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代数幾何学セミナー
過去の記録 ～05/21｜次回の予定｜今後の予定 05/22～
| 開催情報 | 金曜日 13:30～15:00 数理科学研究科棟(駒場) 118号室 |
|---|---|
| 担当者 | 權業 善範、河上 龍郎 、榎園 誠 |
過去の記録
2014年01月22日(水)
15:00-16:30 数理科学研究科棟(駒場) 122号室
いつもと曜日・時間が異なります
Thomas Ducat 氏 (University of Warwick)
Divisorial Extractions from Singular Curves in Smooth 3-Folds (ENGLISH)
いつもと曜日・時間が異なります
Thomas Ducat 氏 (University of Warwick)
Divisorial Extractions from Singular Curves in Smooth 3-Folds (ENGLISH)
[ 講演概要 ]
Consider a singular curve C contained in a smooth 3-fold X.
Assuming the existence of a Du Val general elephant S containing C,
I give a normal form for the equations of C in X and an outline of how to
construct a divisorial extraction from this curve. If the general S is
Du Val of type D_{2k}, E_6 or E_7 then I can give some explicit
conditions for the existence of a terminal extraction. A treatment of
the D_{2k+1} case should be possible by similar means.
Consider a singular curve C contained in a smooth 3-fold X.
Assuming the existence of a Du Val general elephant S containing C,
I give a normal form for the equations of C in X and an outline of how to
construct a divisorial extraction from this curve. If the general S is
Du Val of type D_{2k}, E_6 or E_7 then I can give some explicit
conditions for the existence of a terminal extraction. A treatment of
the D_{2k+1} case should be possible by similar means.
2014年01月20日(月)
15:30-17:00 数理科学研究科棟(駒場) 126号室
いつもと部屋が異なります
佐野 太郎 氏 (University of Warwick)
Deforming elephants of Q-Fano 3-folds (ENGLISH)
いつもと部屋が異なります
佐野 太郎 氏 (University of Warwick)
Deforming elephants of Q-Fano 3-folds (ENGLISH)
[ 講演概要 ]
Shokurov and Reid proved that a Fano 3-fold with canonical
Gorenstein singularities has a Du Val elephant, that is,
a member of the anticanonical linear system with only Du Val singularities.
The classification of Fano 3-folds is based on this fact.
However, for a Fano 3-fold with non-Gorenstein terminal singularities,
the anticanonical system does not contain such a member in general.
Alt{\\i}nok--Brown--Reid conjectured that, if the anticanonical system is non-empty,
a Q-Fano 3-fold can be deformed to that with a Du Val elephant.
In this talk, I will explain how to deform an elephant with isolated
singularities to a Du Val elephant.
Shokurov and Reid proved that a Fano 3-fold with canonical
Gorenstein singularities has a Du Val elephant, that is,
a member of the anticanonical linear system with only Du Val singularities.
The classification of Fano 3-folds is based on this fact.
However, for a Fano 3-fold with non-Gorenstein terminal singularities,
the anticanonical system does not contain such a member in general.
Alt{\\i}nok--Brown--Reid conjectured that, if the anticanonical system is non-empty,
a Q-Fano 3-fold can be deformed to that with a Du Val elephant.
In this talk, I will explain how to deform an elephant with isolated
singularities to a Du Val elephant.
2013年12月09日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
岡田 拓三 氏 (佐賀大学)
On birationally tririgid Q-Fano threefolds (JAPANESE)
岡田 拓三 氏 (佐賀大学)
On birationally tririgid Q-Fano threefolds (JAPANESE)
[ 講演概要 ]
I will talk about birational geometry of Q-Fano threefolds. A Mori
fiber space birational to a given Q-Fano threefold is called a birational Mori fiber structure of the threefold. The existence of Q-Fano threefolds with a unique birational Mori fiber structure (resp. with two birational Mori fiber structures) is known. In this talk I will give an example of Q-Fano threefolds with three birational Mori fiber structures and also discuss about the behavior of birational Mori fiber structures in a family.
I will talk about birational geometry of Q-Fano threefolds. A Mori
fiber space birational to a given Q-Fano threefold is called a birational Mori fiber structure of the threefold. The existence of Q-Fano threefolds with a unique birational Mori fiber structure (resp. with two birational Mori fiber structures) is known. In this talk I will give an example of Q-Fano threefolds with three birational Mori fiber structures and also discuss about the behavior of birational Mori fiber structures in a family.
