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Algebraic Geometry Seminar
Seminar information archive ～03/18｜Next seminar｜Future seminars 03/19～
| Date, time & place | Friday 13:30 - 15:00 ハイブリッド開催/117Room #ハイブリッド開催/117 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | GONGYO Yoshinori, NAKAMURA Yusuke, TANAKA Hiromu |
Seminar information archive
2023/05/10
13:30-15:00 Room #ハイブリッド開催/056 (Graduate School of Math. Sci. Bldg.)
Jakub Witaszek (Princeton University)
Singularities in mixed characteristic via the Riemann-Hilbert correspondence (English)
Jakub Witaszek (Princeton University)
Singularities in mixed characteristic via the Riemann-Hilbert correspondence (English)
[ Abstract ]
In my talk, I will start by reviewing how various properties of characteristic zero singularities can be understood topologically by ways of the Riemann-Hilbert correspondence. After that, I will explain how similar ideas can be applied in the study of mixed characteristic singularities. This is based on a joint work (in progress) with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker, and Joe Waldron.
In my talk, I will start by reviewing how various properties of characteristic zero singularities can be understood topologically by ways of the Riemann-Hilbert correspondence. After that, I will explain how similar ideas can be applied in the study of mixed characteristic singularities. This is based on a joint work (in progress) with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker, and Joe Waldron.
2023/04/28
13:30-15:00 Room #ハイブリッド開催/117 (Graduate School of Math. Sci. Bldg.)
Taro Yoshino (Tokyo)
On the degree of irrationality of complete intersections (Japanese )
Taro Yoshino (Tokyo)
On the degree of irrationality of complete intersections (Japanese )
[ Abstract ]
The degree of irrationality of a variety X is the minimum degree of a dominant, generically finite rational map from X to a rational variety. This invariant gives a measure of how far X is from being rational. There were some varieties whose degree of irrationality was computed. For example, in 2017, Bastianelli, De Poi, Ein, Lazarsfeld, and Ullery computed the degree of irrationality of very general hypersurfaces of general type by using the positivity of the canonical line bundle. On the other hand, in 2020, Chen and Stapleton obtained the lower bound of the degree of irrationality of very general Fano hypersurfaces by using the reduction of modulo p.
In this talk, we will show that we can obtain the lower bound of the degree of irrationality of very general Fano complete intersections. For obtaining the bound, we make a minor adjustment to Chen--Stapleton's method using the trace map of differential modules.
This talk is based on joint work with Lucas Braune.
The degree of irrationality of a variety X is the minimum degree of a dominant, generically finite rational map from X to a rational variety. This invariant gives a measure of how far X is from being rational. There were some varieties whose degree of irrationality was computed. For example, in 2017, Bastianelli, De Poi, Ein, Lazarsfeld, and Ullery computed the degree of irrationality of very general hypersurfaces of general type by using the positivity of the canonical line bundle. On the other hand, in 2020, Chen and Stapleton obtained the lower bound of the degree of irrationality of very general Fano hypersurfaces by using the reduction of modulo p.
In this talk, we will show that we can obtain the lower bound of the degree of irrationality of very general Fano complete intersections. For obtaining the bound, we make a minor adjustment to Chen--Stapleton's method using the trace map of differential modules.
This talk is based on joint work with Lucas Braune.
2023/04/21
14:00-15:30 Room #117 (Graduate School of Math. Sci. Bldg.)
Tatsuro Kawakami (Kyoto University)
Endomorphisms of varieties and Bott vanishing (Japanese)
Tatsuro Kawakami (Kyoto University)
Endomorphisms of varieties and Bott vanishing (Japanese)
[ Abstract ]
In this talk, we show that a projective variety with an int-amplified endomorphism of degree invertible in the base field satisfies Bott vanishing. This is a new way to analyze which varieties have nontrivial endomorphisms. In particular, we extend some classification results on varieties admitting endomorphisms (for Fano threefolds of Picard number one and several other cases) to any characteristic. This talk is based on joint work with Burt Totaro.
