## Algebraic Geometry Seminar

Seminar information archive ～02/07｜Next seminar｜Future seminars 02/08～

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.) |
---|

**Seminar information archive**

### 2012/12/13

10:40-12:10 Room #118 (Graduate School of Math. Sci. Bldg.)

The asymptotic variety of polynomial maps (ENGLISH)

**Jean-Paul Brasselet**(CNRS (Luminy))The asymptotic variety of polynomial maps (ENGLISH)

[ Abstract ]

The asymptotic variety, or set of non-properness has been intensively studied by Zbigniew Jelonek. In a recent paper, Anna and Guillaume Valette associate to a polynomial map $F: {\\mathbb C}^n \\to {\\mathbb C}^n$ a singular variety $N_F$ and relate properness property of $F$ to the vanishing of some intersection homology groups of $N_F$. I will explain how stratifications of the asymptotic variety of $F$ play an important role in the story and how recently, one of my students, Nguyen Thi Bich Thuy, found a nice way to exhibit such a suitable stratification.

The asymptotic variety, or set of non-properness has been intensively studied by Zbigniew Jelonek. In a recent paper, Anna and Guillaume Valette associate to a polynomial map $F: {\\mathbb C}^n \\to {\\mathbb C}^n$ a singular variety $N_F$ and relate properness property of $F$ to the vanishing of some intersection homology groups of $N_F$. I will explain how stratifications of the asymptotic variety of $F$ play an important role in the story and how recently, one of my students, Nguyen Thi Bich Thuy, found a nice way to exhibit such a suitable stratification.

### 2012/12/10

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

A hyperbolic metric and stability conditions on K3 surfaces with $¥rho=1$ (JAPANESE)

**Kotaro Kawatani**(Nagoya University)A hyperbolic metric and stability conditions on K3 surfaces with $¥rho=1$ (JAPANESE)

[ Abstract ]

We introduce a hyperbolic metric on the (normalized) space of stability conditions on projective K3 surfaces $X$ with Picard rank 1. Furthermore we demonstrate how this hyperbolic metric is helpful for us by discussing two or three topics.

We introduce a hyperbolic metric on the (normalized) space of stability conditions on projective K3 surfaces $X$ with Picard rank 1. Furthermore we demonstrate how this hyperbolic metric is helpful for us by discussing two or three topics.

### 2012/11/26

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

A configuration of rational curves on the superspecial K3 surface (JAPANESE)

**Toshiyuki Katsura**(Hosei University)A configuration of rational curves on the superspecial K3 surface (JAPANESE)

### 2012/11/19

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

Stability conditions and birational geometry (JAPANESE)

**Yukinobu Toda**(IPMU)Stability conditions and birational geometry (JAPANESE)

[ Abstract ]

I propose a conjecture which claims that MMP for a smooth projective variety is realized as a variation of Bridgeland moduli spaces of semistable objects in the derived category of coherent sheaves. I will discuss the surface case and extremal contractions for 3-folds. In the former case, the conjecture is completely solved. In the latter case, I will construct the perverse t-structure associated to the extremal contraction, and construct a candidate of the desired stability condition as a double tilting of the perverse heart.

I propose a conjecture which claims that MMP for a smooth projective variety is realized as a variation of Bridgeland moduli spaces of semistable objects in the derived category of coherent sheaves. I will discuss the surface case and extremal contractions for 3-folds. In the former case, the conjecture is completely solved. In the latter case, I will construct the perverse t-structure associated to the extremal contraction, and construct a candidate of the desired stability condition as a double tilting of the perverse heart.

### 2012/11/12

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

On Fano fourfolds with nef vector bundles $Λ^2T_X$ (JAPANESE)

**Kazunori Yasutake**(Kyushu University)On Fano fourfolds with nef vector bundles $Λ^2T_X$ (JAPANESE)

[ Abstract ]

By using results about extremal contractions on smooth fourfolds, we give a classification of fano fourfolds whose the second exterior power of tangent bundles are numerically effective.

