## 代数幾何学セミナー

過去の記録 ～07/14｜次回の予定｜今後の予定 07/15～

開催情報 | 火曜日 15:30～17:00 数理科学研究科棟(駒場) 122号室 |
---|---|

担当者 | 權業 善範・中村 勇哉・田中公 |

**過去の記録**

### 2020年03月02日(月)

15:30-17:00 数理科学研究科棟(駒場) 002号室

諸事情により中止を決定しました。Cancelled.

Semiorthogonal decompositions for singular varieties (English)

諸事情により中止を決定しました。Cancelled.

**Evgeny Shinder 氏**(The University of Sheffield)Semiorthogonal decompositions for singular varieties (English)

[ 講演概要 ]

I will define the semiorthogonal decomposition for derived categories of singular projective varieties due to Professor Kawamata, into finite-dimensional algebras, generalizing the concept of an exceptional collection in the smooth case. I will present known constructions of these for nodal curves (Burban), torsion-free toric surfaces (Karmazyn-Kuznetsov-Shinder) and two nodal threefolds (Kawamata). I will also explain obstructions coming from the K_{-1} group, and how it translates to maximal nonfactoriality in the nodal threefold case. This is joint work with M.Kalck and N.Pavic.

I will define the semiorthogonal decomposition for derived categories of singular projective varieties due to Professor Kawamata, into finite-dimensional algebras, generalizing the concept of an exceptional collection in the smooth case. I will present known constructions of these for nodal curves (Burban), torsion-free toric surfaces (Karmazyn-Kuznetsov-Shinder) and two nodal threefolds (Kawamata). I will also explain obstructions coming from the K_{-1} group, and how it translates to maximal nonfactoriality in the nodal threefold case. This is joint work with M.Kalck and N.Pavic.

### 2020年02月21日(金)

13:30-15:00 数理科学研究科棟(駒場) 370号室

いつもと曜日・時間・部屋が異なります

Keel's theorem and quotients in mixed characteristic (English)

http://www-personal.umich.edu/~jakubw/

いつもと曜日・時間・部屋が異なります

**Jakub Witaszek 氏**(Michigan)Keel's theorem and quotients in mixed characteristic (English)

[ 講演概要 ]

In trying to understand characteristic zero varieties one can apply a wide range of techniques coming from analytic methods such as vanishing theorems. More complicated though they are, positive characteristic varieties come naturally with Frobenius action which sometimes allows for imitating analytic proofs or even showing results which are false over complex numbers. Of all the three classes, the mixed characteristic varieties are the most difficult to understand as they represent the worst of both worlds: one lacks the analytic methods as well the Frobenius action.

What is key for many applications of Frobenius in positive characteristic (to birational geometry, moduli theory, constructing quotients, etc.) is the fact that every universal homeomorphism of algebraic varieties factors through a power of Frobenius. In this talk I will discuss an analogue of this fact (and applications thereof) in mixed characteristic.

[ 参考URL ]In trying to understand characteristic zero varieties one can apply a wide range of techniques coming from analytic methods such as vanishing theorems. More complicated though they are, positive characteristic varieties come naturally with Frobenius action which sometimes allows for imitating analytic proofs or even showing results which are false over complex numbers. Of all the three classes, the mixed characteristic varieties are the most difficult to understand as they represent the worst of both worlds: one lacks the analytic methods as well the Frobenius action.

What is key for many applications of Frobenius in positive characteristic (to birational geometry, moduli theory, constructing quotients, etc.) is the fact that every universal homeomorphism of algebraic varieties factors through a power of Frobenius. In this talk I will discuss an analogue of this fact (and applications thereof) in mixed characteristic.

http://www-personal.umich.edu/~jakubw/

### 2020年01月21日(火)

15:30-17:00 数理科学研究科棟(駒場) 118号室

普段と部屋が異なりますのでご注意ください。The room is different from our usual.

(Few) rational curves on K3 surfaces (English)

普段と部屋が異なりますのでご注意ください。The room is different from our usual.

**Matthias Schütt 氏**(Universität Hannover)(Few) rational curves on K3 surfaces (English)

[ 講演概要 ]

Rational curves play a fundamental role for the structure of a K3 surface. I will first review the general theory before focussing on the case of low degree curves where joint work with S. Rams (Krakow) extends bounds of Miyaoka and Degtyarev. Time permitting, I will also discuss the special case of smooth rational curves as well as applications to Enriques surfaces.

