## Algebraic Geometry Seminar

Seminar information archive ～09/28｜Next seminar｜Future seminars 09/29～

Date, time & place | Tuesday 15:30 - 17:00 122Room #122 (Graduate School of Math. Sci. Bldg.) |
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**Seminar information archive**

### 2021/07/21

15:00-16:00 Room #zoom (Graduate School of Math. Sci. Bldg.)

Cancelled

TBA (日本語)

Cancelled

**Keisuke Miyamoto**(Osaka)TBA (日本語)

[ Abstract ]

TBA

TBA

### 2021/07/05

16:00-17:00 Room #zoom (Graduate School of Math. Sci. Bldg.)

Birational geometry of foliations (English)

**Paolo Cascini**(Imperial College London)Birational geometry of foliations (English)

[ Abstract ]

I will survey about some recent progress towards the Minimal Model Program for foliations on complex varieties, focusing mainly on the case of threefolds and the case of algebraically integrable foliations.

I will survey about some recent progress towards the Minimal Model Program for foliations on complex varieties, focusing mainly on the case of threefolds and the case of algebraically integrable foliations.

### 2021/07/01

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

An O-acyclic variety of even index

**Fumiaki Suzuki**(UCLA)An O-acyclic variety of even index

[ Abstract ]

I will construct a family of Enriques surfaces parametrized by P^1 such that any multi-section has even degree over the base P^1. Over the function field of a complex curve, this gives the first example of an O-acyclic variety (H^i(X,O)=0 for i>0) whose index is not equal to one, and an affirmative answer to a question of Colliot-Thélène and Voisin. I will also discuss applications to related problems, including the integral Hodge conjecture and Murre’s question on universality of the Abel-Jacobi maps. This is joint work with John Christian Ottem.

I will construct a family of Enriques surfaces parametrized by P^1 such that any multi-section has even degree over the base P^1. Over the function field of a complex curve, this gives the first example of an O-acyclic variety (H^i(X,O)=0 for i>0) whose index is not equal to one, and an affirmative answer to a question of Colliot-Thélène and Voisin. I will also discuss applications to related problems, including the integral Hodge conjecture and Murre’s question on universality of the Abel-Jacobi maps. This is joint work with John Christian Ottem.

### 2021/06/14

17:00-18:00 Room # (Graduate School of Math. Sci. Bldg.)

Rank two weak Fano bundles on del Pezzo threefolds of degree 5 (日本語)

Zoom

**Wahei Hara**(University of Glasgow)Rank two weak Fano bundles on del Pezzo threefolds of degree 5 (日本語)

[ Abstract ]

None

[ Reference URL ]None

Zoom

### 2021/06/09

15:00-16:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Rational simple connectedness and Fano threefolds (English)

Zoom

**Andrea Fanelli**(Bordeaux)Rational simple connectedness and Fano threefolds (English)

[ Abstract ]

The notion of rational simple connectedness can be seen as an algebro-geometric analogue of simple connectedness in topology. The work of de Jong, He and Starr has already produced several recent studies to understand this notion.

In this talk I will discuss the joint project with Laurent Gruson and Nicolas Perrin to study rational simple connectedness for Fano threefolds via explicit methods from birational geometry.

[ Reference URL ]The notion of rational simple connectedness can be seen as an algebro-geometric analogue of simple connectedness in topology. The work of de Jong, He and Starr has already produced several recent studies to understand this notion.

In this talk I will discuss the joint project with Laurent Gruson and Nicolas Perrin to study rational simple connectedness for Fano threefolds via explicit methods from birational geometry.

Zoom

### 2021/06/02

15:00-16:00 Room # (Graduate School of Math. Sci. Bldg.)

Quasiexcellence implies strong generation (日本語)

Zoom

**Ko Aoki**(Tokyo)Quasiexcellence implies strong generation (日本語)

[ Abstract ]

BondalとVan den Berghは（小さい）三角圏からの反変関手がいつ表現可能かという問題の考察の中で、対象が三角圏を強生成(strongly generate)することの定義を導入した。強生成する対象が存在するときは良い表現可能性定理が成立する。

どのような有限次元Noetherスキームに対してその連接層の導来圏が強生成であるかについてはBondal–Van den Bergh以降Rouquier, Keller–Van den Bergh, Aihara–Takahashi, Iyengar–Takahashiなどにより多くの結果が得られていたが、最近Neemanは別の手法を用いてそれをalterationが適用できる分離Noetherスキームに対して示した。それを講演者はGabberのweak local uniformizationを用いることでさらに分離的準優秀スキームにまで拡張した。講演ではこの結果およびその証明の手法を紹介する。

