定｜今後の予定 04/01～

開催情報 | 火曜日 10:30～11:30 or 12:00 数理科学研究科棟(駒場) ハイブリッド開催/002号室 |
---|---|

担当者 | 權業 善範・中村 勇哉・田中公 |

**過去の記録**

### 2017年11月07日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Characterizations of projective space and Seshadri constants in arbitrary characteristic

**村山 匠 氏**(ミシガン大学)Characterizations of projective space and Seshadri constants in arbitrary characteristic

[ 講演概要 ]

Mori and Mukai conjectured that projective space should be the only n-dimensional Fano variety whose anti-canonical bundle has degree at least n + 1 along every curve. While this conjecture has been proved in characteristic zero, it remains open in positive characteristic. We will present some progress in this direction by giving another characterization of projective space using Seshadri constants and the Frobenius morphism. The key ingredient is a positive-characteristic analogue of Demailly’s criterion for separation of higher-order jets by adjoint bundles, whose proof gives new results for adjoint bundles even in characteristic zero.

Mori and Mukai conjectured that projective space should be the only n-dimensional Fano variety whose anti-canonical bundle has degree at least n + 1 along every curve. While this conjecture has been proved in characteristic zero, it remains open in positive characteristic. We will present some progress in this direction by giving another characterization of projective space using Seshadri constants and the Frobenius morphism. The key ingredient is a positive-characteristic analogue of Demailly’s criterion for separation of higher-order jets by adjoint bundles, whose proof gives new results for adjoint bundles even in characteristic zero.

### 2017年10月31日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

ACC for log canonical threshold polytopes (English)

**Zhan Li 氏**(Beijing)ACC for log canonical threshold polytopes (English)

[ 講演概要 ]

We show that the log canonical threshold polytopes of varieties with log canonical singularities satisfy the ascending chain condition. This is a joint work with Jingjun Han and Lu Qi.

We show that the log canonical threshold polytopes of varieties with log canonical singularities satisfy the ascending chain condition. This is a joint work with Jingjun Han and Lu Qi.

### 2017年10月30日(月)

10:30-12:00 数理科学研究科棟(駒場) 123号室

Towards birational boundedness of elliptic Calabi-Yau varieties (English)

**Roberto Svaldi 氏**(Cambridge)Towards birational boundedness of elliptic Calabi-Yau varieties (English)

[ 講演概要 ]

I will discuss new results towards the birational boundedness of

low-dimensional elliptic Calabi-Yau varieties, joint work with Gabriele

Di Certo.

Recent work in the minimal model program suggests that pairs with trivial log canonical

class should satisfy some boundedness properties.

I will show that 4-dimensional Calabi-Yau pairs which are not birational to a product are

indeed log birationally bounded. This implies birational boundedness of elliptically fibered

Calabi-Yau manifolds with a section, in dimension up to 5.

If time allows, I will also try to discuss a first approach towards boundedness of rationally

connected CY varieties in low dimension.

I will discuss new results towards the birational boundedness of

low-dimensional elliptic Calabi-Yau varieties, joint work with Gabriele

Di Certo.

Recent work in the minimal model program suggests that pairs with trivial log canonical

class should satisfy some boundedness properties.

I will show that 4-dimensional Calabi-Yau pairs which are not birational to a product are

indeed log birationally bounded. This implies birational boundedness of elliptically fibered

Calabi-Yau manifolds with a section, in dimension up to 5.

If time allows, I will also try to discuss a first approach towards boundedness of rationally

connected CY varieties in low dimension.

### 2017年10月17日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Intersection of currents, dimension excess and complex dynamics (English)

**Tien Cuong Dinh 氏**(Singapore)Intersection of currents, dimension excess and complex dynamics (English)

[ 講演概要 ]

I will discuss dynamical properties of Henon maps in higher dimension, in particular, the equidistribution property of periodic points. Positive closed currents can be seen as an analytic counterpart of effective algebraic cycles. I will explain how a non-generic intersection theory for these currents, possibly with dimension excess, comes into the picture. Other applications of the intersection theory will be also discussed. This is a joint work with Nessim Sibony.

I will discuss dynamical properties of Henon maps in higher dimension, in particular, the equidistribution property of periodic points. Positive closed currents can be seen as an analytic counterpart of effective algebraic cycles. I will explain how a non-generic intersection theory for these currents, possibly with dimension excess, comes into the picture. Other applications of the intersection theory will be also discussed. This is a joint work with Nessim Sibony.