Rational curves play a fundamental role for the structure of a K3 surface. I will first review the general theory before focussing on the case of low degree curves where joint work with S. Rams (Krakow) extends bounds of Miyaoka and Degtyarev. Time permitting, I will also discuss the special case of smooth rational curves as well as applications to Enriques surfaces.

### 2019年12月03日(火)

14:30-16:00 数理科学研究科棟(駒場) 056号室

普段と曜日・時間・部屋が異なりますのでご注意ください。The room and time are different from our usual.

Moduli of K3 surfaces via cubic 4-folds (English)

普段と曜日・時間・部屋が異なりますのでご注意ください。The room and time are different from our usual.

**Gavril Farkas 氏**(Humboldt Univ. Berlin)Moduli of K3 surfaces via cubic 4-folds (English)

[ 講演概要 ]

In a celebrated series of papers, Mukai established structure theorems for polarized K3 surfaces of all genera g<21, with the exception of the case g=14. Using the identification between certain moduli spaces of polarized K3 surfaces of genera 14 and the moduli space of special cubic fourfolds of given discriminant, we discuss a novel approach to moduli spaces of K3 surfaces. As an application, we establish the rationality of the universal K3 surface of these genus 14,22. This is joint work with A. Verra.

In a celebrated series of papers, Mukai established structure theorems for polarized K3 surfaces of all genera g<21, with the exception of the case g=14. Using the identification between certain moduli spaces of polarized K3 surfaces of genera 14 and the moduli space of special cubic fourfolds of given discriminant, we discuss a novel approach to moduli spaces of K3 surfaces. As an application, we establish the rationality of the universal K3 surface of these genus 14,22. This is joint work with A. Verra.

### 2019年10月30日(水)

15:30-17:00 数理科学研究科棟(駒場) 122号室

A Tannakian perspective on rigid analytic geometry (English)

**Andrew Macpherson 氏**(IPMU)A Tannakian perspective on rigid analytic geometry (English)

[ 講演概要 ]

Raynaud's conception of analytic geometry contends that the category of analytic spaces over a non-Archimedean field is a (suitably "geometric") localisation of the category of formal schemes over the ring of integers at a class of modifications "along the central fibre". Unfortunately, as with all existing presentations of non-Archimedean geometry, this viewpoint is confounded by a proliferation of technical difficulties if one does not impose absolute finiteness conditions on the formal schemes under consideration.

I will argue that by combining Raynaud's idea with a Tannakian perspective which prioritises the module category, we can obtain a reasonable framework for rigid analytic geometry with no absolute finiteness hypotheses whatsoever, but which has descent for finitely presented modules.

Raynaud's conception of analytic geometry contends that the category of analytic spaces over a non-Archimedean field is a (suitably "geometric") localisation of the category of formal schemes over the ring of integers at a class of modifications "along the central fibre". Unfortunately, as with all existing presentations of non-Archimedean geometry, this viewpoint is confounded by a proliferation of technical difficulties if one does not impose absolute finiteness conditions on the formal schemes under consideration.

I will argue that by combining Raynaud's idea with a Tannakian perspective which prioritises the module category, we can obtain a reasonable framework for rigid analytic geometry with no absolute finiteness hypotheses whatsoever, but which has descent for finitely presented modules.

### 2019年10月16日(水)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Multidimensional continued fraction for Gorenstein cyclic quotient singularity

**佐藤 悠介 氏**(東大数理/ IPMU)Multidimensional continued fraction for Gorenstein cyclic quotient singularity

[ 講演概要 ]

Let G be a finite cyclic subgroup of GL(n,C). Then Cn/G is a cyclic quotient singularity. In the case n = 2, Cn/G possess the unique minimal resolution, and it is obtained by Hirzubruch-Jung continued fraction. In this talk, we show a sufficient condition of existence of crepant desingularization for Gorenstein abelian quotient singularities in all dimensions by using Ashikaga’s continuous fractions. Moreover, as a corollary, we prove that all three dimensional Gorenstein abelian quotient singularities possess a crepant desingularization.

Let G be a finite cyclic subgroup of GL(n,C). Then Cn/G is a cyclic quotient singularity. In the case n = 2, Cn/G possess the unique minimal resolution, and it is obtained by Hirzubruch-Jung continued fraction. In this talk, we show a sufficient condition of existence of crepant desingularization for Gorenstein abelian quotient singularities in all dimensions by using Ashikaga’s continuous fractions. Moreover, as a corollary, we prove that all three dimensional Gorenstein abelian quotient singularities possess a crepant desingularization.

