## Algebraic Geometry Seminar

Seminar information archive ～06/22｜Next seminar｜Future seminars 06/23～

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

**Seminar information archive**

### 2018/06/19

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

Dormant Miura opers and Tango structures (Japanese (writing in English))

**Yasuhiro Wakabayashi**(TIT)Dormant Miura opers and Tango structures (Japanese (writing in English))

[ Abstract ]

Only Japanese abstract is available.

Only Japanese abstract is available.

### 2018/06/12

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

Ample canonical heights for endomorphisms on projective varieties (English or Japanese)

**Takahiro Shibata**(Kyoto)Ample canonical heights for endomorphisms on projective varieties (English or Japanese)

[ Abstract ]

Given a smooth projective variety on a number field and an

endomorphism on it, we would like to know how the height of a point

grows by iteration of the action of the endomorphism. When the

endomorphism is polarized, Call and Silverman construct the canonical

height, which is an important tool for the calculation of growth of

heights. In this talk, we will give a generalization of the Call-

Silverman canonical heights for not necessarily polarized endomorphisms,

ample canonical heights, and propose an analogue of the Northcott

finiteness theorem as a conjecture. We will see that the conjecture

holds when the variety is an abelian variety or a surface.

Given a smooth projective variety on a number field and an

endomorphism on it, we would like to know how the height of a point

grows by iteration of the action of the endomorphism. When the

endomorphism is polarized, Call and Silverman construct the canonical

height, which is an important tool for the calculation of growth of

heights. In this talk, we will give a generalization of the Call-

Silverman canonical heights for not necessarily polarized endomorphisms,

ample canonical heights, and propose an analogue of the Northcott

finiteness theorem as a conjecture. We will see that the conjecture

holds when the variety is an abelian variety or a surface.

### 2018/05/29

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

Nikulin configurations on Kummer surfaces (English)

**Alessandra Sarti**(Universit\'e de Poitiers)Nikulin configurations on Kummer surfaces (English)

[ Abstract ]

A Nikulin configuration is the data of

16 disjoint smooth rational curves on a K3 surface.

According to results of Nikulin this means that the K3 surface

is a Kummer surface and the abelian surface in the Kummer structure

is determined by the 16 curves. An old question of Shioda is about the

existence of non isomorphic Kummer structures on the same Kummer K3

surface.

The question was positively answered and studied by several authors, and

it was shown that the number of non-isomorphic Kummer structures is

finite,

but no explicit geometric construction of such structures was given.

In the talk I will show how to construct explicitely non isomorphic

Kummer structures on generic Kummer K3 surfaces.

This is a joint work with X. Roulleau.

A Nikulin configuration is the data of

16 disjoint smooth rational curves on a K3 surface.

According to results of Nikulin this means that the K3 surface

is a Kummer surface and the abelian surface in the Kummer structure

is determined by the 16 curves. An old question of Shioda is about the

existence of non isomorphic Kummer structures on the same Kummer K3

surface.

The question was positively answered and studied by several authors, and

it was shown that the number of non-isomorphic Kummer structures is

finite,

but no explicit geometric construction of such structures was given.

In the talk I will show how to construct explicitely non isomorphic

Kummer structures on generic Kummer K3 surfaces.

This is a joint work with X. Roulleau.

### 2018/05/25

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

Endomorphisms of normal projective variety and equivariant-MMP (English)

**De Qi Zhang**(Singapore)Endomorphisms of normal projective variety and equivariant-MMP (English)

[ Abstract ]

We report some recent joint works on polarized or int-amplified endomorphisms f on a normal projective variety X with mild singularities, and prove the pseudo-effectivity of the anti-canonical divisor of X, and the f-equivariance, after replacing f by its power, for every minimal model program starting from X. Fano varieties and Q-abelian varieties turn out to be building blocks having such symmetries. The ground field is closed and of characteristic 0 or at least 7.

We report some recent joint works on polarized or int-amplified endomorphisms f on a normal projective variety X with mild singularities, and prove the pseudo-effectivity of the anti-canonical divisor of X, and the f-equivariance, after replacing f by its power, for every minimal model program starting from X. Fano varieties and Q-abelian varieties turn out to be building blocks having such symmetries. The ground field is closed and of characteristic 0 or at least 7.

