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過去の記録 ～02/15｜次回の予定｜今後の予定 02/16～
| 開催情報 | 金曜日 13:30～15:00 数理科学研究科棟(駒場) ハイブリッド開催/117号室 |
|---|---|
| 担当者 | 權業 善範、中村 勇哉、田中 公 |
過去の記録
2025年01月22日(水)
13:30-15:00 数理科学研究科棟(駒場) 002号室
田中公 氏 (東京大学)
Liftability and vanishing theorems for Fano threefolds in positive characteristic (日本語)
田中公 氏 (東京大学)
Liftability and vanishing theorems for Fano threefolds in positive characteristic (日本語)
[ 講演概要 ]
Smooth Fano threefolds in positive characteristic satisfy Kodaira vanishing and lift to characteristic zero. This is joint work with Tatsuro Kawakami.
Smooth Fano threefolds in positive characteristic satisfy Kodaira vanishing and lift to characteristic zero. This is joint work with Tatsuro Kawakami.
2024年12月20日(金)
13:30-15:00 数理科学研究科棟(駒場) 118号室
榎園誠 氏 (東京大学)
Normal stable degenerations of Noether-Horikawa surfaces
榎園誠 氏 (東京大学)
Normal stable degenerations of Noether-Horikawa surfaces
[ 講演概要 ]
Noether-Horikawa surfaces are surfaces of general type satisfying the equation K2=2pg−4, which represents the boundary of the Noether inequality K2≥2pg−4 for surfaces of general type. In the 1970s, Horikawa conducted a detailed study of smooth Noether-Horikawa surfaces, providing a classification of these surfaces and describing their moduli spaces.
In this talk, I will present an explicit classification of normal stable degenerations of Noether-Horikawa surfaces. Specifically, I will discuss the following results:
(1) A preliminary classification of Noether-Horikawa surfaces with Q-Gorenstein smoothable log canonical singularities.
(2) Several criteria for determining the (global) Q-Gorenstein smoothability of the surfaces described in (1).
(3) Deformation results for Q-Gorenstein smoothable normal stable Noether-Horikawa surfaces, along with a description of the KSBA moduli spaces for these surfaces.
This is joint work with Hiroto Akaike, Masafumi Hattori and Yuki Koto.
Noether-Horikawa surfaces are surfaces of general type satisfying the equation K2=2pg−4, which represents the boundary of the Noether inequality K2≥2pg−4 for surfaces of general type. In the 1970s, Horikawa conducted a detailed study of smooth Noether-Horikawa surfaces, providing a classification of these surfaces and describing their moduli spaces.
In this talk, I will present an explicit classification of normal stable degenerations of Noether-Horikawa surfaces. Specifically, I will discuss the following results:
(1) A preliminary classification of Noether-Horikawa surfaces with Q-Gorenstein smoothable log canonical singularities.
(2) Several criteria for determining the (global) Q-Gorenstein smoothability of the surfaces described in (1).
(3) Deformation results for Q-Gorenstein smoothable normal stable Noether-Horikawa surfaces, along with a description of the KSBA moduli spaces for these surfaces.
This is joint work with Hiroto Akaike, Masafumi Hattori and Yuki Koto.
2024年12月12日(木)
13:30-15:00 数理科学研究科棟(駒場) 128号室
Chenyang Xu 氏 (Princeton University)
Irreducible symplectic varieties with a large second Betti number
Chenyang Xu 氏 (Princeton University)
Irreducible symplectic varieties with a large second Betti number
[ 講演概要 ]
(joint with Yuchen Liu, Zhiyu Liu) We show that the Lagrangian fibration constructed by Iiiev-Manivel using intermediate Jacobians of cubic fivefolds containing a fixed cubic fourfold, admits a compactification as a terminal Q-factorial irreducible symplectic varieties. As far as I know, besides OG10, this is the second family of irreducible symplectic varieties with the second Betti number at least 24.
(joint with Yuchen Liu, Zhiyu Liu) We show that the Lagrangian fibration constructed by Iiiev-Manivel using intermediate Jacobians of cubic fivefolds containing a fixed cubic fourfold, admits a compactification as a terminal Q-factorial irreducible symplectic varieties. As far as I know, besides OG10, this is the second family of irreducible symplectic varieties with the second Betti number at least 24.
2024年11月22日(金)
13:30-15:00 数理科学研究科棟(駒場) 118号室
入谷寛 氏 (京都大学)
Quantum cohomology of blowups
入谷寛 氏 (京都大学)
Quantum cohomology of blowups
[ 講演概要 ]
I will discuss a decomposition theorem for the quantum cohomology of a smooth projective variety blown up along a smooth subvariety. I will start with a general relationship between decomposition of quantum cohomology and extremal contractions, and then specialize to the case of blowups. Applications to birational geometry of this result have been announced by Katzarkov, Kontsevich, Pantev and Yu.