### 2019年07月09日(火)

13:00-14:30 数理科学研究科棟(駒場) 122号室

いつもと曜日・時間・部屋が異なります。

Construction of non-Kähler Calabi-Yau 3-folds by smoothing normal crossing varieties (TBA)

いつもと曜日・時間・部屋が異なります。

**佐野 太郎 氏**(神戸大学)Construction of non-Kähler Calabi-Yau 3-folds by smoothing normal crossing varieties (TBA)

[ 講演概要 ]

It is an open problem whether there are only finitely many diffeomorphism types of projective Calabi-Yau 3-folds. Kawamata--Namikawa developed log deformation theory of normal crossing Calabi-Yau varieties. As an application of their result, one can construct examples of Calabi-Yau manifolds by smoothing SNC varieties. In this talk, I will explain how to construct examples of non-Kähler Calabi-Yau 3-folds with arbitrarily large 2nd Betti numbers. If time permits, I will also explain an example of involutions on a family of K3 surfaces which do not lift biregularly to the total space. This is based on joint work with Kenji Hashimoto.

It is an open problem whether there are only finitely many diffeomorphism types of projective Calabi-Yau 3-folds. Kawamata--Namikawa developed log deformation theory of normal crossing Calabi-Yau varieties. As an application of their result, one can construct examples of Calabi-Yau manifolds by smoothing SNC varieties. In this talk, I will explain how to construct examples of non-Kähler Calabi-Yau 3-folds with arbitrarily large 2nd Betti numbers. If time permits, I will also explain an example of involutions on a family of K3 surfaces which do not lift biregularly to the total space. This is based on joint work with Kenji Hashimoto.

### 2019年07月05日(金)

10:30-12:00 数理科学研究科棟(駒場) 123号室

いつもと曜日・時間・部屋が異なります。

Durfee-type inequality for complete intersection surface singularities

いつもと曜日・時間・部屋が異なります。

**榎園 誠 氏**(東京理科大学)Durfee-type inequality for complete intersection surface singularities

[ 講演概要 ]

Durfee's negativity conjecture says that the signature of the Milnor fiber of a 2-dimensional isolated complete intersection singularity is always negative. In this talk, I will explain that this conjecture is true (more precisely, the signature is bounded above by the negative number determined by the geometric genus, the embedding dimension and the number of irreducible components of the exceptional set of the minimal resolution) by using the theory of invariants of fibered surfaces. If time permits, I will explain the higher dimensional analogue of Durfee's conjecture for isolated complete intersection singularities.

Durfee's negativity conjecture says that the signature of the Milnor fiber of a 2-dimensional isolated complete intersection singularity is always negative. In this talk, I will explain that this conjecture is true (more precisely, the signature is bounded above by the negative number determined by the geometric genus, the embedding dimension and the number of irreducible components of the exceptional set of the minimal resolution) by using the theory of invariants of fibered surfaces. If time permits, I will explain the higher dimensional analogue of Durfee's conjecture for isolated complete intersection singularities.

### 2019年06月28日(金)

15:30-17:00 数理科学研究科棟(駒場) 118号室

いつもと曜日が異なります。

Rational curves on prime Fano 3-folds (TBA)

いつもと曜日が異なります。

**谷本 祥 氏**(熊本)Rational curves on prime Fano 3-folds (TBA)

[ 講演概要 ]

One of important topics in algebraic geometry is the space of rational curves, e.g., the dimension and the number of components of the moduli spaces of rational curves on an algebraic variety X. One of interesting situations where this question is extensively studied is when X is a Fano variety since in this case X is rationally connected so that it does contain a lots of rational curves. In this talk I will talk about my joint work with Brian Lehmann which settles this problem for most Fano 3-folds of Picard rank 1, e.g., a general quartic 3-fold in P^4, and our approach is inspired by Manin’s conjecture which predicts the asymptotic formula for the counting function of rational points on a Fano variety. In particular we systematically use geometric invariants in Manin’s conjecture which have been studied by many mathematicians including Brian and me.