In this talk, we show that a projective variety with an int-amplified endomorphism of degree invertible in the base field satisfies Bott vanishing. This is a new way to analyze which varieties have nontrivial endomorphisms. In particular, we extend some classification results on varieties admitting endomorphisms (for Fano threefolds of Picard number one and several other cases) to any characteristic. This talk is based on joint work with Burt Totaro.
2023/04/21
12:45-13:45 Room #ハイブリッド開催/117 (Graduate School of Math. Sci. Bldg.)
Sung Rak Choi (Yonsei University )
ACC of plc thresholds (English)
Sung Rak Choi (Yonsei University )
ACC of plc thresholds (English)
[ Abstract ]
The notion of potential pairs was developed as a means to bound the singularities while running the anti-MMP. They behave similarly with the usual klt, lc pairs.
We introduce potential log canonical threshold and prove that the set of these thresholds also satisfies the ascending chain condition (ACC). We also study the relation with the complements. This is a joint work with Sungwook Jang.
The notion of potential pairs was developed as a means to bound the singularities while running the anti-MMP. They behave similarly with the usual klt, lc pairs.
We introduce potential log canonical threshold and prove that the set of these thresholds also satisfies the ascending chain condition (ACC). We also study the relation with the complements. This is a joint work with Sungwook Jang.
2023/03/28
10:00-11:30 Room #ハイブリッド開催/056 (Graduate School of Math. Sci. Bldg.)
Paolo Cascini (Imperial College London)
On the canonical bundle formula in positive characteristic (English)
Paolo Cascini (Imperial College London)
On the canonical bundle formula in positive characteristic (English)
[ Abstract ]
In a previous work in collaboration with F. Ambro, V. Shokurov and C. Spicer, we show that algebraically integrable foliations can be used to study the canonical bundle formula for fibrations which are not necessarily lc trivial.
I will discuss a work in progress by M. Benozzo on a generalisation of these results in positive characteristic.
In a previous work in collaboration with F. Ambro, V. Shokurov and C. Spicer, we show that algebraically integrable foliations can be used to study the canonical bundle formula for fibrations which are not necessarily lc trivial.
I will discuss a work in progress by M. Benozzo on a generalisation of these results in positive characteristic.
2023/03/10
13:15-14:45 Room #ハイブリッド開催/123 (Graduate School of Math. Sci. Bldg.)
Paolo Cascini (Imperical College London)
On existence of flips for algebraically integrable foliations. (English)
Paolo Cascini (Imperical College London)
On existence of flips for algebraically integrable foliations. (English)
[ Abstract ]
Assuming termination of (classical) flips in dimension r, we show that flips exist for any algebraically integrable foliation of rank r with log canonical singularities. Joint work with C. Spicer.
Assuming termination of (classical) flips in dimension r, we show that flips exist for any algebraically integrable foliation of rank r with log canonical singularities. Joint work with C. Spicer.
2023/02/20
10:00-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)
The 4th lecture of series talks
Chenyang Xu (Princeton University)
K-stability of Fano varieties. (English)
The 4th lecture of series talks
Chenyang Xu (Princeton University)
K-stability of Fano varieties. (English)
[ Abstract ]
The notion of K-stability of Fano varieties was first introduced to characterize the existence of Kahler-Einstein metric. Recently, a purely algebro-geometric theory has been developed and it has yielded many striking results, such as the solution of the Yau-Tian-Donaldson Conjecture for all Fano varieties, as well as the construction of a projective moduli scheme, called K-moduli, parametrizing K-polystable Fano varieties.
In this lecture series, I will survey the recent progress. The first two lectures will be devoted to explain the evolution of algebraic geometer’s understanding of various aspects of the notion of K-stability. The Lecture 3 and 4 will be devoted to discuss the construction of the K-moduli space.
The notion of K-stability of Fano varieties was first introduced to characterize the existence of Kahler-Einstein metric. Recently, a purely algebro-geometric theory has been developed and it has yielded many striking results, such as the solution of the Yau-Tian-Donaldson Conjecture for all Fano varieties, as well as the construction of a projective moduli scheme, called K-moduli, parametrizing K-polystable Fano varieties.
In this lecture series, I will survey the recent progress. The first two lectures will be devoted to explain the evolution of algebraic geometer’s understanding of various aspects of the notion of K-stability. The Lecture 3 and 4 will be devoted to discuss the construction of the K-moduli space.