2013年11月25日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
小池 貴之 氏 (東大数理)
Minimal singular metrics of some line bundles with infinitely generated section rings (JAPANESE)
小池 貴之 氏 (東大数理)
Minimal singular metrics of some line bundles with infinitely generated section rings (JAPANESE)
[ 講演概要 ]
We consider Hermitian metrics of pseudo-effective line bundles on smooth
projective varieties defined over $\\mathbb{C}$.
Especially we are interested in (possibly singular) Hermitian metrics
with semi-positive curvatures when the section rings are not finitely generated.
We study where and how minimal singular metrics, special Hermitian
metrics with semi-positive curvatures, diverges in the following two situations;
a line bundle admitting no Zariski decomposition even after any
modifications (Nakayama example)
and a nef line bundle $L$ on $X$ satisfying $D \\subset |mL|$ and $|mL-D|
= \\emptyset$ for some divisor $D \\subset X$ and for all $m \\geq 1$ (
Zariski example).
We consider Hermitian metrics of pseudo-effective line bundles on smooth
projective varieties defined over $\\mathbb{C}$.
Especially we are interested in (possibly singular) Hermitian metrics
with semi-positive curvatures when the section rings are not finitely generated.
We study where and how minimal singular metrics, special Hermitian
metrics with semi-positive curvatures, diverges in the following two situations;
a line bundle admitting no Zariski decomposition even after any
modifications (Nakayama example)
and a nef line bundle $L$ on $X$ satisfying $D \\subset |mL|$ and $|mL-D|
= \\emptyset$ for some divisor $D \\subset X$ and for all $m \\geq 1$ (
Zariski example).
2013年11月18日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
この講演はGCOEレクチャーズとの共同セミナーになります
Florin Ambro 氏 (IMAR)
An injectivity theorem (ENGLISH)
この講演はGCOEレクチャーズとの共同セミナーになります
Florin Ambro 氏 (IMAR)
An injectivity theorem (ENGLISH)
[ 講演概要 ]
I will discuss a generalization of the injectivity theorem of Esnault-Viehweg, and an
application to the problem of lifting sections from the non-log canonical locus of a log variety.
I will discuss a generalization of the injectivity theorem of Esnault-Viehweg, and an
application to the problem of lifting sections from the non-log canonical locus of a log variety.
2013年11月11日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
Sung Rak Choi 氏 (POSTECH)
Geography via the base loci (ENGLISH)
Sung Rak Choi 氏 (POSTECH)
Geography via the base loci (ENGLISH)
[ 講演概要 ]
The geography of log model refers to the decomposition of the set of effective adjoint divisors into the cells defined by the resulting models that are obtained by the log minimal model program.
We will describe the geography in terms of the asymptotic base loci and Zariski decompositions of divisors.
As an application, we give a partial answer to a question of B. Totaro concerning the structure of partially ample cones.
The geography of log model refers to the decomposition of the set of effective adjoint divisors into the cells defined by the resulting models that are obtained by the log minimal model program.
We will describe the geography in terms of the asymptotic base loci and Zariski decompositions of divisors.
As an application, we give a partial answer to a question of B. Totaro concerning the structure of partially ample cones.
2013年10月28日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
江 辰 氏 (東京大学数理科学研究科)
Weak Borisov-Alexeev-Borisov conjecture for 3-fold Mori Fiber spaces (ENGLISH)
江 辰 氏 (東京大学数理科学研究科)
Weak Borisov-Alexeev-Borisov conjecture for 3-fold Mori Fiber spaces (ENGLISH)
[ 講演概要 ]
We investigate $\\epsilon$-klt log Fano 3-folds with some Mori fiber space structure, more precisely, with a del Pezzo fibration structure, or a conic bundle structure over projective plane. We give a bound for the log anti-canonical volume of such pair. The method is constructing non-klt centers and using connectedness lemma. This result is related to birational boundedness of log Fano varieties.