One of important topics in algebraic geometry is the space of rational curves, e.g., the dimension and the number of components of the moduli spaces of rational curves on an algebraic variety X. One of interesting situations where this question is extensively studied is when X is a Fano variety since in this case X is rationally connected so that it does contain a lots of rational curves. In this talk I will talk about my joint work with Brian Lehmann which settles this problem for most Fano 3-folds of Picard rank 1, e.g., a general quartic 3-fold in P^4, and our approach is inspired by Manin’s conjecture which predicts the asymptotic formula for the counting function of rational points on a Fano variety. In particular we systematically use geometric invariants in Manin’s conjecture which have been studied by many mathematicians including Brian and me.

### 2019年06月19日(水)

15:30-17:00 数理科学研究科棟(駒場) 118号室

今学期は基本水曜日とします。部屋も去年度と異なります。

A pencil of Enriques surfaces with non-algebraic integral Hodge classes (TBA)

今学期は基本水曜日とします。部屋も去年度と異なります。

**鈴木文顕 氏**(イリノイ州立シカゴ大学)A pencil of Enriques surfaces with non-algebraic integral Hodge classes (TBA)

[ 講演概要 ]

The integral Hodge conjecture is the statement that the integral Hodge classes are algebraic on smooth complex projective varieties. It is known that the conjecture can fail in general. There are two types of counterexamples, ones with non-algebraic integral Hodge classes of torsion-type and of non-torsion type, the first of which were given by Atiyah-Hirzebruch and Kollar, respectively.

In this talk, we exhibit a pencil of Enriques surfaces defined over Q with non-algebraic integral Hodge classes of non-torsion type. This construction relates to certain questions concerning rational points of algebraic varieties.

This gives the first example of a threefold with the trivial Chow group of zero-cycles on which the integral Hodge conjecture fails. As an application, we construct a fourfold which gives the negative answer to a classical question on the universality of the Abel-Jacobi maps.

This is a joint work with John Christian Ottem.

The integral Hodge conjecture is the statement that the integral Hodge classes are algebraic on smooth complex projective varieties. It is known that the conjecture can fail in general. There are two types of counterexamples, ones with non-algebraic integral Hodge classes of torsion-type and of non-torsion type, the first of which were given by Atiyah-Hirzebruch and Kollar, respectively.

In this talk, we exhibit a pencil of Enriques surfaces defined over Q with non-algebraic integral Hodge classes of non-torsion type. This construction relates to certain questions concerning rational points of algebraic varieties.

This gives the first example of a threefold with the trivial Chow group of zero-cycles on which the integral Hodge conjecture fails. As an application, we construct a fourfold which gives the negative answer to a classical question on the universality of the Abel-Jacobi maps.

This is a joint work with John Christian Ottem.

### 2019年05月29日(水)

15:30-17:00 数理科学研究科棟(駒場) 118号室

今学期は基本水曜日とします。部屋も去年度と異なります。

Minimal log discrepancies of 3-dimensional non-canonical singularities (English)

今学期は基本水曜日とします。部屋も去年度と異なります。

**江辰 氏**(Fudan/MSRI)Minimal log discrepancies of 3-dimensional non-canonical singularities (English)

[ 講演概要 ]

Canonical and terminal singularities, introduced by Reid, appear naturally in minimal model program and play important roles in the birational classification of higher dimensional algebraic varieties. Such singularities are well-understood in dimension 3, while the property of non-canonical singularities is still mysterious. We investigate the difference between canonical and non-canonical singularities via minimal log discrepancies (MLD). We show that there is a gap between MLD of 3-dimensional non-canonical singularities and that of 3-dimensional canonical singularities, which is predicted by a conjecture of Shokurov.

This result on local singularities has applications to global geometry of Calabi–Yau 3-folds. We show that the set of all non-canonical klt Calabi–Yau 3-folds are bounded modulo flops, and the global indices of all klt Calabi–Yau 3-folds are bounded from above.

Canonical and terminal singularities, introduced by Reid, appear naturally in minimal model program and play important roles in the birational classification of higher dimensional algebraic varieties. Such singularities are well-understood in dimension 3, while the property of non-canonical singularities is still mysterious. We investigate the difference between canonical and non-canonical singularities via minimal log discrepancies (MLD). We show that there is a gap between MLD of 3-dimensional non-canonical singularities and that of 3-dimensional canonical singularities, which is predicted by a conjecture of Shokurov.