By using results about extremal contractions on smooth fourfolds, we give a classification of fano fourfolds whose the second exterior power of tangent bundles are numerically effective.

### 2012/11/05

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

The rationality of the moduli spaces of trigonal curves (JAPANESE)

**Shouhei Ma**(Nagoya University)The rationality of the moduli spaces of trigonal curves (JAPANESE)

### 2012/10/29

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

The Mukai conjecture for log Fano manifolds (JAPANESE)

**Kento Fujita**(RIMS)The Mukai conjecture for log Fano manifolds (JAPANESE)

[ Abstract ]

The concept of log Fano manifolds is one of the most natural generalization of the concept of Fano manifolds. We will give some structure theorems of log Fano manifolds. For example, we will show that the Mukai conjecture for Fano manifolds implies the `log Mukai conjecture' for log Fano manifolds.

The concept of log Fano manifolds is one of the most natural generalization of the concept of Fano manifolds. We will give some structure theorems of log Fano manifolds. For example, we will show that the Mukai conjecture for Fano manifolds implies the `log Mukai conjecture' for log Fano manifolds.

### 2012/10/15

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

On the moduli b-divisors of lc-trivial fibrations (JAPANESE)

**Yoshinori Gongyo**(University of Tokyo)On the moduli b-divisors of lc-trivial fibrations (JAPANESE)

[ Abstract ]

Roughly speaking, by using the semi-stable minimal model program, we prove that the moduli part of an lc-trivial fibration coincides with that of a klt-trivial fibration induced by adjunction after taking a suitable generically finite cover. As an application, we obtain that the moduli part of an lc-trivial fibration is b-nef and abundant by Ambro's result on klt-trivial fibrations. Moreover I may explain some applications of canonical bundle formulas. These are joint works with Osamu Fujino.

Roughly speaking, by using the semi-stable minimal model program, we prove that the moduli part of an lc-trivial fibration coincides with that of a klt-trivial fibration induced by adjunction after taking a suitable generically finite cover. As an application, we obtain that the moduli part of an lc-trivial fibration is b-nef and abundant by Ambro's result on klt-trivial fibrations. Moreover I may explain some applications of canonical bundle formulas. These are joint works with Osamu Fujino.

### 2012/10/01

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

Weak Lefschetz for divisors (ENGLISH)

**Robert Laterveer**(CNRS, IRMA, Université de Strasbourg)Weak Lefschetz for divisors (ENGLISH)

[ Abstract ]

Let $X$ be a complex projective variety (possibly singular), and $Y\\subset X$ a generic hyperplane section. We prove several weak Lefschetz results concerning the restriction $A^1(X)_{\\qq}\\to A^1(Y)_{\\qq}$, where $A^1$ denotes Fulton--MacPherson's operational Chow cohomology group. In addition, we reprove (and slightly extend) a weak Lefschetz result concerning the Chow group of Weil divisors first proven by Ravindra and Srinivas. As an application of these weak Lefschetz results, we can say something about when the natural map from the Picard group to $A^1$ is an isomorphism.

Let $X$ be a complex projective variety (possibly singular), and $Y\\subset X$ a generic hyperplane section. We prove several weak Lefschetz results concerning the restriction $A^1(X)_{\\qq}\\to A^1(Y)_{\\qq}$, where $A^1$ denotes Fulton--MacPherson's operational Chow cohomology group. In addition, we reprove (and slightly extend) a weak Lefschetz result concerning the Chow group of Weil divisors first proven by Ravindra and Srinivas. As an application of these weak Lefschetz results, we can say something about when the natural map from the Picard group to $A^1$ is an isomorphism.