2023/02/17
10:00-11:30 Room #123 (Graduate School of Math. Sci. Bldg.)
The 3rd lecture of series talks
Chenyang Xu (Princeton University)
K-stability of Fano varieties. ( English)
The 3rd lecture of series talks
Chenyang Xu (Princeton University)
K-stability of Fano varieties. ( English)
[ Abstract ]
The notion of K-stability of Fano varieties was first introduced to characterize the existence of Kahler-Einstein metric. Recently, a purely algebro-geometric theory has been developed and it has yielded many striking results, such as the solution of the Yau-Tian-Donaldson Conjecture for all Fano varieties, as well as the construction of a projective moduli scheme, called K-moduli, parametrizing K-polystable Fano varieties.
In this lecture series, I will survey the recent progress. The first two lectures will be devoted to explain the evolution of algebraic geometer’s understanding of various aspects of the notion of K-stability. The Lecture 3 and 4 will be devoted to discuss the construction of the K-moduli space.
The notion of K-stability of Fano varieties was first introduced to characterize the existence of Kahler-Einstein metric. Recently, a purely algebro-geometric theory has been developed and it has yielded many striking results, such as the solution of the Yau-Tian-Donaldson Conjecture for all Fano varieties, as well as the construction of a projective moduli scheme, called K-moduli, parametrizing K-polystable Fano varieties.
In this lecture series, I will survey the recent progress. The first two lectures will be devoted to explain the evolution of algebraic geometer’s understanding of various aspects of the notion of K-stability. The Lecture 3 and 4 will be devoted to discuss the construction of the K-moduli space.
2023/02/06
13:00-14:30 Room #123 (Graduate School of Math. Sci. Bldg.)
The 2nd lecture of series talks.
Chenyang Xu (Princeton University)
K-stability of Fano varieties. (English)
The 2nd lecture of series talks.
Chenyang Xu (Princeton University)
K-stability of Fano varieties. (English)
[ Abstract ]
The notion of K-stability of Fano varieties was first introduced to characterize the existence of Kahler-Einstein metric. Recently, a purely algebro-geometric theory has been developed and it has yielded many striking results, such as the solution of the Yau-Tian-Donaldson Conjecture for all Fano varieties, as well as the construction of a projective moduli scheme, called K-moduli, parametrizing K-polystable Fano varieties.
In this lecture series, I will survey the recent progress. The first two lectures will be devoted to explain the evolution of algebraic geometer’s understanding of various aspects of the notion of K-stability. The Lecture 3 and 4 will be devoted to discuss the construction of the K-moduli space.
The notion of K-stability of Fano varieties was first introduced to characterize the existence of Kahler-Einstein metric. Recently, a purely algebro-geometric theory has been developed and it has yielded many striking results, such as the solution of the Yau-Tian-Donaldson Conjecture for all Fano varieties, as well as the construction of a projective moduli scheme, called K-moduli, parametrizing K-polystable Fano varieties.
In this lecture series, I will survey the recent progress. The first two lectures will be devoted to explain the evolution of algebraic geometer’s understanding of various aspects of the notion of K-stability. The Lecture 3 and 4 will be devoted to discuss the construction of the K-moduli space.
2023/01/31
14:30-16:00 Room #ハイブリッド開催/002 (Graduate School of Math. Sci. Bldg.)
Shiji Lyu (Princeton University)
Some properties of splinters via ultrapower (English)
Shiji Lyu (Princeton University)
Some properties of splinters via ultrapower (English)
[ Abstract ]
A Noetherian (reduced) ring is called a splinter if it is a direct summand of every finite ring extension of it. This notion is related to various interesting notions of singularities, but far less properties are known about splinters.
In this talk, we will discuss the question of "regular ascent"; in the simplest (but already essential) form, we ask, for a Noetherian splinter R, is the polynomial ring R[X] always a splinter. We will see how ultrapower, a construction mainly belonging to model theory, is involved.
A Noetherian (reduced) ring is called a splinter if it is a direct summand of every finite ring extension of it. This notion is related to various interesting notions of singularities, but far less properties are known about splinters.