### 2017年10月10日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Classification of Mukai pairs with corank 3 (English or Japanese)

**金光 秋博 氏**(東大数理)Classification of Mukai pairs with corank 3 (English or Japanese)

[ 講演概要 ]

A Mukai pair $(X,E)$ is a pair of a Fano manifold $X$ and an ample vector bundle $E$ of rank $r$ on $X$ such that $c_1(X)=c_1(E)$. Study of such pairs was proposed by Mukai. It is known that, for a Mukai pair $(X,E)$, the rank $r$ of the bundle $E$ is at most $\dim X +1$, and Mukai conjectured the explicit

classification with $r \geq \dim X$. The above conjecture was solved independently by Fujita, Peternell and Ye-Zhang. Also the classification of Mukai pairs with $r= \dim X -1$ was given by Peternell-Szurek-Wi\'sniewski. In this talk I will give the classification of Mukai pairs with $r= \dim X -2$ and $\dim X \geq 5$.

A Mukai pair $(X,E)$ is a pair of a Fano manifold $X$ and an ample vector bundle $E$ of rank $r$ on $X$ such that $c_1(X)=c_1(E)$. Study of such pairs was proposed by Mukai. It is known that, for a Mukai pair $(X,E)$, the rank $r$ of the bundle $E$ is at most $\dim X +1$, and Mukai conjectured the explicit

classification with $r \geq \dim X$. The above conjecture was solved independently by Fujita, Peternell and Ye-Zhang. Also the classification of Mukai pairs with $r= \dim X -1$ was given by Peternell-Szurek-Wi\'sniewski. In this talk I will give the classification of Mukai pairs with $r= \dim X -2$ and $\dim X \geq 5$.

### 2017年07月18日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

On a generalization of Frobenius-splitting and a lifting problem of Calabi-Yau varieties (JAPANESE)

**呼子 笛太郎 氏**(東北大理)On a generalization of Frobenius-splitting and a lifting problem of Calabi-Yau varieties (JAPANESE)

[ 講演概要 ]

In this talk, we introduce a notion of Frobenius-splitting height which quantifies Frobenius-splitting varieties and show that a Calabi-Yau variety of finite height over an algebraically closed field of positive characteristic admits a flat lifting to the ring of Witt vectors of length two.

In this talk, we introduce a notion of Frobenius-splitting height which quantifies Frobenius-splitting varieties and show that a Calabi-Yau variety of finite height over an algebraically closed field of positive characteristic admits a flat lifting to the ring of Witt vectors of length two.

### 2017年07月11日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Arithmetic and dynamical degrees of self-maps of algebraic varieties (English or Japanese)

**松澤 陽介 氏**(東大数理)Arithmetic and dynamical degrees of self-maps of algebraic varieties (English or Japanese)

[ 講演概要 ]

The first dynamical degree is an important birational invariant which measures the geometric complexity of dominant rational self-maps of algebraic varieties. On the other hand, when the variety is defined over a number field, one can associate to an orbit an invariant using Weil height function, called arithmetic degree, which measures the arithmetic complexity of the orbit. It is conjectured that the arithmetic degree of a Zariski dense orbit is equal to the first dynamical degree (Kawaguchi-Silverman). I will explain several results related to this conjecture. I will also explain applications to proofs of purely geometric statements.

The first dynamical degree is an important birational invariant which measures the geometric complexity of dominant rational self-maps of algebraic varieties. On the other hand, when the variety is defined over a number field, one can associate to an orbit an invariant using Weil height function, called arithmetic degree, which measures the arithmetic complexity of the orbit. It is conjectured that the arithmetic degree of a Zariski dense orbit is equal to the first dynamical degree (Kawaguchi-Silverman). I will explain several results related to this conjecture. I will also explain applications to proofs of purely geometric statements.

### 2017年07月04日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

The space of rational curves and Manin's conjecture (English)

**谷本 祥 氏**(University of Copenhagen)The space of rational curves and Manin's conjecture (English)

[ 講演概要 ]

Manin's conjecture predicts the asymptotic formula for the counting function of rational points on a Fano variety after removing the exceptional thin set. There are many developments on birational geometry of exceptional sets using MMP, due to Lehmann, myself, Tschinkel, Hacon, and Jiang. Recently we found that the study of exceptional sets has applications to questions regarding the space of rational curves, i.e., its dimension and the number of components. I would like to explain these applications. This is joint work with Brian Lehmann.