### 2012/10/01

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

Frobenius morphisms and derived categories on two dimensional toric Deligne--Mumford stacks (JAPANESE)

**Ryo Ohkawa**(RIMS, Kyoto University)Frobenius morphisms and derived categories on two dimensional toric Deligne--Mumford stacks (JAPANESE)

[ Abstract ]

For a toric Deligne-Mumford (DM) stack over the complex number field, we can consider a certain generalization of the Frobenius endomorphism. For such an endomorphism of a two-dimensional toric DM stack, we show that the push-forward of the structure sheaf generates the bounded derived category of coherent sheaves on the stack. This is joint work with Hokuto Uehara.

For a toric Deligne-Mumford (DM) stack over the complex number field, we can consider a certain generalization of the Frobenius endomorphism. For such an endomorphism of a two-dimensional toric DM stack, we show that the push-forward of the structure sheaf generates the bounded derived category of coherent sheaves on the stack. This is joint work with Hokuto Uehara.

### 2012/07/30

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

Log Bend-and-Break on Deligne-Mumford stacks (ENGLISH)

**Gianluca Pacienza**(Université de Strasbourg)Log Bend-and-Break on Deligne-Mumford stacks (ENGLISH)

[ Abstract ]

We prove a logarithmic Bend-and-Break lemma on a LCI Deligne-Mumford stacks with projective moduli space and integral boundary divisor. As a by-product we obtain a logarithmic version of the Miyaoka-Mori numerical criterion of uniruledness for DM stacks (under additional conditions on the boundary and on the non-schematic locus) and a Cone Theorem for Deligne-Mumford stacks with boundary. These results hold on an algebraically closed field of any characteristic. This is joint work with Michael McQuillan.

We prove a logarithmic Bend-and-Break lemma on a LCI Deligne-Mumford stacks with projective moduli space and integral boundary divisor. As a by-product we obtain a logarithmic version of the Miyaoka-Mori numerical criterion of uniruledness for DM stacks (under additional conditions on the boundary and on the non-schematic locus) and a Cone Theorem for Deligne-Mumford stacks with boundary. These results hold on an algebraically closed field of any characteristic. This is joint work with Michael McQuillan.

### 2012/07/23

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

Derived category of smooth proper Deligne-Mumford stack with p_g>0 (JAPANESE)

**Shinnosuke Okawa**(University of Tokyo)Derived category of smooth proper Deligne-Mumford stack with p_g>0 (JAPANESE)

[ Abstract ]

Semiorthogonal decomposition (SOD) of the derived category of coherent sheaves reflects interesting geometry of varieties (more generally stacks), such as minimal model program. We show that the global sections of the canonical line bundle (if exists) give restrictions on the possible form of SODs. As a special case, we see that the global generation of the canonical line bundle implies the non-existence of SODs. (joint work with Kotaro Kawatani)

Semiorthogonal decomposition (SOD) of the derived category of coherent sheaves reflects interesting geometry of varieties (more generally stacks), such as minimal model program. We show that the global sections of the canonical line bundle (if exists) give restrictions on the possible form of SODs. As a special case, we see that the global generation of the canonical line bundle implies the non-existence of SODs. (joint work with Kotaro Kawatani)

### 2012/06/25

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

Automorphism groups of Calabi-Yau manifolds of Picard number two (JAPANESE)

**Keiji Oguiso**(Osaka University)Automorphism groups of Calabi-Yau manifolds of Picard number two (JAPANESE)

[ Abstract ]

We prove that the automorphism group of an odd dimensional Calabi-Yau manifold of Picard number two is always a finite group. This makes a sharp contrast to the automorphism groups of K3 surfaces and hyperk\\"ahler manifolds and birational automorphism groups, as I shall explain. We also clarify the relation between finiteness of the automorphism group (resp. birational automorphism group) and the rationality of the nef cone (resp. movable cone) for a hyperk\\"ahler manifold of Picard number two. We will also discuss a similar conjectual relation for a Calabi-Yau threefold of Picard number two, together with exsistence of rational curve, expected by the cone conjecture.