In this talk, we will discuss the question of "regular ascent"; in the simplest (but already essential) form, we ask, for a Noetherian splinter R, is the polynomial ring R[X] always a splinter. We will see how ultrapower, a construction mainly belonging to model theory, is involved.
2023/01/27
13:00-14:30 Room #056 (Graduate School of Math. Sci. Bldg.)
4 lectures; 1/27: 13:00―14:30 Room056, 2/6: 13:00―14:30, Room 123, 2/17: 10:00―11:30,Room 123室 2/20 10:00ー11:30, Room:056室
Chenyang Xu (Princeton University)
K-stability of Fano varieties (English)
4 lectures; 1/27: 13:00―14:30 Room056, 2/6: 13:00―14:30, Room 123, 2/17: 10:00―11:30,Room 123室 2/20 10:00ー11:30, Room:056室
Chenyang Xu (Princeton University)
K-stability of Fano varieties (English)
[ Abstract ]
The notion of K-stability of Fano varieties was first introduced to characterize the existence of Kahler-Einstein metric. Recently, a purely algebro-geometric theory has been developed and it has yielded many striking results, such as the solution of the Yau-Tian-Donaldson Conjecture for all Fano varieties, as well as the construction of a projective moduli scheme, called K-moduli, parametrizing K-polystable Fano varieties.
In this lecture series, I will survey the recent progress. The first two lectures will be devoted to explain the evolution of algebraic geometer’s understanding of various aspects of the notion of K-stability. The Lecture 3 and 4 will be devoted to discuss the construction of the K-moduli space.
The notion of K-stability of Fano varieties was first introduced to characterize the existence of Kahler-Einstein metric. Recently, a purely algebro-geometric theory has been developed and it has yielded many striking results, such as the solution of the Yau-Tian-Donaldson Conjecture for all Fano varieties, as well as the construction of a projective moduli scheme, called K-moduli, parametrizing K-polystable Fano varieties.
In this lecture series, I will survey the recent progress. The first two lectures will be devoted to explain the evolution of algebraic geometer’s understanding of various aspects of the notion of K-stability. The Lecture 3 and 4 will be devoted to discuss the construction of the K-moduli space.
2023/01/10
10:30-12:00 Room #ハイブリッド開催/002 (Graduate School of Math. Sci. Bldg.)
Akihiro Higashitani (Osaka/Dept. of Inf. )
Toric Fano varieties arising from posets and their combinatorial mutation equivalence (日本語)
Akihiro Higashitani (Osaka/Dept. of Inf. )
Toric Fano varieties arising from posets and their combinatorial mutation equivalence (日本語)
[ Abstract ]
In 1986, Stanley introduced two polytopes arising from posets, called order polytopes and chain polytopes. Since then, those polytopes have been studied from viewpoints of combinatorics. Projective toric varieties arising from order polytopes are called Hibi toric varieties in these days. On the other hand, combinatorial mutations were introduced by Akhtar-Coates-Galkin-Kasprzyk in 2012 in the context of the classification problem of Fano varieties using mirror symmetry.
In this talk, after surveying two poset polytopes and combinatorial mutations, we discuss the combinatorial mutation equivalence of two poset polytopes. Those equivalence implies qG-deformation equivalence of projective toric varieties arising from two poset polytopes.
Moreover, it turns out that order polytopes, chain polytopes and their intermediate polytopes correspond to some toric Fano varieties.
In 1986, Stanley introduced two polytopes arising from posets, called order polytopes and chain polytopes. Since then, those polytopes have been studied from viewpoints of combinatorics. Projective toric varieties arising from order polytopes are called Hibi toric varieties in these days. On the other hand, combinatorial mutations were introduced by Akhtar-Coates-Galkin-Kasprzyk in 2012 in the context of the classification problem of Fano varieties using mirror symmetry.
In this talk, after surveying two poset polytopes and combinatorial mutations, we discuss the combinatorial mutation equivalence of two poset polytopes. Those equivalence implies qG-deformation equivalence of projective toric varieties arising from two poset polytopes.
Moreover, it turns out that order polytopes, chain polytopes and their intermediate polytopes correspond to some toric Fano varieties.
2022/12/21
13:00-14:00 or 14:30 Room #ハイブリッド開催/056 (Graduate School of Math. Sci. Bldg.)