Manin's conjecture predicts the asymptotic formula for the counting function of rational points on a Fano variety after removing the exceptional thin set. There are many developments on birational geometry of exceptional sets using MMP, due to Lehmann, myself, Tschinkel, Hacon, and Jiang. Recently we found that the study of exceptional sets has applications to questions regarding the space of rational curves, i.e., its dimension and the number of components. I would like to explain these applications. This is joint work with Brian Lehmann.

### 2017年06月27日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Cylinders in del Pezzo fibrations (English )

**岸本 崇 氏**(埼玉大学)Cylinders in del Pezzo fibrations (English )

[ 講演概要 ]

The cylinder is, by definition, an algebraic variety of the form Z × A1 . Certainly it is geometrically a very simple object, but it plays often an important role to connect unipotent group actions on special kinds of affine algebraic varieties to projective geometry. From the point of view of birational geometry, it is essential to look into cylinders found on Mori fiber spaces. In this talk, we shall focus mainly on Mori fiber spaces of relative dimension two or three. One of main results asserts that a del Pezzo fibration π : V → W contains a cylinder respecting the structure of π (so-called a vertical cylinder) if and only if the degree deg π of π is greater than or equal to 5 and π admits a rational section. Especially, in case of dim V = 3, the existence of a vertical cylinder is equivalent to saying deg π ≧ 5 in consideration of Tsen’s theorem, nevertheless, it is worthwhile to note that the affine 3-space A3C is embedded into certains del Pezzo fibrations π : V → P1C of deg π ≦ 4 in a twisted way. This is a joint work with Adrien Dubouloz (Universit ́e de Bourgogne).

The cylinder is, by definition, an algebraic variety of the form Z × A1 . Certainly it is geometrically a very simple object, but it plays often an important role to connect unipotent group actions on special kinds of affine algebraic varieties to projective geometry. From the point of view of birational geometry, it is essential to look into cylinders found on Mori fiber spaces. In this talk, we shall focus mainly on Mori fiber spaces of relative dimension two or three. One of main results asserts that a del Pezzo fibration π : V → W contains a cylinder respecting the structure of π (so-called a vertical cylinder) if and only if the degree deg π of π is greater than or equal to 5 and π admits a rational section. Especially, in case of dim V = 3, the existence of a vertical cylinder is equivalent to saying deg π ≧ 5 in consideration of Tsen’s theorem, nevertheless, it is worthwhile to note that the affine 3-space A3C is embedded into certains del Pezzo fibrations π : V → P1C of deg π ≦ 4 in a twisted way. This is a joint work with Adrien Dubouloz (Universit ́e de Bourgogne).

### 2017年06月12日(月)

17:00-18:30 数理科学研究科棟(駒場) 056号室

普段と曜日・部屋が異なります

Rational and irrational singular quartic threefolds (English)

普段と曜日・部屋が異なります

**Ivan Cheltsov 氏**(The University of Edinburgh)Rational and irrational singular quartic threefolds (English)

[ 講演概要 ]

Burkhardt and Igusa quartics admit a faithful action of the symmetric group of degree 6.

There are other quartic threefolds with this property. All of them are singular.

Beauville proved that all but four of them are irrational. Burkhardt and Igusa quartics are known to be rational.

Two constructions of Todd imply the rationality of the remaining two quartic threefolds.

In this talk, I will give an alternative proof of both these (irrationality and rationality) results.

This proof is based on explicit small resolutions of the so-called Coble fourfold.

This fourfold is the double cover of the four-dimensional projective space branched over Igusa quartic.

This is a joint work with Sasha Kuznetsov and Costya Shramov.

Burkhardt and Igusa quartics admit a faithful action of the symmetric group of degree 6.

There are other quartic threefolds with this property. All of them are singular.

Beauville proved that all but four of them are irrational. Burkhardt and Igusa quartics are known to be rational.

Two constructions of Todd imply the rationality of the remaining two quartic threefolds.

In this talk, I will give an alternative proof of both these (irrationality and rationality) results.

This proof is based on explicit small resolutions of the so-called Coble fourfold.

This fourfold is the double cover of the four-dimensional projective space branched over Igusa quartic.

This is a joint work with Sasha Kuznetsov and Costya Shramov.

### 2017年06月06日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Fano varieties: K-stability and boundedness (English)

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

**Chen Jiang 氏**(IPMU)Fano varieties: K-stability and boundedness (English)

[ 講演概要 ]

There are two interesting problems for Fano varieties, K-stability and boundedness.