We prove that the automorphism group of an odd dimensional Calabi-Yau manifold of Picard number two is always a finite group. This makes a sharp contrast to the automorphism groups of K3 surfaces and hyperk\\"ahler manifolds and birational automorphism groups, as I shall explain. We also clarify the relation between finiteness of the automorphism group (resp. birational automorphism group) and the rationality of the nef cone (resp. movable cone) for a hyperk\\"ahler manifold of Picard number two. We will also discuss a similar conjectual relation for a Calabi-Yau threefold of Picard number two, together with exsistence of rational curve, expected by the cone conjecture.

### 2012/06/18

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

アルバネーゼ次元最大の複素射影多様体の特殊集合について (JAPANESE)

**Katsutoshi Yamanoi**(Tokyo Institute of Technology)アルバネーゼ次元最大の複素射影多様体の特殊集合について (JAPANESE)

[ Abstract ]

アルバネーゼ次元が最大の複素射影多様体の中に含まれる代数的あるいは超越的な複

素曲線について、

高次元ネヴァンリンナ理論の立場からお話します。

アルバネーゼ次元が最大の複素射影多様体の中に含まれる代数的あるいは超越的な複

素曲線について、

高次元ネヴァンリンナ理論の立場からお話します。

### 2012/06/14

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

Vanishing theorems for perverse sheaves on abelian varieties (ENGLISH)

**Christian Schnell**(IPMU)Vanishing theorems for perverse sheaves on abelian varieties (ENGLISH)

[ Abstract ]

I will describe a few results, due to Kraemer-Weissauer and myself, about perverse sheaves on complex abelian varieties; they are natural generalizations of the generic vanishing theorem of Green-Lazarsfeld.

I will describe a few results, due to Kraemer-Weissauer and myself, about perverse sheaves on complex abelian varieties; they are natural generalizations of the generic vanishing theorem of Green-Lazarsfeld.

### 2012/06/04

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

Smooth P1-fibrations and Campana-Peternell conjecture (ENGLISH)

**Kiwamu Watanabe**(Saitama University)Smooth P1-fibrations and Campana-Peternell conjecture (ENGLISH)

[ Abstract ]

We give a complete classification of smooth P1-fibrations

over projective manifolds of Picard number 1 each of which admit another

smooth morphism of relative dimension one.

Furthermore, we consider relations of the result with Campana-Peternell conjecture

on Fano manifolds with nef tangent bundle.

We give a complete classification of smooth P1-fibrations

over projective manifolds of Picard number 1 each of which admit another

smooth morphism of relative dimension one.

Furthermore, we consider relations of the result with Campana-Peternell conjecture

on Fano manifolds with nef tangent bundle.

### 2012/05/28

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

Generic vanishing and linearity via Hodge modules (ENGLISH)

**Mihnea Popa**(University of Illinois at Chicago)Generic vanishing and linearity via Hodge modules (ENGLISH)

[ Abstract ]

I will explain joint work with Christian Schnell, in which we extend the fundamental results of generic vanishing theory (for instance for the canonical bundle of a smooth projective variety) to bundles of holomorphic forms and to rank one local systems, where parts of the theory have eluded previous efforts. To achiever this, we bring all of the old and new results under the same roof by enlarging the scope of generic vanishing theory to the study of filtered D-modules associated to mixed Hodge modules. Besides Saito's vanishing and direct image theorems for Hodge modules, an important input is the Laumon-Rothstein Fourier transform for bundles with integrable connection.

I will explain joint work with Christian Schnell, in which we extend the fundamental results of generic vanishing theory (for instance for the canonical bundle of a smooth projective variety) to bundles of holomorphic forms and to rank one local systems, where parts of the theory have eluded previous efforts. To achiever this, we bring all of the old and new results under the same roof by enlarging the scope of generic vanishing theory to the study of filtered D-modules associated to mixed Hodge modules. Besides Saito's vanishing and direct image theorems for Hodge modules, an important input is the Laumon-Rothstein Fourier transform for bundles with integrable connection.