This result on local singularities has applications to global geometry of Calabi–Yau 3-folds. We show that the set of all non-canonical klt Calabi–Yau 3-folds are bounded modulo flops, and the global indices of all klt Calabi–Yau 3-folds are bounded from above.

### 2019年05月22日(水)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Bogomolov type vanishing on three-dimensional Mori fiber spaces in positive characteristic

**河上 龍郎 氏**(東大数理)Bogomolov type vanishing on three-dimensional Mori fiber spaces in positive characteristic

[ 講演概要 ]

In characteristic zero, cotangent bundle of n(>1)-dimensional smooth projective varieties does not contain a big line bundle. This is a part of Bogomolov vanishing and this vanishing plays an important role in the proof of Miyaoka-Yau inequality. In positive characteristic, it is known that Bogomolov vanishing does not hold. There exists a general type surface whose cotangent bundle contains an ample line bundle. So, it is natural to ask when Bogomolov type vanishing holds in positive characteristic. In this talk, I discuss Bogomolov type vanishing on three-dimensional Mori fiber spaces in positive characteristic.

In characteristic zero, cotangent bundle of n(>1)-dimensional smooth projective varieties does not contain a big line bundle. This is a part of Bogomolov vanishing and this vanishing plays an important role in the proof of Miyaoka-Yau inequality. In positive characteristic, it is known that Bogomolov vanishing does not hold. There exists a general type surface whose cotangent bundle contains an ample line bundle. So, it is natural to ask when Bogomolov type vanishing holds in positive characteristic. In this talk, I discuss Bogomolov type vanishing on three-dimensional Mori fiber spaces in positive characteristic.

### 2019年05月15日(水)

15:30-17:00 数理科学研究科棟(駒場) 118号室

今学期は基本水曜日とします。部屋も去年度と異なります。

On quasi-log canonical pairs

(Japanese)

今学期は基本水曜日とします。部屋も去年度と異なります。

**藤野修 氏**(大阪大学)On quasi-log canonical pairs

(Japanese)

[ 講演概要 ]

The notion of quasi-log canonical pairs was introduced by Florin Ambro. It is a kind of generalizations of that of log canonical pairs. Now we know that quasi-log canonical pairs are ubiquitous in the theory of minimal models. In this talk, I will explain some basic properties and examples of quasi-log canonical pairs. I will also discuss some new developments around quasi-log canonical pairs. Some parts are joint works with Haidong Liu.

The notion of quasi-log canonical pairs was introduced by Florin Ambro. It is a kind of generalizations of that of log canonical pairs. Now we know that quasi-log canonical pairs are ubiquitous in the theory of minimal models. In this talk, I will explain some basic properties and examples of quasi-log canonical pairs. I will also discuss some new developments around quasi-log canonical pairs. Some parts are joint works with Haidong Liu.

### 2019年05月08日(水)

15:30-17:00 数理科学研究科棟(駒場) 118号室

On Minimal model theory for log canonical pairs with big boundary divisors

**橋詰 健太 氏**(東大数理)On Minimal model theory for log canonical pairs with big boundary divisors

[ 講演概要 ]

In 2010, Birkar--Cascini--Hacon--McKernan established the minimal model theory for Kawamata log terminal pairs with big boundary divisors, and a lot of theorems in the birational geometry are

proved by applying this result. It is expected that this result can be generalized to log canonical pairs. Currently, it is known that the minimal model theory for log canonical pairs can be reduced to the case of big boundary divisors. In this talk, we introduce a partial generalization of the result by Birkar--Cascini--Hacon--McKernan. Roughly speaking, we generalized their result to lc pairs with big boundary divisors having only small lc centers. We also explain another generalization, which is originally announced by Hu, and we discuss termination of log minimal model program in a spacial case. This is a joint work with Zhengyu Hu, and the work is in progress.

In 2010, Birkar--Cascini--Hacon--McKernan established the minimal model theory for Kawamata log terminal pairs with big boundary divisors, and a lot of theorems in the birational geometry are

proved by applying this result. It is expected that this result can be generalized to log canonical pairs. Currently, it is known that the minimal model theory for log canonical pairs can be reduced to the case of big boundary divisors. In this talk, we introduce a partial generalization of the result by Birkar--Cascini--Hacon--McKernan. Roughly speaking, we generalized their result to lc pairs with big boundary divisors having only small lc centers. We also explain another generalization, which is originally announced by Hu, and we discuss termination of log minimal model program in a spacial case. This is a joint work with Zhengyu Hu, and the work is in progress.