Significant progress has been made for both problems recently.

In this talk, I will show the boundedness of K-semistable Fano varieties with anti-canonical degree bounded from below, by using methods from birational geometry.

[ 参考URL ]There are two interesting problems for Fano varieties, K-stability and boundedness.

Significant progress has been made for both problems recently.

In this talk, I will show the boundedness of K-semistable Fano varieties with anti-canonical degree bounded from below, by using methods from birational geometry.

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

### 2017年05月30日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Contractible affine threefolds in smooth Fano threefolds (English or Japanese)

**長岡 大 氏**(東大数理)Contractible affine threefolds in smooth Fano threefolds (English or Japanese)

[ 講演概要 ]

By the contribution of M. Furushima, N. Nakayama, Th. Peternell and M.

Schneider, it is completed to classify all projective compactifications

of the affine $3$-space $\mathbb{A}^3$ with Picard number one.

As a similar question, T. Kishimoto raised the problem to classify all

triplets $(V, U, D_1 \cup D_2)$ which consist of smooth Fano threefolds

$V$ of Picard number two, contractible affine threefolds $U$ as open

subsets of $V$, and the complements $D_1 \cup D_2 =V \setminus U$.

He also solved this problem when the log canonical divisors $K_V+D_1+D_2

$ are not nef.

In this talk, I will discuss the triplets $(V, U, D_1 \cup D_2)$ whose

log canonical divisors are linearly equivalent to zero.

I will also explain how to determine all Fano threefolds $V$ which

appear in such triplets.

By the contribution of M. Furushima, N. Nakayama, Th. Peternell and M.

Schneider, it is completed to classify all projective compactifications

of the affine $3$-space $\mathbb{A}^3$ with Picard number one.

As a similar question, T. Kishimoto raised the problem to classify all

triplets $(V, U, D_1 \cup D_2)$ which consist of smooth Fano threefolds

$V$ of Picard number two, contractible affine threefolds $U$ as open

subsets of $V$, and the complements $D_1 \cup D_2 =V \setminus U$.

He also solved this problem when the log canonical divisors $K_V+D_1+D_2

$ are not nef.

In this talk, I will discuss the triplets $(V, U, D_1 \cup D_2)$ whose

log canonical divisors are linearly equivalent to zero.

I will also explain how to determine all Fano threefolds $V$ which

appear in such triplets.

### 2017年05月23日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Perverse coherent sheaves on blow-ups at codimension two loci (English)

**小関 直紀 氏**(東大数理)Perverse coherent sheaves on blow-ups at codimension two loci (English)

[ 講演概要 ]

I would like to talk about my recent work in progress.

Let us consider the blow-up X of Y along a subvariety C.

Then the following natural question arises:

What is the relation between moduli space of sheaves on Y

and that of X?

H.Nakajima and K.Yoshioka answered the above question

in the case when Y is a surface and C is a point. They

showed that the moduli spaces are connected by a sequence

of flip-like diagrams. The key ingredient of the proof is

to use perverse coherent sheaves in the sense of T.Bridgeland

and M.Van den Bergh.

In this talk, I will explain how to generalize their theorem

to the case when Y is a smooth projective variety of arbitrary

dimension and C is its codimension two subvariety.

I would like to talk about my recent work in progress.

Let us consider the blow-up X of Y along a subvariety C.

Then the following natural question arises:

What is the relation between moduli space of sheaves on Y

and that of X?

H.Nakajima and K.Yoshioka answered the above question

in the case when Y is a surface and C is a point. They

showed that the moduli spaces are connected by a sequence

of flip-like diagrams. The key ingredient of the proof is

to use perverse coherent sheaves in the sense of T.Bridgeland

and M.Van den Bergh.

In this talk, I will explain how to generalize their theorem

to the case when Y is a smooth projective variety of arbitrary

dimension and C is its codimension two subvariety.

### 2017年05月16日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

On separable higher Gauss maps (English)

**古川 勝久 氏**(東大数理)On separable higher Gauss maps (English)

[ 講演概要 ]

We study the $m$-th Gauss map in the sense of F. L. Zak of a projective variety $X ¥subset P^N$ over an algebraically closed field in any characteristic, where $m$ is an integer with $n:= ¥dim(X) ¥leq m < N$. It is known that the contact locus on $X$ of a general tangent $m$-plane can be non-linear in positive characteristic, if the $m$-th Gauss map is inseparable.

