## Algebraic Geometry Seminar

Seminar information archive ～09/27｜Next seminar｜Future seminars 09/28～

Date, time & place | Tuesday 10:30 - 11:30 or 12:00 ハイブリッド開催/002Room #ハイブリッド開催/002 (Graduate School of Math. Sci. Bldg.) |
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**Seminar information archive**

### 2019/05/08

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

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

**Kenta Hashizume**(Tokyo)On Minimal model theory for log canonical pairs with big boundary divisors

[ Abstract ]

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.

### 2019/04/24

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

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

endomorphism

**Shou Yoshikawa**(Tokyo)Varieties of dense globally F-split type with a non-invertible polarized

endomorphism

[ Abstract ]

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.

### 2019/01/29

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

Logarithmic good reduction and the index (TBA)

**Kentaro Mitsui**(Kobe)Logarithmic good reduction and the index (TBA)

[ Abstract ]

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.

### 2018/12/21

10:30-11:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Degenerations of p-adic volume forms (English)

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

[ Abstract ]

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.

### 2018/12/14

10:30-11:30 Room #123 (Graduate School of Math. Sci. Bldg.)

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

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

[ Abstract ]

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.

### 2018/11/27

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

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

**Nobuo Hara**(Tokyo University of Agriculture and Technology)Frobenius summands and the finite F-representation type (TBA)

[ Abstract ]

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

### 2018/11/20

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

Artin-Mazur height, Yobuko height and

Hodge-Wittt cohomologies

**Nakkajima Yukiyoshi**(Tokyo Denki University)Artin-Mazur height, Yobuko height and

Hodge-Wittt cohomologies

[ Abstract ]

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 $\dim X=3$.

### 2018/11/13

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

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

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

[ Abstract ]

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.

### 2018/10/16

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

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

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

[ Abstract ]

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.

### 2018/10/09

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

Stability conditions on threefolds with nef tangent bundles (English)

**Naoki Koseki**(Tokyo/IPMU)Stability conditions on threefolds with nef tangent bundles (English)

[ Abstract ]

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.

### 2018/07/18

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

Normal Legendrian singularities (English)

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

[ Abstract ]

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.

### 2018/07/10

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

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

**Ching-Jui Lai**(NCKU)The effective bound of anticanonical volume of Fano threefolds (English)

[ Abstract ]

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.

### 2018/07/03

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

Surface automorphisms and Salem numbers (English)

**Xun Yu**(Tianjin University)Surface automorphisms and Salem numbers (English)

[ Abstract ]

The entropy of a surface automorphism is either zero or the

logarithm of a Salem number.

In this talk, we will discuss which Salem numbers arise in this way. We

will show that any

supersingular K3 surface in odd characteristic has an automorphism the

entropy of which is

the logarithm of a Salem number of degree 22. In particular, such

automorphisms are

not geometrically liftable to characteristic 0.

The entropy of a surface automorphism is either zero or the

logarithm of a Salem number.

In this talk, we will discuss which Salem numbers arise in this way. We

will show that any

supersingular K3 surface in odd characteristic has an automorphism the

entropy of which is

the logarithm of a Salem number of degree 22. In particular, such

automorphisms are

not geometrically liftable to characteristic 0.

### 2018/06/26

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

Varieties with nef diagonal (English)

**Kiwamu Watanabe**(Saitama)Varieties with nef diagonal (English)

[ Abstract ]

For a smooth projective variety $X$, we consider when the diagonal $Δ _X$ is nef as a

cycle on $X \times X$. In particular, we give a classication of complete intersections and smooth

del Pezzo varieties where the diagonal is nef. We also study the nefness of the diagonal for

spherical varieties. This is a joint work with Taku Suzuki.

For a smooth projective variety $X$, we consider when the diagonal $Δ _X$ is nef as a

cycle on $X \times X$. In particular, we give a classication of complete intersections and smooth

del Pezzo varieties where the diagonal is nef. We also study the nefness of the diagonal for

spherical varieties. This is a joint work with Taku Suzuki.

### 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).