代数幾何学セミナー

過去の記録 ~07/18次回の予定今後の予定 07/19~

開催情報 火曜日 15:30~17:00 数理科学研究科棟(駒場) 122号室
担当者 權業 善範・中村 勇哉・田中公

過去の記録

2019年07月09日(火)

13:00-14:30   数理科学研究科棟(駒場) 122号室
いつもと曜日・時間・部屋が異なります。
佐野 太郎 氏 (神戸大学)
Construction of non-Kähler Calabi-Yau 3-folds by smoothing normal crossing varieties (TBA)
[ 講演概要 ]
It is an open problem whether there are only finitely many diffeomorphism types of projective Calabi-Yau 3-folds. Kawamata--Namikawa developed log deformation theory of normal crossing Calabi-Yau varieties. As an application of their result, one can construct examples of Calabi-Yau manifolds by smoothing SNC varieties. In this talk, I will explain how to construct examples of non-Kähler Calabi-Yau 3-folds with arbitrarily large 2nd Betti numbers. If time permits, I will also explain an example of involutions on a family of K3 surfaces which do not lift biregularly to the total space. This is based on joint work with Kenji Hashimoto.

2019年07月05日(金)

10:30-12:00   数理科学研究科棟(駒場) 123号室
いつもと曜日・時間・部屋が異なります。
榎園 誠 氏 (東京理科大学)
Durfee-type inequality for complete intersection surface singularities
[ 講演概要 ]
Durfee's negativity conjecture says that the signature of the Milnor fiber of a 2-dimensional isolated complete intersection singularity is always negative. In this talk, I will explain that this conjecture is true (more precisely, the signature is bounded above by the negative number determined by the geometric genus, the embedding dimension and the number of irreducible components of the exceptional set of the minimal resolution) by using the theory of invariants of fibered surfaces. If time permits, I will explain the higher dimensional analogue of Durfee's conjecture for isolated complete intersection singularities.

2019年06月28日(金)

15:30-17:00   数理科学研究科棟(駒場) 118号室
いつもと曜日が異なります。
谷本 祥 氏 (熊本)
Rational curves on prime Fano 3-folds (TBA)
[ 講演概要 ]
One of important topics in algebraic geometry is the space of rational curves, e.g., the dimension and the number of components of the moduli spaces of rational curves on an algebraic variety X. One of interesting situations where this question is extensively studied is when X is a Fano variety since in this case X is rationally connected so that it does contain a lots of rational curves. In this talk I will talk about my joint work with Brian Lehmann which settles this problem for most Fano 3-folds of Picard rank 1, e.g., a general quartic 3-fold in P^4, and our approach is inspired by Manin’s conjecture which predicts the asymptotic formula for the counting function of rational points on a Fano variety. In particular we systematically use geometric invariants in Manin’s conjecture which have been studied by many mathematicians including Brian and me.

2019年06月19日(水)

15:30-17:00   数理科学研究科棟(駒場) 118号室
今学期は基本水曜日とします。部屋も去年度と異なります。
鈴木文顕 氏 (イリノイ州立シカゴ大学)
A pencil of Enriques surfaces with non-algebraic integral Hodge classes (TBA)
[ 講演概要 ]
The integral Hodge conjecture is the statement that the integral Hodge classes are algebraic on smooth complex projective varieties. It is known that the conjecture can fail in general. There are two types of counterexamples, ones with non-algebraic integral Hodge classes of torsion-type and of non-torsion type, the first of which were given by Atiyah-Hirzebruch and Kollar, respectively.

In this talk, we exhibit a pencil of Enriques surfaces defined over Q with non-algebraic integral Hodge classes of non-torsion type. This construction relates to certain questions concerning rational points of algebraic varieties.

This gives the first example of a threefold with the trivial Chow group of zero-cycles on which the integral Hodge conjecture fails. As an application, we construct a fourfold which gives the negative answer to a classical question on the universality of the Abel-Jacobi maps.