I will discuss a decomposition theorem for the quantum cohomology of a smooth projective variety blown up along a smooth subvariety. I will start with a general relationship between decomposition of quantum cohomology and extremal contractions, and then specialize to the case of blowups. Applications to birational geometry of this result have been announced by Katzarkov, Kontsevich, Pantev and Yu.
2024年11月15日(金)
13:30-15:00 数理科学研究科棟(駒場) 118号室
谷本祥 氏 (名古屋大学)
The spaces of rational curves on del Pezzo surfaces via conic bundles
谷本祥 氏 (名古屋大学)
The spaces of rational curves on del Pezzo surfaces via conic bundles
[ 講演概要 ]
There have been extensive activities on counting functions of rational points of bounded height on del Pezzo surfaces, and one of prominent approaches to this problem is by the usage of conic bundle structures on del Pezzo surfaces. This leads to upper and lower bounds of correct magnitude for quartic del Pezzo surfaces.
In this talk, I will explain how conic bundle structures on del Pezzo surfaces induce fibration structures on the spaces of rational curves on such surfaces. Then I will explain applications of this structure which include:
1. upper bounds of correct magnitude for the counting function of rational curves on quartic del Pezzo surfaces over finite fields.
2. rationality of the space of rational curves on a quartic del Pezzo surface.
Finally, I will explain our ongoing proof of homological stability for the spaces of rational curves on quartic del Pezzo surfaces. This is joint work in progress with Ronno Das, Brian Lehmann, and Philip Tosteson.
There have been extensive activities on counting functions of rational points of bounded height on del Pezzo surfaces, and one of prominent approaches to this problem is by the usage of conic bundle structures on del Pezzo surfaces. This leads to upper and lower bounds of correct magnitude for quartic del Pezzo surfaces.
In this talk, I will explain how conic bundle structures on del Pezzo surfaces induce fibration structures on the spaces of rational curves on such surfaces. Then I will explain applications of this structure which include:
1. upper bounds of correct magnitude for the counting function of rational curves on quartic del Pezzo surfaces over finite fields.
2. rationality of the space of rational curves on a quartic del Pezzo surface.
Finally, I will explain our ongoing proof of homological stability for the spaces of rational curves on quartic del Pezzo surfaces. This is joint work in progress with Ronno Das, Brian Lehmann, and Philip Tosteson.
2024年11月01日(金)
13:30-15:00 数理科学研究科棟(駒場) 118号室
Gerard van der Geer 氏 (University of Amsterdam)
The cycle class of the supersingular locus (English)
Gerard van der Geer 氏 (University of Amsterdam)
The cycle class of the supersingular locus (English)
[ 講演概要 ]
Deuring gave a now classical formula for the number of supersingular elliptic curves
in characteristic p. We generalize this to a formula for the cycle class of the
supersingular locus in the moduli space of principally polarized abelian varieties
of given dimension g in characteristic p. The formula determines the class up to
a multiple and shows that it lies in the tautological ring. We also give the multiple
for g up to 4. This is joint work with S. Harashita.
Deuring gave a now classical formula for the number of supersingular elliptic curves
in characteristic p. We generalize this to a formula for the cycle class of the
supersingular locus in the moduli space of principally polarized abelian varieties
of given dimension g in characteristic p. The formula determines the class up to
a multiple and shows that it lies in the tautological ring. We also give the multiple
for g up to 4. This is joint work with S. Harashita.
2024年10月18日(金)
13:30-15:00 数理科学研究科棟(駒場) 118号室
Jennifer Li 氏 (プリンストン大学)
Rational surfaces with a non-arithmetic automorphism group (英語)
Jennifer Li 氏 (プリンストン大学)
Rational surfaces with a non-arithmetic automorphism group (英語)
[ 講演概要 ]
In [Tot12], Totaro gave examples of a K3 surface such that its automorphism group is not commensurable with an arithmetic group, answering a question of Mazur. We give examples of rational surfaces with the same property. Our examples Y are log Calabi-Yau surfaces, i.e., there is a reduced normal crossing divisor D in Y such that KY+D=0. This is joint work with Sebastián Torres.
In [Tot12], Totaro gave examples of a K3 surface such that its automorphism group is not commensurable with an arithmetic group, answering a question of Mazur. We give examples of rational surfaces with the same property. Our examples Y are log Calabi-Yau surfaces, i.e., there is a reduced normal crossing divisor D in Y such that KY+D=0. This is joint work with Sebastián Torres.