### 2019年04月24日(水)

15:30-17:00 数理科学研究科棟(駒場) 118号室

今学期は基本水曜日とします。部屋も去年度と異なります。

Varieties of dense globally F-split type with a non-invertible polarized

endomorphism

今学期は基本水曜日とします。部屋も去年度と異なります。

**吉川翔 氏**(東大数理)Varieties of dense globally F-split type with a non-invertible polarized

endomorphism

[ 講演概要 ]

Broustet and Gongyo conjectured that if a normal projective variety X has a non-invertible polaried endomorphism, then X is of Calabi-Yau type. Furthermore, Schwede and Smith conjectured that a projective variety is of Calabi-Yau type if and only if of dense globally F-split type. Therefore it is a natural question to ask if a normal projective variety X has a non-invertible polaried endomorphism, then X is of dense globally F-split type. In this talk, I will introduce simple points and difficult points of the question. Furthermore I will give the affirmative answer of my question for 2-dimensional case.

Broustet and Gongyo conjectured that if a normal projective variety X has a non-invertible polaried endomorphism, then X is of Calabi-Yau type. Furthermore, Schwede and Smith conjectured that a projective variety is of Calabi-Yau type if and only if of dense globally F-split type. Therefore it is a natural question to ask if a normal projective variety X has a non-invertible polaried endomorphism, then X is of dense globally F-split type. In this talk, I will introduce simple points and difficult points of the question. Furthermore I will give the affirmative answer of my question for 2-dimensional case.

### 2019年01月29日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Logarithmic good reduction and the index (TBA)

**三井健太郎 氏**(神戸)Logarithmic good reduction and the index (TBA)

[ 講演概要 ]

A proper smooth variety over a complete discrete valuation field is said to have (log) good reduction if it admits a proper (log) smooth model over the valuation ring (the log structure is given by the closed fiber). Monodromy criteria for good reduction and log good reduction have been studied. We study the log case by additional other conditions on geometric invariants such as the index of the variety (the minimal positive degree of a 0-cycle). In particular, we obtain a criterion for log good reduction of curves of genus one.

A proper smooth variety over a complete discrete valuation field is said to have (log) good reduction if it admits a proper (log) smooth model over the valuation ring (the log structure is given by the closed fiber). Monodromy criteria for good reduction and log good reduction have been studied. We study the log case by additional other conditions on geometric invariants such as the index of the variety (the minimal positive degree of a 0-cycle). In particular, we obtain a criterion for log good reduction of curves of genus one.

### 2018年12月21日(金)

10:30-11:30 数理科学研究科棟(駒場) 123号室

Degenerations of p-adic volume forms (English)

**Mattias Jonsson 氏**(Michigan)Degenerations of p-adic volume forms (English)

[ 講演概要 ]

Let X be an n-dimensional smooth projective variety over a non-Archimedean local field K. Also fix a regular n-form on X. This data induces a positive measure on the space of K'-rational points for any finite extension K' of K. We describe the asymptotics, as K' runs through towers of finite extensions of K, in terms of Berkovich analytic geometry. This is joint work with Johannes Nicaise.

Let X be an n-dimensional smooth projective variety over a non-Archimedean local field K. Also fix a regular n-form on X. This data induces a positive measure on the space of K'-rational points for any finite extension K' of K. We describe the asymptotics, as K' runs through towers of finite extensions of K, in terms of Berkovich analytic geometry. This is joint work with Johannes Nicaise.

### 2018年12月14日(金)

10:30-11:30 数理科学研究科棟(駒場) 123号室

On the birationality of quint-canonical systems of irregular threefolds of general type (English)

**Zhi Jiang 氏**(Fudan)On the birationality of quint-canonical systems of irregular threefolds of general type (English)

[ 講演概要 ]

It is well-known that the quint-canonical map of a surface of general type is birational.

We will show that the same result holds for irregular threefolds of general type. The proof is based on

a careful study of the positivity of the pushforwards of pluricanonical bundles on abelian varieties and Severi

type inequalities. This is a joint work with J.A. Chen, J.Chen, and M.Chen.

It is well-known that the quint-canonical map of a surface of general type is birational.

We will show that the same result holds for irregular threefolds of general type. The proof is based on

a careful study of the positivity of the pushforwards of pluricanonical bundles on abelian varieties and Severi

type inequalities. This is a joint work with J.A. Chen, J.Chen, and M.Chen.