In this talk, I will explain that for any $m$, the locus is a linear variety if the $m$-th Gauss map is separable. I will also explain that for smooth $X$ with $n < N-2$, the $(n+1)$-th Gauss

map is birational if it is separable, unless $X$ is the Segre embedding $P^1 ¥times P^n ¥subset P^{2n-1}$. This is related to L. Ein's classification of varieties with small dual varieties in characteristic zero.

This talk is based on a joint work with Atsushi Ito.

We study the $m$-th Gauss map in the sense of F. L. Zak of a projective variety $X ¥subset P^N$ over an algebraically closed field in any characteristic, where $m$ is an integer with $n:= ¥dim(X) ¥leq m < N$. It is known that the contact locus on $X$ of a general tangent $m$-plane can be non-linear in positive characteristic, if the $m$-th Gauss map is inseparable.

In this talk, I will explain that for any $m$, the locus is a linear variety if the $m$-th Gauss map is separable. I will also explain that for smooth $X$ with $n < N-2$, the $(n+1)$-th Gauss

map is birational if it is separable, unless $X$ is the Segre embedding $P^1 ¥times P^n ¥subset P^{2n-1}$. This is related to L. Ein's classification of varieties with small dual varieties in characteristic zero.

This talk is based on a joint work with Atsushi Ito.

### 2017年05月09日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Upper bound of the multiplicity of locally complete intersection singularities (English)

**柴田 康介 氏**(東大数理)Upper bound of the multiplicity of locally complete intersection singularities (English)

[ 講演概要 ]

The multiplicity of a point on a variety is a fundamental invariant to estimate how the singularity is bad. It is introduced in a purely algebraic context. On the other hand, we can also attach to the singularity the log canonical threshold and the minimal log discrepancy, which are introduced in a birational theoretic context. In this talk, we show bounds of the multiplicity by functions of these birational invariants for a singularity of locally a complete intersection. As an application, we obtain the affirmative answer to Watanabe’s conjecture on the multiplicity of canonical singularity of locally a complete intersection up to dimension 32.

The multiplicity of a point on a variety is a fundamental invariant to estimate how the singularity is bad. It is introduced in a purely algebraic context. On the other hand, we can also attach to the singularity the log canonical threshold and the minimal log discrepancy, which are introduced in a birational theoretic context. In this talk, we show bounds of the multiplicity by functions of these birational invariants for a singularity of locally a complete intersection. As an application, we obtain the affirmative answer to Watanabe’s conjecture on the multiplicity of canonical singularity of locally a complete intersection up to dimension 32.

### 2017年04月25日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

On the Picard number of Fano 6-folds with a non-small contraction (English)

**鈴木 拓 氏**(早稲田大学)On the Picard number of Fano 6-folds with a non-small contraction (English)

[ 講演概要 ]

A generalization of S. Mukai's conjecture says that $\rho(i-1) \leq n$ holds for any Fano $n$-fold with Picard number $\rho$ and pseudo-index $i$, with equality if and only if it is isomorphic to $(\mathbb{P}^{i-1})^{\rho}$. In this talk, we consider this conjecture for $n=6$, which is an open problem, and give a proof of some special cases.

A generalization of S. Mukai's conjecture says that $\rho(i-1) \leq n$ holds for any Fano $n$-fold with Picard number $\rho$ and pseudo-index $i$, with equality if and only if it is isomorphic to $(\mathbb{P}^{i-1})^{\rho}$. In this talk, we consider this conjecture for $n=6$, which is an open problem, and give a proof of some special cases.

### 2017年04月18日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

On the existence of almost Fano threefolds with del Pezzo fibrations (English)

**福岡 尊 氏**(東大数理)On the existence of almost Fano threefolds with del Pezzo fibrations (English)

[ 講演概要 ]

We say that a smooth projective 3-fold is almost Fano if its anti-canonical divisor is nef and big but not ample. By Jahnke-Peternell-Radloff and Takeuchi, the numerical classification of such 3-folds was given. Among the classification results, there exists precisely 10 cases such that it was yet to be known whether these have an example or not. The main result of this talk shows the existence of examples of each of 10 cases. In 9 cases of the 10 cases, the degree of del Pezzo fibrations are 6. We will discuss one of the reason of difficulty constructing del Pezzo fibrations of degree 6. After that, we will show that every almost Fano del Pezzo fibration of degree 6 with specific anti-canonical volume can be embedded into some higher dimensional del Pezzo fibration as a relative linear section.