We investigate $\\epsilon$-klt log Fano 3-folds with some Mori fiber space structure, more precisely, with a del Pezzo fibration structure, or a conic bundle structure over projective plane. We give a bound for the log anti-canonical volume of such pair. The method is constructing non-klt centers and using connectedness lemma. This result is related to birational boundedness of log Fano varieties.
2013年07月22日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
Stavros Papadakis 氏 (RIMS)
Equivariant degenerations of spherical modules (ENGLISH)
Stavros Papadakis 氏 (RIMS)
Equivariant degenerations of spherical modules (ENGLISH)
[ 講演概要 ]
Given a reductive algebraic group G and an invariant
Hilbert function h, Alexeev and Brion have defined
a moduli scheme M which parametrizes affine G-schemes X
with the property that the coordinate ring of X decomposes,
as G-module, according to the function h. The talk will
be about joint work with Bart Van Steirteghem (New York)
which studies the moduli scheme M under some additional
assumptions.
Given a reductive algebraic group G and an invariant
Hilbert function h, Alexeev and Brion have defined
a moduli scheme M which parametrizes affine G-schemes X
with the property that the coordinate ring of X decomposes,
as G-module, according to the function h. The talk will
be about joint work with Bart Van Steirteghem (New York)
which studies the moduli scheme M under some additional
assumptions.
2013年04月22日(月)
16:30-18:00 数理科学研究科棟(駒場) 118号室
Professor Igor Reider 氏 (Universite d'Angers / RIMS)
Kodaira-Spencer classes, geometry of surfaces of general type and Torelli
theorem (ENGLISH)
Professor Igor Reider 氏 (Universite d'Angers / RIMS)
Kodaira-Spencer classes, geometry of surfaces of general type and Torelli
theorem (ENGLISH)
[ 講演概要 ]
In this talk I will explain a geometric interpretation of Kodaira-Spencer classes and apply
it to the study of the differential of the period map of weight 2 Hodge structures for surfaces
of general type.
My approach is based on interpreting Kodaira-Spencer classes as higher rank bundles and
then studing their stability. This naturally leads to two parts:
1) unstable case
2) stable case.
I will give a geometric characterization of the first case and show how to relate the second
case to a special family of vector bundles giving rise to a family of rational curves. This family
of rational curves is used to recover the surface in question.
In this talk I will explain a geometric interpretation of Kodaira-Spencer classes and apply
it to the study of the differential of the period map of weight 2 Hodge structures for surfaces
of general type.
My approach is based on interpreting Kodaira-Spencer classes as higher rank bundles and
then studing their stability. This naturally leads to two parts:
1) unstable case
2) stable case.
I will give a geometric characterization of the first case and show how to relate the second
case to a special family of vector bundles giving rise to a family of rational curves. This family
of rational curves is used to recover the surface in question.
2013年01月15日(火)
15:30-17:00 数理科学研究科棟(駒場) 128号室
いつもと曜日・場所が異なります
Jungkai Alfred Chen 氏 (National Taiwan University)
Three Dimensional Birational Geoemtry--updates and problems (ENGLISH)
いつもと曜日・場所が異なります
Jungkai Alfred Chen 氏 (National Taiwan University)
Three Dimensional Birational Geoemtry--updates and problems (ENGLISH)
[ 講演概要 ]
In this talk I will talk about some recent results on
biratioanl classification and biratioanl geoemtry of threefolds.
Given a threefold of general type, we improved our previous result by
showing that $Vol \\ge 1/1680$ and $|mK_X|$ is biratioanl for $m \\ge
61$.
Compare with the worst known example that $X_{46} \\subset
\\mathbb{P}(4,5,6,7,23)$, one also knows that there are only finiteley
many singularities type
for threefolds of general type with $1/1680 \\le Vol \\le 1/420$. It is
then intereting to study threefolds of general type with given basket
of singularities and with given fiber structure.
Concerning threefolds with intermediate Kodaira dimension, we
considered the effective Iitaka fibration. For this purpose, it is
interesting to study threefolds with $\\kappa=1$ with given basket of
singularities and abelian fibration.
For explicit birational geoemtry, we will show our result that each
biratioanl map in minimal model program can be factored into a
sequence of following maps (or its inverse)
1. a divisorial contraction to a point of index r with discrepancy 1/r.