### 2018/05/21

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

Towards the termination of flips. (English)

https://www.math.utah.edu/~hacon/

**Christopher Hacon**(Utah/Kyoto)Towards the termination of flips. (English)

[ Abstract ]

The minimal model program (MMP) predicts that if $X$ is a smooth complex projective variety which is not uniruled, then there is a finite sequence of "elementary" birational maps

$X=X_0-->X_1-->X_2-->...-->X_n$ known as divisorial contractions and flips whose output $\bar X=X_n$ is a minimal model so that $K_{\bar X}$ is a nef $Q$-divisor i.e it intersects all curves $C\subset \bar X$ non-negatively: $K_{\bar X}\cdot C\geq 0$.

The existence of these birational maps has been established, but in order to complete the MMP, it is necessary to show that flips terminate i.e. there are no infinite sequences of flips. In this talk we will discuss recent results towards the termination of flips.

[ Reference URL ]The minimal model program (MMP) predicts that if $X$ is a smooth complex projective variety which is not uniruled, then there is a finite sequence of "elementary" birational maps

$X=X_0-->X_1-->X_2-->...-->X_n$ known as divisorial contractions and flips whose output $\bar X=X_n$ is a minimal model so that $K_{\bar X}$ is a nef $Q$-divisor i.e it intersects all curves $C\subset \bar X$ non-negatively: $K_{\bar X}\cdot C\geq 0$.

The existence of these birational maps has been established, but in order to complete the MMP, it is necessary to show that flips terminate i.e. there are no infinite sequences of flips. In this talk we will discuss recent results towards the termination of flips.

https://www.math.utah.edu/~hacon/

### 2018/05/21

13:30-15:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Perverse sheaves of categories and birational geometry (English)

**Will Donovan**(IPMU)Perverse sheaves of categories and birational geometry (English)

[ Abstract ]

Kapranov and Schechtman have initiated a program to study perverse sheaves of categories, or perverse schobers. It is expected that examples arise from birational geometry, in particular from webs of flops. I explain progress towards constructing these objects for Grothendieck resolutions (work of the above authors with Bondal), and for 3-folds (joint work of myself and Wemyss).

Kapranov and Schechtman have initiated a program to study perverse sheaves of categories, or perverse schobers. It is expected that examples arise from birational geometry, in particular from webs of flops. I explain progress towards constructing these objects for Grothendieck resolutions (work of the above authors with Bondal), and for 3-folds (joint work of myself and Wemyss).

### 2018/05/08

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

Higher order families of lines and Fano manifolds covered by linear

spaces

(Japanese (writing in English))

**Taku Suzuki**(Utsunomiya)Higher order families of lines and Fano manifolds covered by linear

spaces

(Japanese (writing in English))

[ Abstract ]

In this talk, for an embedded Fano manifold $X$, we introduce higher

order families of lines and a new invariant $S_X$. They are line

versions of higher order minimal families of rational curves and the

invariant $N_X$ which were introduced in my previous talk on 4th

November 2016. In addition, $S_X$ is related to the dimension of

covering linear spaces. Our goal is to classify Fano manifolds $X$ which

have large $S_X$.

In this talk, for an embedded Fano manifold $X$, we introduce higher

order families of lines and a new invariant $S_X$. They are line

versions of higher order minimal families of rational curves and the

invariant $N_X$ which were introduced in my previous talk on 4th

November 2016. In addition, $S_X$ is related to the dimension of

covering linear spaces. Our goal is to classify Fano manifolds $X$ which

have large $S_X$.

### 2018/04/24

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

BIRATIONAL BOUNDEDNESS OF RATIONALLY CONNECTED CALABI–YAU 3-FOLDS

(English)

**Wei-Chung Chen**(Tokyo)BIRATIONAL BOUNDEDNESS OF RATIONALLY CONNECTED CALABI–YAU 3-FOLDS

(English)

[ Abstract ]

Firstly, we show that rationally connected Calabi–Yau 3- folds with kawamata log terminal (klt) singularities form a birationally bounded family, or more generally, rationally connected 3-folds of ε-CY type form a birationally bounded family for ε > 0. Then we focus on ε-lc log Calabi–Yau pairs (X, B) such that coefficients of B are bounded from below away from zero. We show that such pairs are log bounded modulo flops. As a consequence, we show that rationally connected klt Calabi–Yau 3-folds with mld bounding away from 1 are bounded modulo flops.