2024年10月04日(金)
13:30-15:00 数理科学研究科棟(駒場) 118号室
高松哲平 氏 (京都大学)
Arithmetic finiteness of Mukai varieties of genus 7 (日本語)
高松哲平 氏 (京都大学)
Arithmetic finiteness of Mukai varieties of genus 7 (日本語)
[ 講演概要 ]
Fano threefolds over C with Picard number and index equal to 1 are known to be classified by their genus g (where 2≤g≤12 and g≠11). In particular, Mukai has shown that those with genus 7 can be described as hyperplane sections of a connected component of the 10-dimensional orthogonal Grassmannian.
In this talk, we discuss the arithmetic properties of these genus 7 threefolds and their higher-dimensional generalizations (called Mukai varieties of genus 7). More precisely, we consider the finiteness problem of varieties over a ring of S-integers (so called the Shafarevich conjecture), and the existence problem of varieties over the rational integer ring Z.
This talk is based on a joint work with Tetsushi Ito, Akihiro Kanemitsu, and Yuuji Tanaka.
Fano threefolds over C with Picard number and index equal to 1 are known to be classified by their genus g (where 2≤g≤12 and g≠11). In particular, Mukai has shown that those with genus 7 can be described as hyperplane sections of a connected component of the 10-dimensional orthogonal Grassmannian.
In this talk, we discuss the arithmetic properties of these genus 7 threefolds and their higher-dimensional generalizations (called Mukai varieties of genus 7). More precisely, we consider the finiteness problem of varieties over a ring of S-integers (so called the Shafarevich conjecture), and the existence problem of varieties over the rational integer ring Z.
This talk is based on a joint work with Tetsushi Ito, Akihiro Kanemitsu, and Yuuji Tanaka.
2024年07月04日(木)
13:00-14:30 数理科学研究科棟(駒場) ハイブリッド開催/118号室
Stefan Reppen 氏 (東京大学)
On a principle of Ogus: the Hasse invariant's order of vanishing and "Frobenius and the Hodge filtration'' (English)
Stefan Reppen 氏 (東京大学)
On a principle of Ogus: the Hasse invariant's order of vanishing and "Frobenius and the Hodge filtration'' (English)
[ 講演概要 ]
In joint work with W. Goldring we generalize a result of Ogus that, under certain technical conditions, the vanishing order of the Hasse invariant of a family $Y/X$ of $n$-dimensional Calabi-Yau varieties in characteristic $p$ at a point $x$ of $X$ equals the "conjugate line position" of $H^n_{\dR}(Y/X)$ at $x$, i.e. the largest $i$ such that the line of the conjugate filtration is contained in $\text{Fil}^i$ of the Hodge filtration. For every triple $(G,\mu,r)$ consisting of a connected, reductive $\mathbb{F}_p$-group $G$, a cocharacter $\mu \in X_*(G)$ and an $\mathbb{F}_p$-representation $r$ of $G$, we state a generalized Ogus Principle. If $\zeta:X \to \GZip^{\mu}$ is a smooth morphism, then the group theoretic Ogus Principle implies an Ogus Principle on $X$. We deduce an Ogus Principle for several Hodge and abelian-type Shimura varieties and the moduli space of K3 surfaces. In the talk I will present this work.
In joint work with W. Goldring we generalize a result of Ogus that, under certain technical conditions, the vanishing order of the Hasse invariant of a family $Y/X$ of $n$-dimensional Calabi-Yau varieties in characteristic $p$ at a point $x$ of $X$ equals the "conjugate line position" of $H^n_{\dR}(Y/X)$ at $x$, i.e. the largest $i$ such that the line of the conjugate filtration is contained in $\text{Fil}^i$ of the Hodge filtration. For every triple $(G,\mu,r)$ consisting of a connected, reductive $\mathbb{F}_p$-group $G$, a cocharacter $\mu \in X_*(G)$ and an $\mathbb{F}_p$-representation $r$ of $G$, we state a generalized Ogus Principle. If $\zeta:X \to \GZip^{\mu}$ is a smooth morphism, then the group theoretic Ogus Principle implies an Ogus Principle on $X$. We deduce an Ogus Principle for several Hodge and abelian-type Shimura varieties and the moduli space of K3 surfaces. In the talk I will present this work.