2. a blowup along a smooth curve
3. a flop
In this talk I will talk about some recent results on
biratioanl classification and biratioanl geoemtry of threefolds.
Given a threefold of general type, we improved our previous result by
showing that $Vol \\ge 1/1680$ and $|mK_X|$ is biratioanl for $m \\ge
61$.
Compare with the worst known example that $X_{46} \\subset
\\mathbb{P}(4,5,6,7,23)$, one also knows that there are only finiteley
many singularities type
for threefolds of general type with $1/1680 \\le Vol \\le 1/420$. It is
then intereting to study threefolds of general type with given basket
of singularities and with given fiber structure.
Concerning threefolds with intermediate Kodaira dimension, we
considered the effective Iitaka fibration. For this purpose, it is
interesting to study threefolds with $\\kappa=1$ with given basket of
singularities and abelian fibration.
For explicit birational geoemtry, we will show our result that each
biratioanl map in minimal model program can be factored into a
sequence of following maps (or its inverse)
1. a divisorial contraction to a point of index r with discrepancy 1/r.
2. a blowup along a smooth curve
3. a flop
2012年12月13日(木)
10:40-12:10 数理科学研究科棟(駒場) 118号室
いつもと曜日・時間・場所が異なります
Jean-Paul Brasselet 氏 (CNRS (Luminy))
The asymptotic variety of polynomial maps (ENGLISH)
いつもと曜日・時間・場所が異なります
Jean-Paul Brasselet 氏 (CNRS (Luminy))
The asymptotic variety of polynomial maps (ENGLISH)
[ 講演概要 ]
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年12月10日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
川谷 康太郎 氏 (名古屋大学多元数理科学研究科)
A hyperbolic metric and stability conditions on K3 surfaces with $¥rho=1$ (JAPANESE)
川谷 康太郎 氏 (名古屋大学多元数理科学研究科)
A hyperbolic metric and stability conditions on K3 surfaces with $¥rho=1$ (JAPANESE)
[ 講演概要 ]
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年11月26日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
桂 利行 氏 (法政大学理工学部)
超特殊K3曲面上の有理曲線の配置について (JAPANESE)
桂 利行 氏 (法政大学理工学部)
超特殊K3曲面上の有理曲線の配置について (JAPANESE)
[ 講演概要 ]
正標数の代数的閉体$k$上の超特異K3曲面のArtin不変量が1のとき超特殊K3曲面という。標数が3以上であれば、このようなK3曲面は、2つの超特異楕円曲線の直積であるアーべル曲面からつくられるKummer曲面になることが知られている。この講演では$S$上の有理曲線の配置をアーベル曲面の因子の構造を用いて考察し、標数が2ならば$(21)_5$-symmetric configurationが存在すること、また標数3ならば$(16)_{10}$-symmetric configurationと$(280_{4}, 112_{10})$-configurationが存在することを示す。また、後者は、$p^{a} + 1$次のFermat hypersurfaceのline configurationや、N\\'eron-S\\'everi群${\\rm NS}(S)$がLeech latticeを用いて捉えられることと関係することを述べる。
正標数の代数的閉体$k$上の超特異K3曲面のArtin不変量が1のとき超特殊K3曲面という。標数が3以上であれば、このようなK3曲面は、2つの超特異楕円曲線の直積であるアーべル曲面からつくられるKummer曲面になることが知られている。この講演では$S$上の有理曲線の配置をアーベル曲面の因子の構造を用いて考察し、標数が2ならば$(21)_5$-symmetric configurationが存在すること、また標数3ならば$(16)_{10}$-symmetric configurationと$(280_{4}, 112_{10})$-configurationが存在することを示す。また、後者は、$p^{a} + 1$次のFermat hypersurfaceのline configurationや、N\\'eron-S\\'everi群${\\rm NS}(S)$がLeech latticeを用いて捉えられることと関係することを述べる。
2012年11月19日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
戸田 幸伸 氏 (IPMU)
Stability conditions and birational geometry (JAPANESE)
戸田 幸伸 氏 (IPMU)
Stability conditions and birational geometry (JAPANESE)
[ 講演概要 ]
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年11月12日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
安武 和範 氏 (九州大学大学院数理学研究院)
On Fano fourfolds with nef vector bundles $Λ^2T_X$ (JAPANESE)
安武 和範 氏 (九州大学大学院数理学研究院)