Firstly, we show that rationally connected Calabi–Yau 3- folds with kawamata log terminal (klt) singularities form a birationally bounded family, or more generally, rationally connected 3-folds of ε-CY type form a birationally bounded family for ε > 0. Then we focus on ε-lc log Calabi–Yau pairs (X, B) such that coefficients of B are bounded from below away from zero. We show that such pairs are log bounded modulo flops. As a consequence, we show that rationally connected klt Calabi–Yau 3-folds with mld bounding away from 1 are bounded modulo flops.

### 2018/04/17

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

SNC log symplectic structures on Fano products (English/Japanese)

**Katsuhiko Okumura**(Waseda Univ. )SNC log symplectic structures on Fano products (English/Japanese)

[ Abstract ]

In 2014, Lima and Pereira gave a characterization of the even-dimensional projective space in terms of log symplectic Poisson structures. After that Pym gave an another more algebraic proof. In this talk, we will extend the result of Lima and Pereira to the case that the variety is a product of Fano varieties with the cyclic Picard group. This will be proved by extending Pym's proof. As a corollary, we will obtain a characterization of the projective space of all dimensions.

In 2014, Lima and Pereira gave a characterization of the even-dimensional projective space in terms of log symplectic Poisson structures. After that Pym gave an another more algebraic proof. In this talk, we will extend the result of Lima and Pereira to the case that the variety is a product of Fano varieties with the cyclic Picard group. This will be proved by extending Pym's proof. As a corollary, we will obtain a characterization of the projective space of all dimensions.

### 2018/04/09

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

Adjoint forms on algebraic varieties (English)

**Luca Rizzi**(Udine)Adjoint forms on algebraic varieties (English)

[ Abstract ]

The so called adjoint theory was introduced by A. Collino and G.P. Pirola in the case of smooth algebraic curves and then generalized by G.P. Pirola and F. Zucconi in the case of smooth algebraic varieties of arbitrary dimension.

The main idea of this theory is to study particular differential forms, called adjoint forms, on an algebraic variety to obtain information on the infinitesimal deformations of the variety itself.

The natural context for the application of this theory is given by Torelli-type problems, in particular infinitesimal Torelli problems.

The so called adjoint theory was introduced by A. Collino and G.P. Pirola in the case of smooth algebraic curves and then generalized by G.P. Pirola and F. Zucconi in the case of smooth algebraic varieties of arbitrary dimension.

The main idea of this theory is to study particular differential forms, called adjoint forms, on an algebraic variety to obtain information on the infinitesimal deformations of the variety itself.

The natural context for the application of this theory is given by Torelli-type problems, in particular infinitesimal Torelli problems.

### 2018/04/09

13:30-15:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Commuting nilpotents, punctual Hilbert schemes and jet bundles (ENGLISH)

**David Hyeon**(Seoul National University)Commuting nilpotents, punctual Hilbert schemes and jet bundles (ENGLISH)

[ Abstract ]

Pairs of commuting nilpotent matrices have been extensively studied, especially from the view point of quivers, but the space of commuting nilpotents modulo simultaneous conjugation has not received any attention at all despite its moduli theory flavor. I will explain how a 'moduli space' can be constructed via two different methods and demonstrate many interesting properties of the space:

- It is isomorphic to an open subscheme of a punctual Hilbert scheme.

- Over the field of complex numbers, it is diffeomorphic to a direct sum of twisted tangent bundles over a projective space.

- It is isomorphic to a bundle of regular jets.

- It gives examples of affine space bundles that are not vector bundles.

This is a joint work with W. Haboush (Illinois) and G. Bérczi (Zurich).

Pairs of commuting nilpotent matrices have been extensively studied, especially from the view point of quivers, but the space of commuting nilpotents modulo simultaneous conjugation has not received any attention at all despite its moduli theory flavor. I will explain how a 'moduli space' can be constructed via two different methods and demonstrate many interesting properties of the space:

- It is isomorphic to an open subscheme of a punctual Hilbert scheme.

- Over the field of complex numbers, it is diffeomorphic to a direct sum of twisted tangent bundles over a projective space.

- It is isomorphic to a bundle of regular jets.

- It gives examples of affine space bundles that are not vector bundles.

This is a joint work with W. Haboush (Illinois) and G. Bérczi (Zurich).