We say that a smooth projective 3-fold is almost Fano if its anti-canonical divisor is nef and big but not ample. By Jahnke-Peternell-Radloff and Takeuchi, the numerical classification of such 3-folds was given. Among the classification results, there exists precisely 10 cases such that it was yet to be known whether these have an example or not. The main result of this talk shows the existence of examples of each of 10 cases. In 9 cases of the 10 cases, the degree of del Pezzo fibrations are 6. We will discuss one of the reason of difficulty constructing del Pezzo fibrations of degree 6. After that, we will show that every almost Fano del Pezzo fibration of degree 6 with specific anti-canonical volume can be embedded into some higher dimensional del Pezzo fibration as a relative linear section.

### 2017年02月10日(金)

14:00-15:30 数理科学研究科棟(駒場) 002号室

Stability theory of a klt singularity II (English)

**Chenyang Xu 氏**(Beijing International Center of Mathematics Research)Stability theory of a klt singularity II (English)

[ 講演概要 ]

In higher dimensional geometry, it has been known that from many perspectives a log terminal singularity is a local analogue of Fano varieties. Many statements of Fano varieties have a counterpart for log terminal singularities. One central topic on the geometry of a Fano variety is its stability which in particular reflects whether the Fano variety carries a canonical metric. In the talks, we will discuss a series of recent works started by Chi Li, and then by Harold Blum, Yuchen Liu and myself, in which we want to establish an algebro-geometric stability theory of a fixed log terminal singularity. Inspired by the study from differential geometry, (e.g. metric tangent cone, Sasakian-Einstein metric), for any log terminal singularity, we investigate the valuation which has the minimal normalized volume. Our goal is to prove various properties of this valuation which enable us to degenerate the singularity to a K-semistable T-singularity (with a torus action) in the Sasakian-Einstein sense.

In higher dimensional geometry, it has been known that from many perspectives a log terminal singularity is a local analogue of Fano varieties. Many statements of Fano varieties have a counterpart for log terminal singularities. One central topic on the geometry of a Fano variety is its stability which in particular reflects whether the Fano variety carries a canonical metric. In the talks, we will discuss a series of recent works started by Chi Li, and then by Harold Blum, Yuchen Liu and myself, in which we want to establish an algebro-geometric stability theory of a fixed log terminal singularity. Inspired by the study from differential geometry, (e.g. metric tangent cone, Sasakian-Einstein metric), for any log terminal singularity, we investigate the valuation which has the minimal normalized volume. Our goal is to prove various properties of this valuation which enable us to degenerate the singularity to a K-semistable T-singularity (with a torus action) in the Sasakian-Einstein sense.

### 2017年02月07日(火)

15:30-17:00 数理科学研究科棟(駒場) 117号室

Stability theory of a klt singularity I (English)

**Chenyang Xu 氏**(Beijing International Center of Mathematics Research)Stability theory of a klt singularity I (English)

[ 講演概要 ]

In higher dimensional geometry, it has been known that from many perspectives a log terminal singularity is a local analogue of Fano varieties. Many statements of Fano varieties have a counterpart for log terminal singularities. One central topic on the geometry of a Fano variety is its stability which in particular reflects whether the Fano variety carries a canonical metric. In the talks, we will discuss a series of recent works started by Chi Li, and then by Harold Blum, Yuchen Liu and myself, in which we want to establish an algebro-geometric stability theory of a fixed log terminal singularity. Inspired by the study from differential geometry, (e.g. metric tangent cone, Sasakian-Einstein metric), for any log terminal singularity, we investigate the valuation which has the minimal normalized volume. Our goal is to prove various properties of this valuation which enable us to degenerate the singularity to a K-semistable T-singularity (with a torus action) in the Sasakian-Einstein sense.

In higher dimensional geometry, it has been known that from many perspectives a log terminal singularity is a local analogue of Fano varieties. Many statements of Fano varieties have a counterpart for log terminal singularities. One central topic on the geometry of a Fano variety is its stability which in particular reflects whether the Fano variety carries a canonical metric. In the talks, we will discuss a series of recent works started by Chi Li, and then by Harold Blum, Yuchen Liu and myself, in which we want to establish an algebro-geometric stability theory of a fixed log terminal singularity. Inspired by the study from differential geometry, (e.g. metric tangent cone, Sasakian-Einstein metric), for any log terminal singularity, we investigate the valuation which has the minimal normalized volume. Our goal is to prove various properties of this valuation which enable us to degenerate the singularity to a K-semistable T-singularity (with a torus action) in the Sasakian-Einstein sense.