[ Reference URL ]BondalとVan den Berghは（小さい）三角圏からの反変関手がいつ表現可能かという問題の考察の中で、対象が三角圏を強生成(strongly generate)することの定義を導入した。強生成する対象が存在するときは良い表現可能性定理が成立する。

どのような有限次元Noetherスキームに対してその連接層の導来圏が強生成であるかについてはBondal–Van den Bergh以降Rouquier, Keller–Van den Bergh, Aihara–Takahashi, Iyengar–Takahashiなどにより多くの結果が得られていたが、最近Neemanは別の手法を用いてそれをalterationが適用できる分離Noetherスキームに対して示した。それを講演者はGabberのweak local uniformizationを用いることでさらに分離的準優秀スキームにまで拡張した。講演ではこの結果およびその証明の手法を紹介する。

Zoom

### 2021/05/26

15:00-16:00 Room #zoom (Graduate School of Math. Sci. Bldg.)

Multiplier ideals via ultraproducts (日本語)

**Itsuki Yamaguchi**(Tokyo)Multiplier ideals via ultraproducts (日本語)

[ Abstract ]

正標数の可換環と複素数体上の可換環の性質を比較する方法の一つにultraproductを用いた手法がある. このultraproductは超準解析において超実数の構成などに用いられているものである. これを可換環論へ応用する研究としてSchoutensによるnon-standard hullがある. この手法は等標数0の局所環に対するbig Cohen-Macaulay 代数の構成などにも応用がある. 彼の研究の一つに川又対数端末特異点のultraproductを用いた特徴付けがある. 本講演では, この結果の一般化として乗数イデアルがultraproductを用いて記述できることを説明する.

正標数の可換環と複素数体上の可換環の性質を比較する方法の一つにultraproductを用いた手法がある. このultraproductは超準解析において超実数の構成などに用いられているものである. これを可換環論へ応用する研究としてSchoutensによるnon-standard hullがある. この手法は等標数0の局所環に対するbig Cohen-Macaulay 代数の構成などにも応用がある. 彼の研究の一つに川又対数端末特異点のultraproductを用いた特徴付けがある. 本講演では, この結果の一般化として乗数イデアルがultraproductを用いて記述できることを説明する.

### 2021/05/17

17:00-18:00 Room #zoom (Graduate School of Math. Sci. Bldg.)

Calabi problem for smooth Fano threefolds (English)

**Ivan Cheltsov**(Edinburgh)Calabi problem for smooth Fano threefolds (English)

[ Abstract ]

In this talk I will explain which three-dimensional complex Fano manifolds admit Kahler-Einstein metrics.

In this talk I will explain which three-dimensional complex Fano manifolds admit Kahler-Einstein metrics.

### 2021/05/13

9:00-10:00 Room #zoom (Graduate School of Math. Sci. Bldg.)

いつもと日時が異なります。京大と共催

Relative vanishing theorems for schemes of equal characteristic zero (Englishg)

いつもと日時が異なります。京大と共催

**Takumi Murayama**(Princeton)Relative vanishing theorems for schemes of equal characteristic zero (Englishg)

[ Abstract ]

In 1953, Kodaira proved the Kodaira vanishing theorem, which states that if L is an ample divisor on a complex projective manifold X, then H^i(X,-L) = 0 for all i < dim(X). Since then, Kodaira's theorem and its generalizations have become indispensable tools in algebraic geometry over fields of characteristic zero. Even in this context, however, it is often necessary to work with schemes of finite type over power series rings, and a fundamental problem has been the lack of vanishing theorems in this setting.

We prove the analogue of the Kawamata-Viehweg vanishing theorem for proper morphisms of schemes of equal characteristic zero, which implies Kodaira's vanishing theorem in this context. This result resolves conjectures of Boutot and Kawakita, and is an important ingredient toward establishing the minimal model program for excellent schemes of equal characteristic zero.

In 1953, Kodaira proved the Kodaira vanishing theorem, which states that if L is an ample divisor on a complex projective manifold X, then H^i(X,-L) = 0 for all i < dim(X). Since then, Kodaira's theorem and its generalizations have become indispensable tools in algebraic geometry over fields of characteristic zero. Even in this context, however, it is often necessary to work with schemes of finite type over power series rings, and a fundamental problem has been the lack of vanishing theorems in this setting.

We prove the analogue of the Kawamata-Viehweg vanishing theorem for proper morphisms of schemes of equal characteristic zero, which implies Kodaira's vanishing theorem in this context. This result resolves conjectures of Boutot and Kawakita, and is an important ingredient toward establishing the minimal model program for excellent schemes of equal characteristic zero.

### 2021/04/28

15:00-16:00 Room #Zoom (Graduate School of Math. Sci. Bldg.)