This is a joint work with John Christian Ottem.

2019年05月29日(水)

15:30-17:00   数理科学研究科棟(駒場) 118号室
今学期は基本水曜日とします。部屋も去年度と異なります。
江辰 氏 (Fudan/MSRI)
Minimal log discrepancies of 3-dimensional non-canonical singularities (English)
[ 講演概要 ]
Canonical and terminal singularities, introduced by Reid, appear naturally in minimal model program and play important roles in the birational classification of higher dimensional algebraic varieties. Such singularities are well-understood in dimension 3, while the property of non-canonical singularities is still mysterious. We investigate the difference between canonical and non-canonical singularities via minimal log discrepancies (MLD). We show that there is a gap between MLD of 3-dimensional non-canonical singularities and that of 3-dimensional canonical singularities, which is predicted by a conjecture of Shokurov.
This result on local singularities has applications to global geometry of Calabi–Yau 3-folds. We show that the set of all non-canonical klt Calabi–Yau 3-folds are bounded modulo flops, and the global indices of all klt Calabi–Yau 3-folds are bounded from above.

2019年05月22日(水)

15:30-17:00   数理科学研究科棟(駒場) 122号室
河上 龍郎 氏 (東大数理)
Bogomolov type vanishing on three-dimensional Mori fiber spaces in positive characteristic

[ 講演概要 ]
In characteristic zero, cotangent bundle of n(>1)-dimensional smooth projective varieties does not contain a big line bundle. This is a part of Bogomolov vanishing and this vanishing plays an important role in the proof of Miyaoka-Yau inequality. In positive characteristic, it is known that Bogomolov vanishing does not hold. There exists a general type surface whose cotangent bundle contains an ample line bundle. So, it is natural to ask when Bogomolov type vanishing holds in positive characteristic. In this talk, I discuss Bogomolov type vanishing on three-dimensional Mori fiber spaces in positive characteristic.

2019年05月15日(水)

15:30-17:00   数理科学研究科棟(駒場) 118号室
今学期は基本水曜日とします。部屋も去年度と異なります。
藤野修 氏 (大阪大学)
On quasi-log canonical pairs
(Japanese)
[ 講演概要 ]
The notion of quasi-log canonical pairs was introduced by Florin Ambro. It is a kind of generalizations of that of log canonical pairs. Now we know that quasi-log canonical pairs are ubiquitous in the theory of minimal models. In this talk, I will explain some basic properties and examples of quasi-log canonical pairs. I will also discuss some new developments around quasi-log canonical pairs. Some parts are joint works with Haidong Liu.

2019年05月08日(水)

15:30-17:00   数理科学研究科棟(駒場) 118号室
橋詰 健太 氏 (東大数理)
On Minimal model theory for log canonical pairs with big boundary divisors
[ 講演概要 ]
In 2010, Birkar--Cascini--Hacon--McKernan established the minimal model theory for Kawamata log terminal pairs with big boundary divisors, and a lot of theorems in the birational geometry are
proved by applying this result. It is expected that this result can be generalized to log canonical pairs. Currently, it is known that the minimal model theory for log canonical pairs can be reduced to the case of big boundary divisors. In this talk, we introduce a partial generalization of the result by Birkar--Cascini--Hacon--McKernan. Roughly speaking, we generalized their result to lc pairs with big boundary divisors having only small lc centers. We also explain another generalization, which is originally announced by Hu, and we discuss termination of log minimal model program in a spacial case. This is a joint work with Zhengyu Hu, and the work is in progress.