The room is different from the usual place. This is a joint seminar with Kyoto University.
Hsueh-Yung Lin (NTU)
Towards a geometric origin of the dynamical filtrations (English)
The room is different from the usual place. This is a joint seminar with Kyoto University.
Hsueh-Yung Lin (NTU)
Towards a geometric origin of the dynamical filtrations (English)
[ Abstract ]
Let X be a smooth projective variety with an automorphism f. When X is a threefold, Serge Cantat asked whether X has a non-trivial equivariant rational fibration, if the action of f on the Néron-Severi space is non-trivial and unipotent. We will propose a precise conjecture related to Cantat's question for minimal varieties in arbitrary dimension, in light of the "dynamical filtrations" arising in the study of zero entropy group actions. This conjecture also suggests a geometric origin of dynamical filtrations, whose definition is purely cohomological. We will provide some heuristic evidence from the relative abundance conjecture.
If time permits, we will also explain how the study of dynamical filtrations leads to new results about solvable group actions, which are not necessarily of zero entropy.
Let X be a smooth projective variety with an automorphism f. When X is a threefold, Serge Cantat asked whether X has a non-trivial equivariant rational fibration, if the action of f on the Néron-Severi space is non-trivial and unipotent. We will propose a precise conjecture related to Cantat's question for minimal varieties in arbitrary dimension, in light of the "dynamical filtrations" arising in the study of zero entropy group actions. This conjecture also suggests a geometric origin of dynamical filtrations, whose definition is purely cohomological. We will provide some heuristic evidence from the relative abundance conjecture.
If time permits, we will also explain how the study of dynamical filtrations leads to new results about solvable group actions, which are not necessarily of zero entropy.
2022/12/20
9:30-10:30 Room #オンラインZoom (Graduate School of Math. Sci. Bldg.)
Takumi Murayama (Purdue)
The relative minimal model program for excellent algebraic spaces and analytic spaces in equal characteristic zero (English)
Takumi Murayama (Purdue)
The relative minimal model program for excellent algebraic spaces and analytic spaces in equal characteristic zero (English)
[ Abstract ]
In 2010, Birkar, Cascini, Hacon, and McKernan proved a relative version of the minimal model program for projective morphisms of complex quasi-projective varieties, called the relative minimal model program with scaling. Their result is now fundamental to our understanding of the birational classification of quasi-projective varieties and has numerous applications.
In this talk, I will discuss recent joint work with Shiji Lyu that establishes the relative minimal model program with scaling for excellent schemes, excellent algebraic spaces, and analytic spaces simultaneously in equal characteristic zero. This not only recovers previous results for complex varieties, complex algebraic spaces, and complex analytic spaces, but also greatly extends the scope of the relative minimal model program with scaling to a broader class of geometric spaces, including formal schemes, rigid analytic spaces, and Berkovich spaces, all in equal characteristic zero. Our results for (non-algebraic) schemes and rigid analytic spaces were previously only known in dimensions ≤3 and ≤2, respectively, and our results for formal schemes and Berkovich spaces are completely new.
In 2010, Birkar, Cascini, Hacon, and McKernan proved a relative version of the minimal model program for projective morphisms of complex quasi-projective varieties, called the relative minimal model program with scaling. Their result is now fundamental to our understanding of the birational classification of quasi-projective varieties and has numerous applications.
In this talk, I will discuss recent joint work with Shiji Lyu that establishes the relative minimal model program with scaling for excellent schemes, excellent algebraic spaces, and analytic spaces simultaneously in equal characteristic zero. This not only recovers previous results for complex varieties, complex algebraic spaces, and complex analytic spaces, but also greatly extends the scope of the relative minimal model program with scaling to a broader class of geometric spaces, including formal schemes, rigid analytic spaces, and Berkovich spaces, all in equal characteristic zero. Our results for (non-algebraic) schemes and rigid analytic spaces were previously only known in dimensions ≤3 and ≤2, respectively, and our results for formal schemes and Berkovich spaces are completely new.
2022/12/13
10:30-11:30 Room #ハイブリッド開催/002 (Graduate School of Math. Sci. Bldg.)