2024年06月28日(金)
13:30-15:00 数理科学研究科棟(駒場) ハイブリッド開催/117号室
吉野太郎 氏 (東京大学)
Stable rationality of hypersurfaces in mock toric varieties (日本語)
吉野太郎 氏 (東京大学)
Stable rationality of hypersurfaces in mock toric varieties (日本語)
[ 講演概要 ]
In recent years, there has been a development in approaching rationality problems through motivic methods. This approach requires the explicit construction of degeneration families over curves with favorable properties. However, the specific construction is generally difficult. Nicaise and Ottem combined combinatorial methods to construct degeneration families of hypersurfaces in toric varieties and mentioned the stable rationality of a very general hypersurface in projective spaces. In this talk, we mention the following two points: First, I introduce the notion of mock toric varieties, which are generalizations of toric varieties. Second, I combinatorially construct degeneration families of hypersurfaces in mock toric varieties, and I mention the irrationality of a very general hypersurface in the complex Grassmannian variety Gr(2, n).
In recent years, there has been a development in approaching rationality problems through motivic methods. This approach requires the explicit construction of degeneration families over curves with favorable properties. However, the specific construction is generally difficult. Nicaise and Ottem combined combinatorial methods to construct degeneration families of hypersurfaces in toric varieties and mentioned the stable rationality of a very general hypersurface in projective spaces. In this talk, we mention the following two points: First, I introduce the notion of mock toric varieties, which are generalizations of toric varieties. Second, I combinatorially construct degeneration families of hypersurfaces in mock toric varieties, and I mention the irrationality of a very general hypersurface in the complex Grassmannian variety Gr(2, n).
2024年06月21日(金)
13:30-15:00 数理科学研究科棟(駒場) ハイブリッド開催/056号室
Kien Nguyen Huu 氏 (Normandie Université/KU Leuven)
ON THE POWER SERIES OF DENEF AND LOESER'S MOTIVIC VANISHING CYCLES OF JET POLYNOMIALS (English)
Kien Nguyen Huu 氏 (Normandie Université/KU Leuven)
ON THE POWER SERIES OF DENEF AND LOESER'S MOTIVIC VANISHING CYCLES OF JET POLYNOMIALS (English)
[ 講演概要 ]
Let f be a non-constant polynomial in n variables over a field k of characteristic
0. Denef and Loeser introduced the notion of motivic vanishing cycles of f as an element in
the localization Mμˆ of the Grothendieck ring Kμˆ(Var ) of k-varieties with a good action of k0k
μˆ := lim μm by inverting the affne line equipped with the trivial action of μˆ, where μm
is the group scheme over k of mth roots of unity. In particular, if k is the field of complex
numbers then Denef and Loeser showed that their motivic vanishing cycles and the complex
φf [n − 1] has the same Hodge characteristic, where φf is the complex of vanishing cycles
in the usual sense. Motivated by the Igusa conjecture for exponential sums and the strong
monodromy conjecture, we introduce the notion of Poincaré series of Denef-Loeser's van-
ishing cycles of jet polynomials of f, where jet polynomials of f are polynomials appearing
naturally when we compute the jet schemes of f. By using Davison-Meinhardt's conjecture
which was proved by Nicaise and Payne in 2019, we can show that our Poincaré series is a
rational function over a quotient ring of Mμˆ by very natural relations. In particular, we can k
recovery Denef and Loeser's motivic vanishing cycles from our Poincaré series. Moreover, we can show that our Poincaré series owns a universal property in the sense that if k is a number field then the Igusa local zeta functions, the motivic Igusa zeta functions, the Poincaré series of exponential sums modulo pm of f can be obtained from our Poincaré se- ries by suitable specialization maps preserving the rationality. If time permits, I will present some initial consequences that have arisen during the study of our Poincaré series.
Let f be a non-constant polynomial in n variables over a field k of characteristic
0. Denef and Loeser introduced the notion of motivic vanishing cycles of f as an element in
the localization Mμˆ of the Grothendieck ring Kμˆ(Var ) of k-varieties with a good action of k0k
μˆ := lim μm by inverting the affne line equipped with the trivial action of μˆ, where μm
is the group scheme over k of mth roots of unity. In particular, if k is the field of complex
numbers then Denef and Loeser showed that their motivic vanishing cycles and the complex
φf [n − 1] has the same Hodge characteristic, where φf is the complex of vanishing cycles
in the usual sense. Motivated by the Igusa conjecture for exponential sums and the strong
monodromy conjecture, we introduce the notion of Poincaré series of Denef-Loeser's van-
ishing cycles of jet polynomials of f, where jet polynomials of f are polynomials appearing
naturally when we compute the jet schemes of f. By using Davison-Meinhardt's conjecture
which was proved by Nicaise and Payne in 2019, we can show that our Poincaré series is a
rational function over a quotient ring of Mμˆ by very natural relations. In particular, we can k
recovery Denef and Loeser's motivic vanishing cycles from our Poincaré series. Moreover, we can show that our Poincaré series owns a universal property in the sense that if k is a number field then the Igusa local zeta functions, the motivic Igusa zeta functions, the Poincaré series of exponential sums modulo pm of f can be obtained from our Poincaré se- ries by suitable specialization maps preserving the rationality. If time permits, I will present some initial consequences that have arisen during the study of our Poincaré series.