### 2018/01/26

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

On classification of prime Q-Fano 3-folds with only 1/2(1,1,1)-singularities and of genus less than 2

**Hiromichi Takagi**(The University of Tokyo)On classification of prime Q-Fano 3-folds with only 1/2(1,1,1)-singularities and of genus less than 2

[ Abstract ]

I classified prime Q-Fano threefolds with only 1/2(1,1,1)-singularities and of genus greater than 1 (2002, Nagoya Math. J.).

In this talk, I will explain how the method in that paper can be extended to the case of genus less than 2. The method is so called two ray game. By this method, I can classify the possibilities of such Q-Fano's. The classification is not yet completed since constructions of examples in certain cases are difficult. I will also explain some pretty examples in this talk.

I classified prime Q-Fano threefolds with only 1/2(1,1,1)-singularities and of genus greater than 1 (2002, Nagoya Math. J.).

In this talk, I will explain how the method in that paper can be extended to the case of genus less than 2. The method is so called two ray game. By this method, I can classify the possibilities of such Q-Fano's. The classification is not yet completed since constructions of examples in certain cases are difficult. I will also explain some pretty examples in this talk.

### 2017/12/26

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

K-stability of log Fano hyperplane arrangements (English)

**Kento Fujita**(RIMS)K-stability of log Fano hyperplane arrangements (English)

[ Abstract ]

We completely determine which log Fano hyperplane arrangements are uniformly K-stable, K-stable, K-polystable, K-semistable or not.

We completely determine which log Fano hyperplane arrangements are uniformly K-stable, K-stable, K-polystable, K-semistable or not.

### 2017/12/14

15:30-17:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Algebraic curves and modular forms of low degree (English)

**Gerard van der Geer**(Universiteit van Amsterdam)Algebraic curves and modular forms of low degree (English)

[ Abstract ]

For genus 2 and 3 modular forms are intimately connected with the moduli of curves of genus 2 and 3. We give an explicit way to describe such modular forms for genus 2 and 3 using invariant theory and give some applications. This is based on joint work with Fabien Clery and Carel Faber.

For genus 2 and 3 modular forms are intimately connected with the moduli of curves of genus 2 and 3. We give an explicit way to describe such modular forms for genus 2 and 3 using invariant theory and give some applications. This is based on joint work with Fabien Clery and Carel Faber.

### 2017/12/14

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

Perfectoid test ideals (English)

**Linquan Ma**(University of Utah)Perfectoid test ideals (English)

[ Abstract ]

Inspired by the recent solution of the direct summand conjecture

of Andre and Bhatt, we introduce perfectoid multiplier/test ideals in mixed

characteristic. As an application, we obtain a uniform bound on the growth

of symbolic powers in regular local rings of mixed characteristic analogous

to results of Ein--Lazarsfeld--Smith and Hochster--Huneke in equal

characteristic. This is joint work with Karl Schwede.

Inspired by the recent solution of the direct summand conjecture

of Andre and Bhatt, we introduce perfectoid multiplier/test ideals in mixed

characteristic. As an application, we obtain a uniform bound on the growth

of symbolic powers in regular local rings of mixed characteristic analogous

to results of Ein--Lazarsfeld--Smith and Hochster--Huneke in equal

characteristic. This is joint work with Karl Schwede.

### 2017/12/05

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

Ascending chain condition for F-pure thresholds on a fixed strongly F-regular germ (English or Japanese)

**Kenta Sato**(The University of Tokyo)Ascending chain condition for F-pure thresholds on a fixed strongly F-regular germ (English or Japanese)

[ Abstract ]

For a germ of a variety in positive characteristic and a non-zero ideal sheaf on the variety, we can define the F-pure threshold of the ideal by using Frobenius morphisms, which measures the singularities of the pair. In this talk, I will show that the set of all F-pure thresholds on a fixed strongly F-regular germ satisfies the ascending chain condition. This is a positive characteristic analogue of the "ascending chain condition for log canonical thresholds" in characteristic 0, which was recently proved by Hacon, McKernan, and Xu.

For a germ of a variety in positive characteristic and a non-zero ideal sheaf on the variety, we can define the F-pure threshold of the ideal by using Frobenius morphisms, which measures the singularities of the pair. In this talk, I will show that the set of all F-pure thresholds on a fixed strongly F-regular germ satisfies the ascending chain condition. This is a positive characteristic analogue of the "ascending chain condition for log canonical thresholds" in characteristic 0, which was recently proved by Hacon, McKernan, and Xu.