The speaker will give his talk by Zoom
山岸亮 (NTU)
Moduli of G-constellations and crepant resolutions (日本語)
The speaker will give his talk by Zoom
山岸亮 (NTU)
Moduli of G-constellations and crepant resolutions (日本語)
[ Abstract ]
For a finite subgroup G of SL_n(C), a moduli space of G-constellations is a generalization of the G-Hilbert scheme and is important from the viewpoint of McKay correspondence. In this talk I will explain its basic properties and show that every projective crepant resolution of C^3/G is isomorphic to such a moduli space.
For a finite subgroup G of SL_n(C), a moduli space of G-constellations is a generalization of the G-Hilbert scheme and is important from the viewpoint of McKay correspondence. In this talk I will explain its basic properties and show that every projective crepant resolution of C^3/G is isomorphic to such a moduli space.
2022/11/29
10:30-11:30 Room #ハイブリッド開催/002 (Graduate School of Math. Sci. Bldg.)
Thomas Hall (University of Nottingham)
The behaviour of Kahler-Einstein polygons under combinatorial mutation
(English)
Thomas Hall (University of Nottingham)
The behaviour of Kahler-Einstein polygons under combinatorial mutation
(English)
[ Abstract ]
Combinatorial mutations play an important role in the mirror symmetry approach to the classification of Fano varieties. Another important notion for Fano varieties is that of K-polystability, which turns out to have a nice combinatorial characterisation in the toric case. In this talk, I will give an overview of how mutations work and sketch the key ideas used to explore its interaction with Kahler-Einstein polygons (i.e. the Fano polygons whose associated toric variety is K-polystable).
Combinatorial mutations play an important role in the mirror symmetry approach to the classification of Fano varieties. Another important notion for Fano varieties is that of K-polystability, which turns out to have a nice combinatorial characterisation in the toric case. In this talk, I will give an overview of how mutations work and sketch the key ideas used to explore its interaction with Kahler-Einstein polygons (i.e. the Fano polygons whose associated toric variety is K-polystable).
2022/11/22
10:30-12:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Sho Tanimoto (Nagoya)
Non-free sections of Fano fibrations (日本語)
Sho Tanimoto (Nagoya)
Non-free sections of Fano fibrations (日本語)
[ Abstract ]
Manin’s Conjecture predicts the asymptotic formula for the counting function of rational points over number fields or global function fields. In the late 80’s, Batyrev developed a heuristic argument for Manin’s Conjecture over global function fields, and the assumptions underlying Batyrev’s heuristics are refined and formulated as Geometric Manin’s Conjecture. Geometric Manin’s Conjecture is a set of conjectures regarding properties of the space of sections of Fano fibrations, and it consists of three conjectures: (i) Pathological components are controlled by Fujita invariants; (ii) For each nef algebraic class, a non-pathological component which should be counted in Manin’s Conjecture is unique (This component is called as Manin components); (iii) Manin components exhibit homological or motivic stability. In this talk we discuss our proofs of GMC (i) over complex numbers using theory of foliations and the minimal model program. Using this result, we prove that these pathological components are coming from a bounded family of accumulating maps. This is joint work in progress with Brian Lehmann and Eric Riedl.
Manin’s Conjecture predicts the asymptotic formula for the counting function of rational points over number fields or global function fields. In the late 80’s, Batyrev developed a heuristic argument for Manin’s Conjecture over global function fields, and the assumptions underlying Batyrev’s heuristics are refined and formulated as Geometric Manin’s Conjecture. Geometric Manin’s Conjecture is a set of conjectures regarding properties of the space of sections of Fano fibrations, and it consists of three conjectures: (i) Pathological components are controlled by Fujita invariants; (ii) For each nef algebraic class, a non-pathological component which should be counted in Manin’s Conjecture is unique (This component is called as Manin components); (iii) Manin components exhibit homological or motivic stability. In this talk we discuss our proofs of GMC (i) over complex numbers using theory of foliations and the minimal model program. Using this result, we prove that these pathological components are coming from a bounded family of accumulating maps. This is joint work in progress with Brian Lehmann and Eric Riedl.
2022/11/15
10:30-12:00 Room #ハイブリッド開催/002 (Graduate School of Math. Sci. Bldg.)