2024年06月07日(金)
13:30-15:00 数理科学研究科棟(駒場) ハイブリッド開催/056号室
Ivan Cheltsov 氏 (University of Edinburgh)
K-stability of pointless Fano 3-folds (English)
Ivan Cheltsov 氏 (University of Edinburgh)
K-stability of pointless Fano 3-folds (English)
[ 講演概要 ]
In this talk we will show how to prove that all pointless smooth Fano 3-folds defined over a subfield of the field of complex numbers are Kahler-Einstein unless they belong to 8 exceptional deformation families. This is a joint work in progress with Hamid Abban (Nottingham) and Frederic Mangolte (Marseille).
In this talk we will show how to prove that all pointless smooth Fano 3-folds defined over a subfield of the field of complex numbers are Kahler-Einstein unless they belong to 8 exceptional deformation families. This is a joint work in progress with Hamid Abban (Nottingham) and Frederic Mangolte (Marseille).
2024年05月24日(金)
13:30-15:00 数理科学研究科棟(駒場) ハイブリッド開催/056号室
佐藤謙太 氏 (九州大学)
Boundedness of weak Fano threefolds with fixed Gorenstein index in positive characteristic
佐藤謙太 氏 (九州大学)
Boundedness of weak Fano threefolds with fixed Gorenstein index in positive characteristic
[ 講演概要 ]
In this talk, we give a partial affirmative answer to the BAB conjecture for 3-folds in characteristic p>5. Specifically, we prove that a set of weak Fano 3-folds over an uncountable algebraically closed field is bounded, if each element X satisfies certain conditions regarding the Gorenstein index, a complement and Kodaira type vanishing. In the course of the proof, we also study a uniform lower bound for Seshadri constants of nef and big invertible sheaves on projective 3-folds.
In this talk, we give a partial affirmative answer to the BAB conjecture for 3-folds in characteristic p>5. Specifically, we prove that a set of weak Fano 3-folds over an uncountable algebraically closed field is bounded, if each element X satisfies certain conditions regarding the Gorenstein index, a complement and Kodaira type vanishing. In the course of the proof, we also study a uniform lower bound for Seshadri constants of nef and big invertible sheaves on projective 3-folds.
2024年05月17日(金)
13:30-15:00 数理科学研究科棟(駒場) ハイブリッド開催/056号室
松本 雄也 氏 (東京理科大学)
非分離Kummer曲面 (日本語)
松本 雄也 氏 (東京理科大学)
非分離Kummer曲面 (日本語)
[ 講演概要 ]
Kummer曲面Km(A)とは，+-1倍写像によるアーベル曲面Aの商の最小特異点解消として得られる曲面である．Aが標数≠2の場合（resp. 標数2で，超特異ではない場合）は，Km(A)はK3曲面であり，例外曲線は互いに交わらない（resp. 所定の交わり方をする）16本の有理曲線である．Aが標数2で超特異の場合はKm(A)はK3曲面にならない．また，Km(A)が標数2の超特異K3曲面になることはない．
本講演では，標数2の超特異K3曲面とその上の16本の有理曲線で所定の交わり方をするものに対し，非分離2重被覆Aを構成することができること，Aは非特異部分に群構造が入り「アーベル曲面もどき」になることを示す．Aの分類のために，RDP K3曲面のRDPの補集合から最小特異点解消への B_n \Omega^1（Cartier作用素を何回か適用すると消える1次微分形式の層）の延長に関する結果を用いるので，これにも言及したい．
プレプリントは https://arxiv.org/abs/2403.02770 でご覧いただけます．
Kummer曲面Km(A)とは，+-1倍写像によるアーベル曲面Aの商の最小特異点解消として得られる曲面である．Aが標数≠2の場合（resp. 標数2で，超特異ではない場合）は，Km(A)はK3曲面であり，例外曲線は互いに交わらない（resp. 所定の交わり方をする）16本の有理曲線である．Aが標数2で超特異の場合はKm(A)はK3曲面にならない．また，Km(A)が標数2の超特異K3曲面になることはない．
本講演では，標数2の超特異K3曲面とその上の16本の有理曲線で所定の交わり方をするものに対し，非分離2重被覆Aを構成することができること，Aは非特異部分に群構造が入り「アーベル曲面もどき」になることを示す．Aの分類のために，RDP K3曲面のRDPの補集合から最小特異点解消への B_n \Omega^1（Cartier作用素を何回か適用すると消える1次微分形式の層）の延長に関する結果を用いるので，これにも言及したい．
プレプリントは https://arxiv.org/abs/2403.02770 でご覧いただけます．
2024年04月26日(金)
14:00-15:30 数理科学研究科棟(駒場) 056号室
河上 龍郎 氏 (京都大学)
Frobenius stable Grauert-Riemenschneider vanishing fails (日本語)
河上 龍郎 氏 (京都大学)
Frobenius stable Grauert-Riemenschneider vanishing fails (日本語)
[ 講演概要 ]
We show that the Frobenius stable version of Grauert-Riemenschneider vanishing fails for a terminal 3-fold in characteristic 2. To prove this, we introduce the notion of $F_p$-rationality for singularities in positive characteristic, and prove that 3-dimensional klt singularities are $\mathbb F_p$-rational. I will also talk about the vanishing of $F_p$-cohomologies of log Fano threefolds. This is joint work with Jefferson Baudin and Fabio Bernasconi.