### 2012/05/21

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

Characterizations of projective spaces and hyperquadrics

(JAPANESE)

**Taku Suzuki**(Waseda University)Characterizations of projective spaces and hyperquadrics

(JAPANESE)

[ Abstract ]

After Mori's works on Hartshorne's conjecture, many results to

characterize projective spaces and hyperquadrics in terms of

positivity properties of the tangent bundle have been provided.

Kov\\'acs' conjecture states that smooth complex projective

varieties are projective spaces or hyperquadrics if the $p$-th

exterior product of their tangent bundle contains the $p$-th

exterior product of an ample vector bundle. This conjecture is

the generalization of many preceding results. In this talk, I will

explain the idea of the proof of Kov\\'acs' conjecture for varieties

with Picard number one by using a method of slope-stabilities

of sheaves.

After Mori's works on Hartshorne's conjecture, many results to

characterize projective spaces and hyperquadrics in terms of

positivity properties of the tangent bundle have been provided.

Kov\\'acs' conjecture states that smooth complex projective

varieties are projective spaces or hyperquadrics if the $p$-th

exterior product of their tangent bundle contains the $p$-th

exterior product of an ample vector bundle. This conjecture is

the generalization of many preceding results. In this talk, I will

explain the idea of the proof of Kov\\'acs' conjecture for varieties

with Picard number one by using a method of slope-stabilities

of sheaves.

### 2012/05/07

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

Algebro-geometric characterization of Cayley polytopes (JAPANESE)

**Atsushi Ito**(University of Tokyo)Algebro-geometric characterization of Cayley polytopes (JAPANESE)

[ Abstract ]

A lattice polytope is called a Cayley polytope if it is "small" in some

sense.

In this talk, I will explain an algebro-geometric characterization of

Cayley polytopes

by considering whether or not the corresponding polarized toric

varieties are covered by lines, planes, etc.

We can apply this characterization to the study of Seshadri constants,

which are invariants measuring the positivity of ample line bundles.

That is, we can obtain an explicit description of a polarized toric

variety whose Seshadri constant is one.

A lattice polytope is called a Cayley polytope if it is "small" in some

sense.

In this talk, I will explain an algebro-geometric characterization of

Cayley polytopes

by considering whether or not the corresponding polarized toric

varieties are covered by lines, planes, etc.

We can apply this characterization to the study of Seshadri constants,

which are invariants measuring the positivity of ample line bundles.

That is, we can obtain an explicit description of a polarized toric

variety whose Seshadri constant is one.

### 2012/04/23

17:10-18:40 Room #122 (Graduate School of Math. Sci. Bldg.)

Motivic integration and wild group actions (JAPANESE)

**Takehiko Yasuda**(Osaka University)Motivic integration and wild group actions (JAPANESE)

[ Abstract ]

The cohomological McKay correspondence proved by Batyrev is the equality of an orbifold invariant

and a stringy invariant. The former is an invariant of a smooth variety with a finite group action and the latter is

an invariant of its quotient variety. Denef and Loeser gave an alternative proof of it which uses the motivic integration theory developped by themselves.

Then I pushed forward with their study by generalizing the motivic integration to

Deligne-Mumford stacks and reformulating the cohomological McKay correspondence from the viewpoint of

the birational geometry of stacks.

However all of these are about tame group actions (the order of a group is not divisible by the characteristic of the base field),

and the wild (= not tame) case has remained unexplored.

In this talk, I will explain my attempt to examine the simplest situation of the wild case. Namely linear actions of a cyclic group

of order equal to the characteristic of the base field are treated. A remarkable new phenomenon is that the space of generalized

arcs is a fibration over an infinite dimensional space with infinite dimensional fibers, where the base space is the space of

Artin-Schreier extensions of $k((t))$, the field of Laurent series.