Chi-Kang Chang (NTU/Tokyo)
Positivity of anticanonical divisors in algebraic fibre spaces (日本語)
Chi-Kang Chang (NTU/Tokyo)
Positivity of anticanonical divisors in algebraic fibre spaces (日本語)
[ Abstract ]
It is known that the positivity of the anti-canonical divisor is an important property that is closely related to the geometric structure of a variety. Given an algebraic fibre space f : X → Y between normal projective varieties with mild singularities, and let F be a general fibre of f. In this talk, we will introduce results relating the positivity of −KX and −KY under some conditions on the asymptotic base loci of −KX. In particular, we will obtain an inequality between the anti-canonical Iitaka dimensions κ(X, −KX) ≤ κ(F, −KF ) + κ(Y, −KY ) under the assumption that the stable base locus B(−KX) does not dominant over Y .
It is known that the positivity of the anti-canonical divisor is an important property that is closely related to the geometric structure of a variety. Given an algebraic fibre space f : X → Y between normal projective varieties with mild singularities, and let F be a general fibre of f. In this talk, we will introduce results relating the positivity of −KX and −KY under some conditions on the asymptotic base loci of −KX. In particular, we will obtain an inequality between the anti-canonical Iitaka dimensions κ(X, −KX) ≤ κ(F, −KF ) + κ(Y, −KY ) under the assumption that the stable base locus B(−KX) does not dominant over Y .
2022/11/01
10:30-12:00 Room #ハイブリッド開催/002 (Graduate School of Math. Sci. Bldg.)
Tatsuro Kawakami (Kyoto)
Extendability of differential forms via Cartier operators (Japanese)
Tatsuro Kawakami (Kyoto)
Extendability of differential forms via Cartier operators (Japanese)
[ Abstract ]
For a normal variety X, we say X satisfies the extension theorem if, for every proper birational morphism from Y, every differential form on the regular locus of X extends to Y. This is a basic property relating differential forms and singularities, and many results are known over the field of complex numbers.
In this talk, we discuss the extension theorem in positive characteristic. Existing studies depend on geometric tools such as log resolutions, (mixed) Hodge theory, the minimal model program, and vanishing theorems, which are not expected to be true or are not known for higher-dimensional varieties in positive characteristic.
For this reason, I introduce a new algebraic approach to the extension theorem using Cartier operators. I also talk about an application of the theory of quasi-F-splitting, which is studied in joint work with Takamatsu-Tanaka-Witaszek-Yobuko-Yoshikawa, to the extension problem.
For a normal variety X, we say X satisfies the extension theorem if, for every proper birational morphism from Y, every differential form on the regular locus of X extends to Y. This is a basic property relating differential forms and singularities, and many results are known over the field of complex numbers.
In this talk, we discuss the extension theorem in positive characteristic. Existing studies depend on geometric tools such as log resolutions, (mixed) Hodge theory, the minimal model program, and vanishing theorems, which are not expected to be true or are not known for higher-dimensional varieties in positive characteristic.
For this reason, I introduce a new algebraic approach to the extension theorem using Cartier operators. I also talk about an application of the theory of quasi-F-splitting, which is studied in joint work with Takamatsu-Tanaka-Witaszek-Yobuko-Yoshikawa, to the extension problem.
2022/10/25
10:30-11:45 Room #ハイブリッド開催/002 (Graduate School of Math. Sci. Bldg.)
Atsushi Ito (Okayama)
Projective normality of general polarized abelian varieties (Japanese)
Atsushi Ito (Okayama)
Projective normality of general polarized abelian varieties (Japanese)
[ Abstract ]
Projective normality is an important property of polarized varieties. Hwang and To prove that a general polarized abelian variety $(X,L)$ of dimension $g$ is projectively normal if $\chi(X,L) \geq (8g)^g/2g!$. In this talk, I will explain that their bound can be weaken as $\chi(X,L) > 2^{2g-1}$, which is sharp. A key tool in the proof is an invariant introduced by Jiang and Pareschi, which measures the basepoint freeness of $\mathbb{Q}$-divisors on abelian varieties.