We show that the Frobenius stable version of Grauert-Riemenschneider vanishing fails for a terminal 3-fold in characteristic 2. To prove this, we introduce the notion of $F_p$-rationality for singularities in positive characteristic, and prove that 3-dimensional klt singularities are $\mathbb F_p$-rational. I will also talk about the vanishing of $F_p$-cohomologies of log Fano threefolds. This is joint work with Jefferson Baudin and Fabio Bernasconi.
2023年12月15日(金)
13:30-15:00 数理科学研究科棟(駒場) 118号室
石井 志保子 氏 (東京大学)
On a pair of a smooth variety and a multi-ideal with a real exponent in positive characteristic (日本語)
石井 志保子 氏 (東京大学)
On a pair of a smooth variety and a multi-ideal with a real exponent in positive characteristic (日本語)
[ 講演概要 ]
In birational geometry, the behaviors of the invariants, mld (minimal log discrepancy) and lct (log canonical threshold), play important roles. These invariants are studied well in case the base field is characteristic zero, but not so in positive characteristic case. In this talk, I work on a pair consisting of smooth variety and a multi-ideal with a real exponent over an algebraically closed field of positive characteristic. We reduce some behaviors of the invariants for such pairs in positive characteristic case into characteristic zero.
In birational geometry, the behaviors of the invariants, mld (minimal log discrepancy) and lct (log canonical threshold), play important roles. These invariants are studied well in case the base field is characteristic zero, but not so in positive characteristic case. In this talk, I work on a pair consisting of smooth variety and a multi-ideal with a real exponent over an algebraically closed field of positive characteristic. We reduce some behaviors of the invariants for such pairs in positive characteristic case into characteristic zero.
2023年11月24日(金)
14:00-15:30 数理科学研究科棟(駒場) ハイブリッド開催/056号室
Haidong Liu 氏 (Sun Yat-sen University)
On Kawamata-Miyaoka type inequality
Haidong Liu 氏 (Sun Yat-sen University)
On Kawamata-Miyaoka type inequality
[ 講演概要 ]
For klt projective varieties with nef and big canonical divisors, there exists a Miyaoka-Yau type inequality concerning the first and the second Chern classes. In this talk, I will present a Kawamata-Miyaoka type inequality for terminal Q-Fano varieties, which is a mirror version of the Miyaoka-Yau type inequality. This is a joint work with Jie Liu.
For klt projective varieties with nef and big canonical divisors, there exists a Miyaoka-Yau type inequality concerning the first and the second Chern classes. In this talk, I will present a Kawamata-Miyaoka type inequality for terminal Q-Fano varieties, which is a mirror version of the Miyaoka-Yau type inequality. This is a joint work with Jie Liu.
2023年10月16日(月)
14:00-15:30 数理科学研究科棟(駒場) 002号室
Lena Ji 氏 (University of Michigan)
Symmetries of Fano varieties
Lena Ji 氏 (University of Michigan)
Symmetries of Fano varieties
[ 講演概要 ]
Prokhorov and Shramov proved that the BAB conjecture (which Birkar later proved) implies the uniform Jordan property for automorphism groups of complex Fano varieties of fixed dimension. This property in particular gives an upper bound on the size of semi-simple groups (meaning those with no non-trivial normal abelian subgroups) acting faithfully on n-dimensional complex Fano varieties, and this bound only depends on n. In this talk, we investigate the consequences of a large action by a particular semi-simple group: the symmetric group. This work is joint with Louis Esser and Joaquín Moraga.
Prokhorov and Shramov proved that the BAB conjecture (which Birkar later proved) implies the uniform Jordan property for automorphism groups of complex Fano varieties of fixed dimension. This property in particular gives an upper bound on the size of semi-simple groups (meaning those with no non-trivial normal abelian subgroups) acting faithfully on n-dimensional complex Fano varieties, and this bound only depends on n. In this talk, we investigate the consequences of a large action by a particular semi-simple group: the symmetric group. This work is joint with Louis Esser and Joaquín Moraga.