### 2017/11/28

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

Kodaira vanishing theorem for Witt canonical sheaves (English)

**Hiromu Tanaka**(Tokyo)Kodaira vanishing theorem for Witt canonical sheaves (English)

[ Abstract ]

We establish an analogue of the Kodaira vanishing theorem in terms of de Rham-Witt complex. More specifically, given a smooth projective variety over a perfect field of positive characteristic, we prove that the higher cohomologies vanish for the tensor product of the Witt canonical sheaf and the Teichmuller lift of an ample invertible sheaf.

We establish an analogue of the Kodaira vanishing theorem in terms of de Rham-Witt complex. More specifically, given a smooth projective variety over a perfect field of positive characteristic, we prove that the higher cohomologies vanish for the tensor product of the Witt canonical sheaf and the Teichmuller lift of an ample invertible sheaf.

### 2017/11/21

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

Orbifold rational connectedness (English)

**Frédéric Campana**(Université de Lorraine/KIAS)Orbifold rational connectedness (English)

[ Abstract ]

The first step in the decomposition by canonical fibrations with fibres of `signed' canonical bundle of an arbitrary complex projective manifolds $X$ is its `rational quotient' (also called `MRC' fibration): it has rationally connected fibres and non-uniruled base. In general, the further steps (such as the Moishezon-Iitaka fibration) of this decomposition will require the consideration of 'orbifold base' of fibrations in order to deal with the multiple fibres (as seen already for elliptic surfaces). One thus needs to work in the larger category of (smooth) `orbifold pairs' $(X,D)$ to achieve this decomposition. The aim of the talk is thus to introduce the notions of Rational Connectedness and 'rational quotient' in this context, by means of suitable equivalent notions of negativity for the orbifold cotangent bundle (suitably defined. When $D$ is reduced, this is just the usual Log-version). The expected equivalence with connecting families of `orbifold rational curves' remains however presently open.

The first step in the decomposition by canonical fibrations with fibres of `signed' canonical bundle of an arbitrary complex projective manifolds $X$ is its `rational quotient' (also called `MRC' fibration): it has rationally connected fibres and non-uniruled base. In general, the further steps (such as the Moishezon-Iitaka fibration) of this decomposition will require the consideration of 'orbifold base' of fibrations in order to deal with the multiple fibres (as seen already for elliptic surfaces). One thus needs to work in the larger category of (smooth) `orbifold pairs' $(X,D)$ to achieve this decomposition. The aim of the talk is thus to introduce the notions of Rational Connectedness and 'rational quotient' in this context, by means of suitable equivalent notions of negativity for the orbifold cotangent bundle (suitably defined. When $D$ is reduced, this is just the usual Log-version). The expected equivalence with connecting families of `orbifold rational curves' remains however presently open.

### 2017/11/14

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

A characterization of the birationality of 4-canonical maps of minimal 3-folds (English)

**Meng Chen**(Fudan)A characterization of the birationality of 4-canonical maps of minimal 3-folds (English)

[ Abstract ]

We explain the following theorem: For any minimal 3-fold X of general type with p_g>4, the 4-canonical map is non-birational if and only if X is birationally fibred by a pencil of (1,2) surfaces. The statement fails in the case of p_g=4.

We explain the following theorem: For any minimal 3-fold X of general type with p_g>4, the 4-canonical map is non-birational if and only if X is birationally fibred by a pencil of (1,2) surfaces. The statement fails in the case of p_g=4.

### 2017/11/07

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

Characterizations of projective space and Seshadri constants in arbitrary characteristic

**Takumi Murayama**(University of Michigan)Characterizations of projective space and Seshadri constants in arbitrary characteristic

[ Abstract ]

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.

### 2017/10/31

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

ACC for log canonical threshold polytopes (English)

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

[ Abstract ]

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.

### 2017/10/30

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

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

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

[ Abstract ]

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.

### 2017/10/17

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

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

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

[ Abstract ]

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.

### 2017/10/10

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

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

**Akihiro Kanemitsu**(The University of Tokyo)Classification of Mukai pairs with corank 3 (English or Japanese)

[ Abstract ]

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 $\dim X \geq 5$.

### 2017/07/18

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

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

**Fuetaro Yobuko**(Tohoku University)On a generalization of Frobenius-splitting and a lifting problem of Calabi-Yau varieties (JAPANESE)

[ Abstract ]

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.