Projective normality is an important property of polarized varieties. Hwang and To prove that a general polarized abelian variety $(X,L)$ of dimension $g$ is projectively normal if $\chi(X,L) \geq (8g)^g/2g!$. In this talk, I will explain that their bound can be weaken as $\chi(X,L) > 2^{2g-1}$, which is sharp. A key tool in the proof is an invariant introduced by Jiang and Pareschi, which measures the basepoint freeness of $\mathbb{Q}$-divisors on abelian varieties.
2022/10/05
13:00-14:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Yuri Tschinkel (Mathematics and Physical Sciences Division, Simons Foundation/ Courant Institute, New York University)
Equivariant birational geometry (joint with A. Kresch) (English)
Yuri Tschinkel (Mathematics and Physical Sciences Division, Simons Foundation/ Courant Institute, New York University)
Equivariant birational geometry (joint with A. Kresch) (English)
[ Abstract ]
Ideas from motivic integration led to the introduction of new invariants in equivariant birational geometry, the study of actions of finite groups on algebraic varieties, up to equivariant birational transformations.
These invariants allow us to distinguish actions in many new cases, shedding light on the structure of the Cremona group. The structure of the invariants themselves is also interesting: there are unexpected connections to modular curves and cohomology of arithmetic groups.
Ideas from motivic integration led to the introduction of new invariants in equivariant birational geometry, the study of actions of finite groups on algebraic varieties, up to equivariant birational transformations.
These invariants allow us to distinguish actions in many new cases, shedding light on the structure of the Cremona group. The structure of the invariants themselves is also interesting: there are unexpected connections to modular curves and cohomology of arithmetic groups.
2021/07/21
15:00-16:00 Room #zoom (Graduate School of Math. Sci. Bldg.)
Cancelled
Keisuke Miyamoto (Osaka)
TBA (日本語)
Cancelled
Keisuke Miyamoto (Osaka)
TBA (日本語)
[ Abstract ]
TBA
TBA
2021/07/05
16:00-17:00 Room #zoom (Graduate School of Math. Sci. Bldg.)
Paolo Cascini (Imperial College London)
Birational geometry of foliations (English)
Paolo Cascini (Imperial College London)
Birational geometry of foliations (English)
[ Abstract ]
I will survey about some recent progress towards the Minimal Model Program for foliations on complex varieties, focusing mainly on the case of threefolds and the case of algebraically integrable foliations.
I will survey about some recent progress towards the Minimal Model Program for foliations on complex varieties, focusing mainly on the case of threefolds and the case of algebraically integrable foliations.
2021/07/01
10:00-11:00 Room # (Graduate School of Math. Sci. Bldg.)
Fumiaki Suzuki (UCLA)
An O-acyclic variety of even index
Fumiaki Suzuki (UCLA)
An O-acyclic variety of even index
[ Abstract ]
I will construct a family of Enriques surfaces parametrized by P^1 such that any multi-section has even degree over the base P^1. Over the function field of a complex curve, this gives the first example of an O-acyclic variety (H^i(X,O)=0 for i>0) whose index is not equal to one, and an affirmative answer to a question of Colliot-Thélène and Voisin. I will also discuss applications to related problems, including the integral Hodge conjecture and Murre’s question on universality of the Abel-Jacobi maps. This is joint work with John Christian Ottem.
I will construct a family of Enriques surfaces parametrized by P^1 such that any multi-section has even degree over the base P^1. Over the function field of a complex curve, this gives the first example of an O-acyclic variety (H^i(X,O)=0 for i>0) whose index is not equal to one, and an affirmative answer to a question of Colliot-Thélène and Voisin. I will also discuss applications to related problems, including the integral Hodge conjecture and Murre’s question on universality of the Abel-Jacobi maps. This is joint work with John Christian Ottem.
2021/06/14
17:00-18:00 Room # (Graduate School of Math. Sci. Bldg.)
Wahei Hara (University of Glasgow)
Rank two weak Fano bundles on del Pezzo threefolds of degree 5 (日本語)
Zoom
Wahei Hara (University of Glasgow)
Rank two weak Fano bundles on del Pezzo threefolds of degree 5 (日本語)
[ Abstract ]
None
[ Reference URL ]None
Zoom