### 2017年01月27日(金)

14:00-17:30 数理科学研究科棟(駒場) 122号室

今週は金曜日に2つの講演があります。Two lectures will be delivered on Friday.

On the number and boundedness of minimal models of general type (English)

Adjoint dimension of foliations (English)

https://www.dpmms.cam.ac.uk/~rs872/

今週は金曜日に2つの講演があります。Two lectures will be delivered on Friday.

**Luca Tasin 氏**(Roma Tre University) 14:00-15:30On the number and boundedness of minimal models of general type (English)

[ 講演概要 ]

In this talk I will explain that the number of minimal models yields a constructible function on the base of any family of varieties of general type. From this it follows that the number of minimal models of a variety of general type can be bounded in terms of its volume. I will also show that in any dimension minimal models of general type and bounded volume form a bounded family. This is based on a joint work with D. Martinelli and S. Schreieder.

In this talk I will explain that the number of minimal models yields a constructible function on the base of any family of varieties of general type. From this it follows that the number of minimal models of a variety of general type can be bounded in terms of its volume. I will also show that in any dimension minimal models of general type and bounded volume form a bounded family. This is based on a joint work with D. Martinelli and S. Schreieder.

**Roberto Svaldi 氏**(University of Cambridge/SISSA) 16:00-17:30Adjoint dimension of foliations (English)

[ 講演概要 ]

The classification of foliated surfaces by Brunella, McQuillan and Mendes carries many similarities with Enriques-Kodaira classification of surfaces but also many important differences. I will discuss an alternative classification scheme where the role of differential forms along the leaves is replaced by differential forms along the leaves with values in fractional powers of the conormal bundle of the foliation. In this alternative setup one obtains a classification of foliated surfaces closer to the usual Enriques-Kodaira classification. If time permits, I will show how to apply this alternative classification to describe the Zariski closure of the set foliations which admit rational first integral of bounded genus in families of foliated surfaces. Joint work with Jorge Vitorio Pereira.

[ 参考URL ]The classification of foliated surfaces by Brunella, McQuillan and Mendes carries many similarities with Enriques-Kodaira classification of surfaces but also many important differences. I will discuss an alternative classification scheme where the role of differential forms along the leaves is replaced by differential forms along the leaves with values in fractional powers of the conormal bundle of the foliation. In this alternative setup one obtains a classification of foliated surfaces closer to the usual Enriques-Kodaira classification. If time permits, I will show how to apply this alternative classification to describe the Zariski closure of the set foliations which admit rational first integral of bounded genus in families of foliated surfaces. Joint work with Jorge Vitorio Pereira.

https://www.dpmms.cam.ac.uk/~rs872/

### 2016年11月29日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Etale fundamental groups of F-regular schemes (English)

**Karl Schwede 氏**(University of Utah)Etale fundamental groups of F-regular schemes (English)

[ 講演概要 ]

I will discuss recent work studying etale fundamental groups of the regular locus of F-regular schemes. I will describe how to use F-signature to bound the size of the fundamental group of an F-regular scheme, similar to a result of Xu. I will then discuss a recent extension showing that every F-regular scheme X has a finite cover Y, etale over the regular lcous of X, so that the etale fundamental groups of Y and the regular locus of Y agree. This is analogous to results of Greb-Kebekus-Peternell.

All the work discussed is joint with Carvajal-Rojas and Tucker or with with Bhatt, Carvajal-Rojas, Graf and Tucker.

I will discuss recent work studying etale fundamental groups of the regular locus of F-regular schemes. I will describe how to use F-signature to bound the size of the fundamental group of an F-regular scheme, similar to a result of Xu. I will then discuss a recent extension showing that every F-regular scheme X has a finite cover Y, etale over the regular lcous of X, so that the etale fundamental groups of Y and the regular locus of Y agree. This is analogous to results of Greb-Kebekus-Peternell.

All the work discussed is joint with Carvajal-Rojas and Tucker or with with Bhatt, Carvajal-Rojas, Graf and Tucker.