The cohomological McKay correspondence proved by Batyrev is the equality of an orbifold invariant

and a stringy invariant. The former is an invariant of a smooth variety with a finite group action and the latter is

an invariant of its quotient variety. Denef and Loeser gave an alternative proof of it which uses the motivic integration theory developped by themselves.

Then I pushed forward with their study by generalizing the motivic integration to

Deligne-Mumford stacks and reformulating the cohomological McKay correspondence from the viewpoint of

the birational geometry of stacks.

However all of these are about tame group actions (the order of a group is not divisible by the characteristic of the base field),

and the wild (= not tame) case has remained unexplored.

In this talk, I will explain my attempt to examine the simplest situation of the wild case. Namely linear actions of a cyclic group

of order equal to the characteristic of the base field are treated. A remarkable new phenomenon is that the space of generalized

arcs is a fibration over an infinite dimensional space with infinite dimensional fibers, where the base space is the space of

Artin-Schreier extensions of $k((t))$, the field of Laurent series.

### 2012/04/16

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

Toric degenerations of minuscule Schubert varieties and mirror symmetry (JAPANESE)

**Makoto Miura**(University of Tokyo)Toric degenerations of minuscule Schubert varieties and mirror symmetry (JAPANESE)

[ Abstract ]

Minuscule Schubert varieties admit the flat degenerations to projective

Hibi toric varieties, whose combinatorial structure is explicitly

described by finite posets. In this talk, I will explain these toric

degenerations and discuss the mirror symmetry for complete intersection

Calabi-Yau varieties in Gorenstein minuscule Schubert varieties.

Minuscule Schubert varieties admit the flat degenerations to projective

Hibi toric varieties, whose combinatorial structure is explicitly

described by finite posets. In this talk, I will explain these toric

degenerations and discuss the mirror symmetry for complete intersection

Calabi-Yau varieties in Gorenstein minuscule Schubert varieties.

### 2012/04/09

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

On mirror symmetry for weighted Calabi-Yau hypersurfaces (JAPANESE)

**Kazushi Ueda**(Osaka University)On mirror symmetry for weighted Calabi-Yau hypersurfaces (JAPANESE)

[ Abstract ]

In the talk, I will discuss relation between homological mirror symmetry for weighted projective spaces, their Calabi-Yau hypersurfaces, and weighted homogeneous singularities.

If the time permits, I will also discuss an application to monodromy of hypergeometric functions.

In the talk, I will discuss relation between homological mirror symmetry for weighted projective spaces, their Calabi-Yau hypersurfaces, and weighted homogeneous singularities.

If the time permits, I will also discuss an application to monodromy of hypergeometric functions.

### 2012/01/30

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

On varieties of globally F-regular type (JAPANESE)

**Yoshinori Gongyo**(University of Tokyo)On varieties of globally F-regular type (JAPANESE)

[ Abstract ]

I will talk about recent topics on varieties of globally F-regular type.

I will talk about recent topics on varieties of globally F-regular type.

### 2012/01/23

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

Sigularities and Kodaira dimension of the moduli of stable sheaves on Enriques surfaces (JAPANESE)

**Kimiko Yamada**(Okayama university)Sigularities and Kodaira dimension of the moduli of stable sheaves on Enriques surfaces (JAPANESE)

[ Abstract ]

We shall estimate singularities of moduli of stable sheaves on Enriques/hyper-elliptic surfaces via the Kuranishi theory, consider when its singularities are canonical, and calculate its Kodaira dimension.

We shall estimate singularities of moduli of stable sheaves on Enriques/hyper-elliptic surfaces via the Kuranishi theory, consider when its singularities are canonical, and calculate its Kodaira dimension.

### 2012/01/16

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

TBA(Cancelled) (JAPANESE)

**Mihai Paun**(Institut Élie Cartan and KIAS)TBA(Cancelled) (JAPANESE)