2023年07月28日(金)
13:30-15:00 数理科学研究科棟(駒場) ハイブリッド開催/002号室
(7/11更新) 中止となりました。来学期への延期を検討しています。
石井 志保子 氏 (東京大学)
TBA
(7/11更新) 中止となりました。来学期への延期を検討しています。
石井 志保子 氏 (東京大学)
TBA
2023年07月21日(金)
13:30-15:00 数理科学研究科棟(駒場) ハイブリッド開催/056号室
普段と部屋が異なります。
江尻 祥 氏 (大阪公立大学)
The Demailly--Peternell--Schneider conjecture is true in positive characteristic
普段と部屋が異なります。
江尻 祥 氏 (大阪公立大学)
The Demailly--Peternell--Schneider conjecture is true in positive characteristic
[ 講演概要 ]
In 1993, Demailly, Peternell and Schneider conjectured that the Albanese morphism of a compact K\"{a}hler manifold with nef anti-canonical divisor is surjective. For smooth projective varieties of characteristic zero, the conjecture was verified by Zhang in 1996. In positive characteristic, the conjecture was solved under the assumption that the geometric generic fiber F of the Albanese morphism has only mild singularities. However, F may have bad singularities even if we restrict ourselves to the case when the anti-canonical divisor is nef. In this talk, we prove the conjecture in positive characteristic without any extra assumption. We also discuss properties of the Albanese morphism, such as flatness or local isotriviality. This talk is based on joint work with Zsolt Patakfalvi.
In 1993, Demailly, Peternell and Schneider conjectured that the Albanese morphism of a compact K\"{a}hler manifold with nef anti-canonical divisor is surjective. For smooth projective varieties of characteristic zero, the conjecture was verified by Zhang in 1996. In positive characteristic, the conjecture was solved under the assumption that the geometric generic fiber F of the Albanese morphism has only mild singularities. However, F may have bad singularities even if we restrict ourselves to the case when the anti-canonical divisor is nef. In this talk, we prove the conjecture in positive characteristic without any extra assumption. We also discuss properties of the Albanese morphism, such as flatness or local isotriviality. This talk is based on joint work with Zsolt Patakfalvi.
2023年06月28日(水)
13:30-15:00 数理科学研究科棟(駒場) ハイブリッド開催/056号室
(6/27更新) 講演者の都合で中止となりました。
松澤 陽介 氏 (大阪公立大学)
Preimages question and dynamical cancellation
(6/27更新) 講演者の都合で中止となりました。
松澤 陽介 氏 (大阪公立大学)
Preimages question and dynamical cancellation
[ 講演概要 ]
Pulling back an invariant subvariety by a self-morphism on projective variety, you will get a tower of increasing closed subsets. Working over a number field, we expect that the set of rational points contained in this increasing subsets eventually stabilizes. I am planning to discuss several results on this problem, such as the case of etale morphisms, morphisms on the product of two P^1. I will also present some counter examples that occur when we drop some of the assumptions. This work is based on a joint work with Matt Satriano and Jason Bell, and recent work in progress with Kaoru Sano.
Pulling back an invariant subvariety by a self-morphism on projective variety, you will get a tower of increasing closed subsets. Working over a number field, we expect that the set of rational points contained in this increasing subsets eventually stabilizes. I am planning to discuss several results on this problem, such as the case of etale morphisms, morphisms on the product of two P^1. I will also present some counter examples that occur when we drop some of the assumptions. This work is based on a joint work with Matt Satriano and Jason Bell, and recent work in progress with Kaoru Sano.
2023年06月23日(金)
13:30-15:00 数理科学研究科棟(駒場) ハイブリッド開催/117号室
柴田 康介 氏 (東京電機大学)
Minimal log discrepnacies for quotient singularities
柴田 康介 氏 (東京電機大学)
Minimal log discrepnacies for quotient singularities
[ 講演概要 ]
In this talk, I will discuss recent joint work with Yusuke Nakamura on minimal log discrepancies for quotient singularities. The minimal log discrepancy is an important invariant of singularities in birational geometry. The denominator of the minimal log discrepancy of a variety depends on the Gorenstein index. On the other hand, Shokurov conjectured that the Gorenstein index of a Q-Gorenstein germ can be bounded in terms of its dimension and minimal log discrepancy. In this talk, I will explain basic properties for quotient singularities and show Shokurov's index conjecture for quotient singularities.