### 2016年11月08日(火)

15:30-17:00 数理科学研究科棟(駒場) 118号室

いつもと部屋が異なります。

Minimal model theory for relatively trivial log canonical pairs

いつもと部屋が異なります。

**橋詰 健太 氏**(京都大学数学教室)Minimal model theory for relatively trivial log canonical pairs

[ 講演概要 ]

The good minimal model conjecture is an important open problem in the birational geometry, and inductive arguments on the dimension of varieties are useful when we work on this conjecture. In fibration with a log canoinical pair having some good properties, it is expected that the above conjecture for the log canonical pair on total space can be proved by investigating the general fiber and base variety of the fibration. In this talk, I will explain an inductive argument in fibrations with relatively trivial log canonical pairs and introduce some applications of the inductive argument.

The good minimal model conjecture is an important open problem in the birational geometry, and inductive arguments on the dimension of varieties are useful when we work on this conjecture. In fibration with a log canoinical pair having some good properties, it is expected that the above conjecture for the log canonical pair on total space can be proved by investigating the general fiber and base variety of the fibration. In this talk, I will explain an inductive argument in fibrations with relatively trivial log canonical pairs and introduce some applications of the inductive argument.

### 2016年10月25日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Q-Gorenstein deformation theory and it applications to algebraic surfaces (English)

**Yongnam Lee 氏**(KAIST/RIMS)Q-Gorenstein deformation theory and it applications to algebraic surfaces (English)

[ 講演概要 ]

The notion of Q-Gorenstein variety is important for the minimal model theory and the compact moduli theory of algebraic varieties in characteristic 0. Also Q-Gorenstein deformation theory can be applied to construct (simply connected) surfaces of general type with geometric genus 0 over the field of any characteristic. In this talk, some applications of Q-Gorenstein deformation theory to algebraic surfaces and some interesting examples related to Q-Gorenstein morphisms will be presented.

The notion of Q-Gorenstein variety is important for the minimal model theory and the compact moduli theory of algebraic varieties in characteristic 0. Also Q-Gorenstein deformation theory can be applied to construct (simply connected) surfaces of general type with geometric genus 0 over the field of any characteristic. In this talk, some applications of Q-Gorenstein deformation theory to algebraic surfaces and some interesting examples related to Q-Gorenstein morphisms will be presented.

### 2016年10月11日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

On varieties with splittings of relative Frobenius morphisms of Albanese maps

**江尻 祥 氏**(東大数理)On varieties with splittings of relative Frobenius morphisms of Albanese maps

[ 講演概要 ]

Varieties with splittings of Frobenius morphisms are called F-split varieties, which satisfy strong properties such as Kodaira vanishing. However, some important varieties are not F-split. For example, an abelian variety is F-split if and only if its p-rank is maximum. In this talk, we discuss the class of varieties with splittings of relative Frobenius morphisms of Albanese maps, which includes abelian varieties. As a consequence of Sannai and Tanaka's characterization of ordinary abelian varieties, we see that this class also includes F-split varieties. Furthermore, for varieties in this class, we show that the Kodaira vanishing theorem holds, and that Albanese maps are algebraic fiber spaces.

Varieties with splittings of Frobenius morphisms are called F-split varieties, which satisfy strong properties such as Kodaira vanishing. However, some important varieties are not F-split. For example, an abelian variety is F-split if and only if its p-rank is maximum. In this talk, we discuss the class of varieties with splittings of relative Frobenius morphisms of Albanese maps, which includes abelian varieties. As a consequence of Sannai and Tanaka's characterization of ordinary abelian varieties, we see that this class also includes F-split varieties. Furthermore, for varieties in this class, we show that the Kodaira vanishing theorem holds, and that Albanese maps are algebraic fiber spaces.

### 2016年10月04日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Higher order minimal families of rational curves and Fano manifolds with nef Chern characters (Japanese. Writing in English. )

**鈴木拓 氏**(早稲田大学)Higher order minimal families of rational curves and Fano manifolds with nef Chern characters (Japanese. Writing in English. )

[ 講演概要 ]

In this talk, we introduce higher order minimal families $H_i$ of rational curves

associated to Fano manifolds $X$. We prove that $H_i$ is also a Fano manifold

if the Chern characters of $X$ satisfy some positivity conditions. We also provide

a sufficient condition for Fano manifolds to be covered by higher rational manifolds.

In this talk, we introduce higher order minimal families $H_i$ of rational curves

associated to Fano manifolds $X$. We prove that $H_i$ is also a Fano manifold

if the Chern characters of $X$ satisfy some positivity conditions. We also provide

a sufficient condition for Fano manifolds to be covered by higher rational manifolds.