Dimensional reduction in cohomological Donaldson-Thomas theory (日本語)

**Tasuki Kinjo**(Tokyo/IPMU)Dimensional reduction in cohomological Donaldson-Thomas theory (日本語)

[ Abstract ]

None

None

### 2021/04/21

15:00-16:00 Room #ZOOM (Graduate School of Math. Sci. Bldg.)

A decomposition formula for J-stability and its applications (日本語)

**Masafumi Hattori**(Kyoto)A decomposition formula for J-stability and its applications (日本語)

[ Abstract ]

J-stability is an analog of K-stability and plays an important role in K-stability for general polarized varieties (not only for Kahler-Einstein metrics). Strikingly, G.Chen proved uniform J-stability and slope uniform J-stability are equivalent, analogous to Ross-Thomas slope theory and Mumford-Takemoto slope theory for vector bundles, by differential geometric arguments recently. However, this fact has not been proved in algebro-geometric way before. In this talk, I would like to explain a decomposition formula of non-Archimedean J-functional, the (n+1)-dimensional intersection number into n-dimensional intersection numbers and its applications to prove the fact for surfaces and to construct a K-stable but not uniformly K-stable lc pair. Based on arXiv:2103.04603

J-stability is an analog of K-stability and plays an important role in K-stability for general polarized varieties (not only for Kahler-Einstein metrics). Strikingly, G.Chen proved uniform J-stability and slope uniform J-stability are equivalent, analogous to Ross-Thomas slope theory and Mumford-Takemoto slope theory for vector bundles, by differential geometric arguments recently. However, this fact has not been proved in algebro-geometric way before. In this talk, I would like to explain a decomposition formula of non-Archimedean J-functional, the (n+1)-dimensional intersection number into n-dimensional intersection numbers and its applications to prove the fact for surfaces and to construct a K-stable but not uniformly K-stable lc pair. Based on arXiv:2103.04603

### 2021/04/14

15:00-16:00 Room #Zoom (Graduate School of Math. Sci. Bldg.)

Arithmetic deformation of F-singularities (日本語)

**Kenta Sato**(Kyushu)Arithmetic deformation of F-singularities (日本語)

[ Abstract ]

None

None

### 2020/03/02

15:30-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Semiorthogonal decompositions for singular varieties (English)

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

[ Abstract ]

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.

### 2020/02/21

13:30-15:00 Room #370 (Graduate School of Math. Sci. Bldg.)

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

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

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

[ Abstract ]

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.

[ Reference 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/

### 2020/01/21

15:30-17:00 Room #118 (Graduate School of Math. Sci. Bldg.)

(Few) rational curves on K3 surfaces (English)

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

[ Abstract ]

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.

### 2019/12/03

14:30-16:00 Room #056 (Graduate School of Math. Sci. Bldg.)

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

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

[ Abstract ]

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.

### 2019/10/30

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

A Tannakian perspective on rigid analytic geometry (English)

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

[ Abstract ]

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.

### 2019/10/16

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Multidimensional continued fraction for Gorenstein cyclic quotient singularity

**Yusuke Sato**(University of Tokyo/ IPMU)Multidimensional continued fraction for Gorenstein cyclic quotient singularity

[ Abstract ]

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.

### 2019/07/09

13:00-14:30 Room #122 (Graduate School of Math. Sci. Bldg.)

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

**Taro Sano**(Kobe university)Construction of non-Kähler Calabi-Yau 3-folds by smoothing normal crossing varieties (TBA)

[ Abstract ]

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.

### 2019/07/05

10:30-12:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Durfee-type inequality for complete intersection surface singularities

**Makoto Enokizono**(Tokyo university of science)Durfee-type inequality for complete intersection surface singularities

[ Abstract ]

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.

### 2019/06/28

15:30-17:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Rational curves on prime Fano 3-folds (TBA)

**Sho Tanimoto**(Kumamoto)Rational curves on prime Fano 3-folds (TBA)

[ Abstract ]

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.

### 2019/06/19

15:30-17:00 Room #118 (Graduate School of Math. Sci. Bldg.)

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

**Fumiaki Suzuki**(UIC)A pencil of Enriques surfaces with non-algebraic integral Hodge classes (TBA)

[ Abstract ]

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.

### 2019/05/29

15:30-17:00 Room #118 (Graduate School of Math. Sci. Bldg.)

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

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

[ Abstract ]

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.

### 2019/05/22

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

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

**Tatsuro Kawakami**(Tokyo)Bogomolov type vanishing on three-dimensional Mori fiber spaces in positive characteristic

[ Abstract ]

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.

### 2019/05/15

15:30-17:00 Room #118 (Graduate School of Math. Sci. Bldg.)

On quasi-log canonical pairs

(Japanese)

**Osamu Fujino**(Osaka)On quasi-log canonical pairs

(Japanese)

[ Abstract ]

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.