### 2018年11月27日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Frobenius summands and the finite F-representation type (TBA)

**原伸生 氏**(東京農工大)Frobenius summands and the finite F-representation type (TBA)

[ 講演概要 ]

We are motivated by a question arising from commutative algebra, asking what kind of

graded rings in positive characteristic p have finite F-representation type. In geometric

setting, this is related to the problem to looking out for Frobenius summands. Namely,

given aline bundle L on a projective variety X, we want to know how many and what

kind of indecomposable direct summands appear in the direct sum decomposition of

the iterated Frobenius push-forwards of L. We will consider the problem in the following

two cases, although the present situation in (2) is far from satisfactory.

(1) two-dimensional normal graded rings (a joint work with Ryo Ohkawa)

(2) the anti-canonical ring of a quintic del Pezzo surface

We are motivated by a question arising from commutative algebra, asking what kind of

graded rings in positive characteristic p have finite F-representation type. In geometric

setting, this is related to the problem to looking out for Frobenius summands. Namely,

given aline bundle L on a projective variety X, we want to know how many and what

kind of indecomposable direct summands appear in the direct sum decomposition of

the iterated Frobenius push-forwards of L. We will consider the problem in the following

two cases, although the present situation in (2) is far from satisfactory.

(1) two-dimensional normal graded rings (a joint work with Ryo Ohkawa)

(2) the anti-canonical ring of a quintic del Pezzo surface

### 2018年11月20日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Artin-Mazur height, Yobuko height and

Hodge-Wittt cohomologies

**中島 幸喜 氏**(東京電機大)Artin-Mazur height, Yobuko height and

Hodge-Wittt cohomologies

[ 講演概要 ]

A few years ago Yobuko has introduced the notion of

a delicate invariant for a proper smooth scheme over a perfect field $k$

of finite characteristic. (We call this invariant Yobuko height.)

This generalize the notion of the F-splitness due to Mehta-Srinivas.

In this talk we give relations between Artin-Mazur heights

and Yobuko heights. We also give a finiteness result on

Hodge-Witt cohomologies of a proper smooth scheme $X$ over $k$

with finite Yobuko height. If time permits, we give a cofinite type result on

the $p$-primary torsion part of Chow group of of $X$

of codimension 2 if $\dim X=3$.

A few years ago Yobuko has introduced the notion of

a delicate invariant for a proper smooth scheme over a perfect field $k$

of finite characteristic. (We call this invariant Yobuko height.)

This generalize the notion of the F-splitness due to Mehta-Srinivas.

In this talk we give relations between Artin-Mazur heights

and Yobuko heights. We also give a finiteness result on

Hodge-Witt cohomologies of a proper smooth scheme $X$ over $k$

with finite Yobuko height. If time permits, we give a cofinite type result on

the $p$-primary torsion part of Chow group of of $X$

of codimension 2 if $\dim X=3$.

### 2018年11月13日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Boundedness of varieties of Fano type with alpha-invariants and volumes bounded below (English)

**陳韋中 氏**(東大数理)Boundedness of varieties of Fano type with alpha-invariants and volumes bounded below (English)

[ 講演概要 ]

We show that fixed dimensional klt weak Fano pairs with alpha-invariants and volumes bounded away from 0 and the coefficients of the boundaries belong to the set of hyperstandard multiplicities Φ(R) associated to a fixed finite set R form a bounded family. We also show α(X, B)d−1vol(−(KX + B)) are bounded from above for all klt weak Fano pairs (X, B) of a fixed dimension d.

We show that fixed dimensional klt weak Fano pairs with alpha-invariants and volumes bounded away from 0 and the coefficients of the boundaries belong to the set of hyperstandard multiplicities Φ(R) associated to a fixed finite set R form a bounded family. We also show α(X, B)d−1vol(−(KX + B)) are bounded from above for all klt weak Fano pairs (X, B) of a fixed dimension d.

### 2018年10月16日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

A countable characterisation of smooth algebraic plane curves, and generalisations (English)

**Tuyen Truong 氏**(Oslo)A countable characterisation of smooth algebraic plane curves, and generalisations (English)

[ 講演概要 ]

Given a smooth algebraic curve X in C^3, I will present a way to construct a sequence of algebraic varieties (whose ideals are explicitly determined from the ideal defining X), whose solution set is non-empty iff the curve X can be algebraically embedded into C^2.