On Fano fourfolds with nef vector bundles $Λ^2T_X$ (JAPANESE)
[ 講演概要 ]
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年11月05日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
馬 昭平 氏 (名古屋大学多元数理科学研究科)
トリゴナル曲線のモジュライの有理性 (JAPANESE)
馬 昭平 氏 (名古屋大学多元数理科学研究科)
トリゴナル曲線のモジュライの有理性 (JAPANESE)
[ 講演概要 ]
射影直線を3対1で被覆するような曲線はトリゴナル曲線と呼ばれ、超楕円曲線に次いで特殊な曲線である。Shepherd-Barron氏は種数が4で割って2余る場合にトリゴナル曲線のモジュライ空間が射影空間と双有理同型であることを証明した。本講演ではその結果を全ての種数に拡張する。
射影直線を3対1で被覆するような曲線はトリゴナル曲線と呼ばれ、超楕円曲線に次いで特殊な曲線である。Shepherd-Barron氏は種数が4で割って2余る場合にトリゴナル曲線のモジュライ空間が射影空間と双有理同型であることを証明した。本講演ではその結果を全ての種数に拡張する。
2012年10月29日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
藤田 健人 氏 (京都大学数理解析研究所)
The Mukai conjecture for log Fano manifolds (JAPANESE)
藤田 健人 氏 (京都大学数理解析研究所)
The Mukai conjecture for log Fano manifolds (JAPANESE)
[ 講演概要 ]
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年10月15日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
權業善範 氏 (東京大学数理科学研究科)
On the moduli b-divisors of lc-trivial fibrations (JAPANESE)
權業善範 氏 (東京大学数理科学研究科)
On the moduli b-divisors of lc-trivial fibrations (JAPANESE)
[ 講演概要 ]
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年10月01日(月)
13:30-15:00 数理科学研究科棟(駒場) 122号室
Robert Laterveer 氏 (CNRS, IRMA, Université de Strasbourg)
Weak Lefschetz for divisors (ENGLISH)
Robert Laterveer 氏 (CNRS, IRMA, Université de Strasbourg)
Weak Lefschetz for divisors (ENGLISH)
[ 講演概要 ]
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年10月01日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
大川 領 氏 (京都大学数理解析研究所)
Frobenius morphisms and derived categories on two dimensional toric Deligne--Mumford stacks (JAPANESE)
大川 領 氏 (京都大学数理解析研究所)
Frobenius morphisms and derived categories on two dimensional toric Deligne--Mumford stacks (JAPANESE)
[ 講演概要 ]
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年07月30日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
Gianluca Pacienza 氏 (Université de Strasbourg)
Log Bend-and-Break on Deligne-Mumford stacks (ENGLISH)
Gianluca Pacienza 氏 (Université de Strasbourg)
Log Bend-and-Break on Deligne-Mumford stacks (ENGLISH)
[ 講演概要 ]
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年07月23日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
講演者が変更になりました
大川 新之介 氏 (東京大学)
Derived category of smooth proper Deligne-Mumford stack with p_g>0 (JAPANESE)
講演者が変更になりました
大川 新之介 氏 (東京大学)
Derived category of smooth proper Deligne-Mumford stack with p_g>0 (JAPANESE)
[ 講演概要 ]
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年06月25日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
小木曽啓示 氏 (大阪大学)
Automorphism groups of Calabi-Yau manifolds of Picard number two (JAPANESE)
小木曽啓示 氏 (大阪大学)
Automorphism groups of Calabi-Yau manifolds of Picard number two (JAPANESE)
[ 講演概要 ]
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年06月18日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
山ノ井克俊 氏 (東京工業大学)
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)
山ノ井克俊 氏 (東京工業大学)
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)
[ 講演概要 ]
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年06月14日(木)
13:30-15:00 数理科学研究科棟(駒場) 122号室
Christian Schnell 氏 (IPMU)
Vanishing theorems for perverse sheaves on abelian varieties (ENGLISH)
Christian Schnell 氏 (IPMU)
Vanishing theorems for perverse sheaves on abelian varieties (ENGLISH)
[ 講演概要 ]
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.