2019年04月24日(水)

15:30-17:00   数理科学研究科棟(駒場) 118号室
今学期は基本水曜日とします。部屋も去年度と異なります。
吉川翔 氏 (東大数理)
Varieties of dense globally F-split type with a non-invertible polarized
endomorphism
[ 講演概要 ]
Broustet and Gongyo conjectured that if a normal projective variety X has a non-invertible polaried endomorphism, then X is of Calabi-Yau type. Furthermore, Schwede and Smith conjectured that a projective variety is of Calabi-Yau type if and only if of dense globally F-split type. Therefore it is a natural question to ask if a normal projective variety X has a non-invertible polaried endomorphism, then X is of dense globally F-split type. In this talk, I will introduce simple points and difficult points of the question. Furthermore I will give the affirmative answer of my question for 2-dimensional case.

2019年01月29日(火)

15:30-17:00   数理科学研究科棟(駒場) 122号室
三井健太郎 氏 (神戸)
Logarithmic good reduction and the index (TBA)
[ 講演概要 ]
A proper smooth variety over a complete discrete valuation field is said to have (log) good reduction if it admits a proper (log) smooth model over the valuation ring (the log structure is given by the closed fiber). Monodromy criteria for good reduction and log good reduction have been studied. We study the log case by additional other conditions on geometric invariants such as the index of the variety (the minimal positive degree of a 0-cycle). In particular, we obtain a criterion for log good reduction of curves of genus one.

2018年12月21日(金)

10:30-11:30   数理科学研究科棟(駒場) 123号室
Mattias Jonsson 氏 (Michigan)
Degenerations of p-adic volume forms (English)
[ 講演概要 ]
Let X be an n-dimensional smooth projective variety over a non-Archimedean local field K. Also fix a regular n-form on X. This data induces a positive measure on the space of K'-rational points for any finite extension K' of K. We describe the asymptotics, as K' runs through towers of finite extensions of K, in terms of Berkovich analytic geometry. This is joint work with Johannes Nicaise.

2018年12月14日(金)

10:30-11:30   数理科学研究科棟(駒場) 123号室
Zhi Jiang 氏 (Fudan)
On the birationality of quint-canonical systems of irregular threefolds of general type (English)
[ 講演概要 ]
It is well-known that the quint-canonical map of a surface of general type is birational.
We will show that the same result holds for irregular threefolds of general type. The proof is based on
a careful study of the positivity of the pushforwards of pluricanonical bundles on abelian varieties and Severi
type inequalities. This is a joint work with J.A. Chen, J.Chen, and M.Chen.

2018年11月27日(火)

15:30-17:00   数理科学研究科棟(駒場) 122号室
原伸生 氏 (東京農工大)
Frobenius summands and the finite F-representation type (TBA)
[ 講演概要 ]
We are motivated by a question arising from commutative algebra, asking what kind of
graded rings in positive characteristic p have finite F-representation type. In geometric
setting, this is related to the problem to looking out for Frobenius summands. Namely,
given aline bundle L on a projective variety X, we want to know how many and what
kind of indecomposable direct summands appear in the direct sum decomposition of
the iterated Frobenius push-forwards of L. We will consider the problem in the following
two cases, although the present situation in (2) is far from satisfactory.
 (1) two-dimensional normal graded rings (a joint work with Ryo Ohkawa)
 (2) the anti-canonical ring of a quintic del Pezzo surface

2018年11月20日(火)

15:30-17:00   数理科学研究科棟(駒場) 122号室
中島 幸喜 氏 (東京電機大)
Artin-Mazur height, Yobuko height and
Hodge-Wittt cohomologies

[ 講演概要 ]
A few years ago Yobuko has introduced the notion of
a delicate invariant for a proper smooth scheme over a perfect field $k$
of finite characteristic. (We call this invariant Yobuko height.)
This generalize the notion of the F-splitness due to Mehta-Srinivas.

In this talk we give relations between Artin-Mazur heights
and Yobuko heights. We also give a finiteness result on
Hodge-Witt cohomologies of a proper smooth scheme $X$ over $k$
with finite Yobuko height. If time permits, we give a cofinite type result on
the $p$-primary torsion part of Chow group of of $X$
of codimension 2 if $\dim X=3$.