Various other questions, such as whether two given algebraic varieties are birational, can be similarly treated. Some related conjectures are stated.

Given a smooth algebraic curve X in C^3, I will present a way to construct a sequence of algebraic varieties (whose ideals are explicitly determined from the ideal defining X), whose solution set is non-empty iff the curve X can be algebraically embedded into C^2.

Various other questions, such as whether two given algebraic varieties are birational, can be similarly treated. Some related conjectures are stated.

### 2018年10月09日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Stability conditions on threefolds with nef tangent bundles (English)

**小関直紀 氏**(東大数理/IPMU)Stability conditions on threefolds with nef tangent bundles (English)

[ 講演概要 ]

The construction of Bridgeland stability conditions on threefolds

is an open problem in general.

The problem is reduced to proving

the so-called Bogomolov-Gieseker (BG) type inequality conjecture,

proposed by Bayer, Macrí, and Toda.

In this talk, I will explain how to prove the BG type inequality

conjecture

for threefolds in the title.

The construction of Bridgeland stability conditions on threefolds

is an open problem in general.

The problem is reduced to proving

the so-called Bogomolov-Gieseker (BG) type inequality conjecture,

proposed by Bayer, Macrí, and Toda.

In this talk, I will explain how to prove the BG type inequality

conjecture

for threefolds in the title.

### 2018年07月18日(水)

15:30-17:00 数理科学研究科棟(駒場) 122号室

普段と違う水曜日にセミナーを行います。The seminar will be held on Wednesday. This is a different day from usual.

Normal Legendrian singularities (English)

普段と違う水曜日にセミナーを行います。The seminar will be held on Wednesday. This is a different day from usual.

**Jun-Muk Hwang 氏**(KIAS)Normal Legendrian singularities (English)

[ 講演概要 ]

A germ of a Legendrian subvariety in a holomorphic contact manifold

is called a Legendrian singularity. Legendrian singularities are usually not normal.

We look at some examples of normal Legendrian singularities and discuss their rigidity under deformation.

A germ of a Legendrian subvariety in a holomorphic contact manifold

is called a Legendrian singularity. Legendrian singularities are usually not normal.

We look at some examples of normal Legendrian singularities and discuss their rigidity under deformation.

### 2018年07月10日(火)

15:30-17:00 数理科学研究科棟(駒場) 002号室

いつもと部屋が違います。The room is different from usual.

The effective bound of anticanonical volume of Fano threefolds (English)

いつもと部屋が違います。The room is different from usual.

**賴青瑞 氏**(国立成功大学)The effective bound of anticanonical volume of Fano threefolds (English)

[ 講演概要 ]

According to Mori's program, varieties covered by rational curves are

built up from anti-canonically polarized varieties, aka Fano varieties. After fixed the

dimension and singularity type, Fano varieties form a bounded family by Birkar's proof (2016)

of Borisov-Alexeev-Borisov conjecture, which In particular implies that the anticanonical

volume -K^\dim is bounded. In this talk, we focus on canonical Fano threefolds,

where boundedness was established by Koll\'ar-Miyaoka-Mori-Takagi (2000).

Our aim is to find an effective bound of the anticanonical volume -K^3, which is

not explicit either from the work of Koll\'ar-Miyaoka-Mori-Takagi or Birkar. We will discuss

some effectiveness results related to this problem and prove that -K_X^3\leq 72 if \rho(X)\leq 2.

This partially extends early work of Mori, Mukai, Y. Prokhorov, et al.

According to Mori's program, varieties covered by rational curves are

built up from anti-canonically polarized varieties, aka Fano varieties. After fixed the

dimension and singularity type, Fano varieties form a bounded family by Birkar's proof (2016)

of Borisov-Alexeev-Borisov conjecture, which In particular implies that the anticanonical

volume -K^\dim is bounded. In this talk, we focus on canonical Fano threefolds,

where boundedness was established by Koll\'ar-Miyaoka-Mori-Takagi (2000).

Our aim is to find an effective bound of the anticanonical volume -K^3, which is

not explicit either from the work of Koll\'ar-Miyaoka-Mori-Takagi or Birkar. We will discuss

some effectiveness results related to this problem and prove that -K_X^3\leq 72 if \rho(X)\leq 2.

This partially extends early work of Mori, Mukai, Y. Prokhorov, et al.