In this talk, I will discuss recent joint work with Yusuke Nakamura on minimal log discrepancies for quotient singularities. The minimal log discrepancy is an important invariant of singularities in birational geometry. The denominator of the minimal log discrepancy of a variety depends on the Gorenstein index. On the other hand, Shokurov conjectured that the Gorenstein index of a Q-Gorenstein germ can be bounded in terms of its dimension and minimal log discrepancy. In this talk, I will explain basic properties for quotient singularities and show Shokurov's index conjecture for quotient singularities.
2023年06月14日(水)
14:00-15:30 数理科学研究科棟(駒場) ハイブリッド開催/056号室
普段と曜日・時間・場所が異なります。
Wenliang Zhang 氏 (University of Illinois Chicago)
Vanishing of local cohomology modules
普段と曜日・時間・場所が異なります。
Wenliang Zhang 氏 (University of Illinois Chicago)
Vanishing of local cohomology modules
[ 講演概要 ]
Studying the vanishing of local cohomology modules has a long and rich history, and is still an active research area. In this talk, we will discuss classic theorems (due to Grothendieck, Hartshorne, Peskine-Szpiro, and Ogus), recent developments, and some open problems.
Studying the vanishing of local cohomology modules has a long and rich history, and is still an active research area. In this talk, we will discuss classic theorems (due to Grothendieck, Hartshorne, Peskine-Szpiro, and Ogus), recent developments, and some open problems.
2023年06月07日(水)
13:30-15:00 数理科学研究科棟(駒場) ハイブリッド開催/056号室
普段と曜日と部屋が異なります.
呼子 笛太郎 氏 (名古屋大学)
Quasi-F-splitting and Hodge-Witt
普段と曜日と部屋が異なります.
呼子 笛太郎 氏 (名古屋大学)
Quasi-F-splitting and Hodge-Witt
[ 講演概要 ]
Quasi-F-splitting is an extension of F-splitting, which is defined for schemes in positive characteristic. On the other hand, Hodge-Wittness is defined for smooth proper schemes over a perfect field using the de Rham-Witt complex and ordinarity implies Hodge-Wittness. In this talk, I will explain (unexpected) relations between F-split/quasi-F-split and ordinary/Hodge-Witt via examples and properties.
Quasi-F-splitting is an extension of F-splitting, which is defined for schemes in positive characteristic. On the other hand, Hodge-Wittness is defined for smooth proper schemes over a perfect field using the de Rham-Witt complex and ordinarity implies Hodge-Wittness. In this talk, I will explain (unexpected) relations between F-split/quasi-F-split and ordinary/Hodge-Witt via examples and properties.
2023年05月26日(金)
13:30-15:00 数理科学研究科棟(駒場) ハイブリッド開催/117号室
吉川 翔 氏 (東京工業大学, 理研)
Varieties in positive characteristic with numerically flat tangent bundle
吉川 翔 氏 (東京工業大学, 理研)
Varieties in positive characteristic with numerically flat tangent bundle
[ 講演概要 ]
The positivity condition imposed on the tangent bundle of a smooth projective variety is known to restrict the geometric structure of the variety. Demailly, Peternell and Schneider established a decomposition theorem for a smooth projective complex variety with nef tangent bundle. The theorem states that, up to an etale cover, such a variety has a smooth fibration admitting a smooth algebraic fiber space over an abelian variety whose fibers are Fano varieties, so one can say that such a variety decomposes into the "positive” part and the "flat” part. A positive characteristic analog of the above decomposition theorem was proved by Kanemitsu and Watanabe. The "flat” part of their theorem is a smooth projective variety with numerically flat tangent bundle. In this talk, I will introduce the result that every ordinary variety with numerically flat tangent bundle is an etale quotient of an ordinary Abelian variety. In particular, we obtain the decomposition theorem for Frobenius splitting varieties with nef tangent bundle. This talk is based on joint work with Sho Ejiri.
The positivity condition imposed on the tangent bundle of a smooth projective variety is known to restrict the geometric structure of the variety. Demailly, Peternell and Schneider established a decomposition theorem for a smooth projective complex variety with nef tangent bundle. The theorem states that, up to an etale cover, such a variety has a smooth fibration admitting a smooth algebraic fiber space over an abelian variety whose fibers are Fano varieties, so one can say that such a variety decomposes into the "positive” part and the "flat” part. A positive characteristic analog of the above decomposition theorem was proved by Kanemitsu and Watanabe. The "flat” part of their theorem is a smooth projective variety with numerically flat tangent bundle. In this talk, I will introduce the result that every ordinary variety with numerically flat tangent bundle is an etale quotient of an ordinary Abelian variety. In particular, we obtain the decomposition theorem for Frobenius splitting varieties with nef tangent bundle. This talk is based on joint work with Sho Ejiri.