2018年11月13日(火)

15:30-17:00   数理科学研究科棟(駒場) 122号室
陳韋中 氏 (東大数理)
Boundedness of varieties of Fano type with alpha-invariants and volumes bounded below (English)
[ 講演概要 ]
We show that fixed dimensional klt weak Fano pairs with alpha-invariants and volumes bounded away from 0 and the coefficients of the boundaries belong to the set of hyperstandard multiplicities Φ(R) associated to a fixed finite set R form a bounded family. We also show α(X, B)d−1vol(−(KX + B)) are bounded from above for all klt weak Fano pairs (X, B) of a fixed dimension d.

2018年10月16日(火)

15:30-17:00   数理科学研究科棟(駒場) 122号室
Tuyen Truong 氏 (Oslo)
A countable characterisation of smooth algebraic plane curves, and generalisations (English)
[ 講演概要 ]
Given a smooth algebraic curve X in C^3, I will present a way to construct a sequence of algebraic varieties (whose ideals are explicitly determined from the ideal defining X), whose solution set is non-empty iff the curve X can be algebraically embedded into C^2.
Various other questions, such as whether two given algebraic varieties are birational, can be similarly treated. Some related conjectures are stated.

2018年10月09日(火)

15:30-17:00   数理科学研究科棟(駒場) 122号室
小関直紀 氏 (東大数理/IPMU)
Stability conditions on threefolds with nef tangent bundles (English)
[ 講演概要 ]
The construction of Bridgeland stability conditions on threefolds
is an open problem in general.
The problem is reduced to proving
the so-called Bogomolov-Gieseker (BG) type inequality conjecture,
proposed by Bayer, Macrí, and Toda.
In this talk, I will explain how to prove the BG type inequality
conjecture
for threefolds in the title.

2018年07月18日(水)

15:30-17:00   数理科学研究科棟(駒場) 122号室
普段と違う水曜日にセミナーを行います。The seminar will be held on Wednesday. This is a different day from usual.
Jun-Muk Hwang 氏 (KIAS)
Normal Legendrian singularities (English)
[ 講演概要 ]
A germ of a Legendrian subvariety in a holomorphic contact manifold
is called a Legendrian singularity. Legendrian singularities are usually not normal.
We look at some examples of normal Legendrian singularities and discuss their rigidity under deformation.

2018年07月10日(火)

15:30-17:00   数理科学研究科棟(駒場) 002号室
いつもと部屋が違います。The room is different from usual.
賴青瑞 氏 (国立成功大学)
The effective bound of anticanonical volume of Fano threefolds (English)
[ 講演概要 ]
According to Mori's program, varieties covered by rational curves are
built up from anti-canonically polarized varieties, aka Fano varieties. After fixed the
dimension and singularity type, Fano varieties form a bounded family by Birkar's proof (2016)
of Borisov-Alexeev-Borisov conjecture, which In particular implies that the anticanonical
volume -K^\dim is bounded. In this talk, we focus on canonical Fano threefolds,
where boundedness was established by Koll\'ar-Miyaoka-Mori-Takagi (2000).
Our aim is to find an effective bound of the anticanonical volume -K^3, which is
not explicit either from the work of Koll\'ar-Miyaoka-Mori-Takagi or Birkar. We will discuss
some effectiveness results related to this problem and prove that -K_X^3\leq 72 if \rho(X)\leq 2.
This partially extends early work of Mori, Mukai, Y. Prokhorov, et al.

2018年07月03日(火)

15:30-17:00   数理科学研究科棟(駒場) 122号室
Xun Yu 氏 (Tianjin University)
Surface automorphisms and Salem numbers (English)
[ 講演概要 ]
The entropy of a surface automorphism is either zero or the
logarithm of a Salem number.
In this talk, we will discuss which Salem numbers arise in this way. We
will show that any
supersingular K3 surface in odd characteristic has an automorphism the
entropy of which is
the logarithm of a Salem number of degree 22. In particular, such
automorphisms are
not geometrically liftable to characteristic 0.

2018年06月26日(火)

15:30-17:00   数理科学研究科棟(駒場) 122号室
渡辺究 氏 (埼玉)
Varieties with nef diagonal (English)
[ 講演概要 ]
For a smooth projective variety $X$, we consider when the diagonal $Δ _X$ is nef as a
cycle on $X \times X$. In particular, we give a classication of complete intersections and smooth
del Pezzo varieties where the diagonal is nef. We also study the nefness of the diagonal for
spherical varieties. This is a joint work with Taku Suzuki.

2018年06月19日(火)

15:30-17:00   数理科学研究科棟(駒場) 122号室
若林 泰央 氏 (東工大)
Dormant Miura opers and Tango structures (Japanese (writing in English))
[ 講演概要 ]
Tango(丹後)構造とは, 正標数の代数曲線で定義された然るべき直線束であり, (小平消滅定理の反例となるなどの, いわゆる)「病理的な」正標数の代数多様体を構成するうえで基本的な概念です. 本発表では, このTango構造と, 一見すると無関係にも見える幾つかのトピック(dormant Miura oper, 3正則グラフのナンバリング, Gaudin模型の対角化etc.)との結びつきや関連する結果について説明させていただきたく予定です.(各トピックについて発表者はそれほど詳しくありませんので, この機会にぜひ皆さまの知見を伺えればと思っております.)

2018年06月12日(火)

15:30-17:00   数理科学研究科棟(駒場) 122号室
柴田 崇広 氏 (京都大学)
Ample canonical heights for endomorphisms on projective varieties (English or Japanese)
[ 講演概要 ]
Given a smooth projective variety on a number field and an
endomorphism on it, we would like to know how the height of a point
grows by iteration of the action of the endomorphism. When the
endomorphism is polarized, Call and Silverman construct the canonical
height, which is an important tool for the calculation of growth of
heights. In this talk, we will give a generalization of the Call-
Silverman canonical heights for not necessarily polarized endomorphisms,
ample canonical heights, and propose an analogue of the Northcott
finiteness theorem as a conjecture. We will see that the conjecture
holds when the variety is an abelian variety or a surface.

2018年05月29日(火)

15:30-17:00   数理科学研究科棟(駒場) 122号室
Alessandra Sarti 氏 (Universit\'e de Poitiers)
Nikulin configurations on Kummer surfaces (English)
[ 講演概要 ]
A Nikulin configuration is the data of
16 disjoint smooth rational curves on a K3 surface.
According to results of Nikulin this means that the K3 surface
is a Kummer surface and the abelian surface in the Kummer structure
is determined by the 16 curves. An old question of Shioda is about the
existence of non isomorphic Kummer structures on the same Kummer K3
surface.
The question was positively answered and studied by several authors, and
it was shown that the number of non-isomorphic Kummer structures is
finite,
but no explicit geometric construction of such structures was given.
In the talk I will show how to construct explicitely non isomorphic
Kummer structures on generic Kummer K3 surfaces.
This is a joint work with X. Roulleau.

2018年05月25日(金)

15:30-17:00   数理科学研究科棟(駒場) 122号室
普段と違う金曜日にセミナーを行います。The seminar will be held on Friday. This is a different day from usual.
De Qi Zhang 氏 (Singapore)
Endomorphisms of normal projective variety and equivariant-MMP (English)
[ 講演概要 ]
We report some recent joint works on polarized or int-amplified endomorphisms f on a normal projective variety X with mild singularities, and prove the pseudo-effectivity of the anti-canonical divisor of X, and the f-equivariance, after replacing f by its power, for every minimal model program starting from X. Fano varieties and Q-abelian varieties turn out to be building blocks having such symmetries. The ground field is closed and of characteristic 0 or at least 7.

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