## 代数幾何学セミナー

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

### 2010年05月17日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室

On the GIT stability of Polarized Varieties (JAPANESE)
[ 講演概要 ]
Background:
Original GIT-stability notion for polarized variety is
"asymptotic stability", studied by Mumford, Gieseker etc around 1970s.
Recently a version appeared, so-called "K-stability", introduced by
Tian(1997) and reformulated by Donaldson(2002), by the way of seeking
the analogue of Kobayashi-Hitchin correspondence, which gives
"differential geometric" interpretation of "stability". These two have
subtle but interesting differences in dimension higher than 1.

Contents:
(1*) Any semistable (in any sense) polarized variety should have only
"semi-log-canonical" singularities. (Partly observed around 1970s)
(2) On the other hand, we proved some stabilities, which corresponds to
"Calabi conjecture", also with admitting mild singularities.

As applications these yield
(3*) Compact moduli spaces with GIT interpretations.
(4) Many counterexamples (as orbifolds) to folklore conjecture:
"K-stability implies asymptotic stability".

(*: Some technical points are yet to be settled.
Some parts for (1)(2) are available on arXiv:0910.1794.)

### 2010年05月10日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室

Grassmann多様体のトーリック退化とミラー対称性 (JAPANESE)
[ 講演概要 ]
Grassmann多様体のトーリック退化と、
それを用いたGrassmann多様体の完全交叉カラビ・ヤウ多様体に対するミラー構

とくに、項順序によるトーリック退化に着目すれば、

A型Grassmann多様体やスピノル多様体などの例に関してこの条件を考察する。

### 2010年04月26日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室

The unirationality of the moduli spaces of 2-elementary K3
surfaces (JAPANESE)
[ 講演概要 ]
We prove the unirationality of the moduli spaces of K3 surfaces
with non-symplectic involution. As a by-product, we describe the
configuration spaces of 5, 6, 7, 8 points in the projective plane as
arithmetic quotients of type IV.

### 2010年04月19日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室

[ 講演概要 ]

(部分多様体に沿った)自己交点数の巨大な因子への一般化である制限型体積は,

また, 時間が許せば, 元々の問題意識であった制限型体積の複素解析的な側面に
ついても触れたい.

### 2010年04月05日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室
Alexandru Dimca 氏 (Université Nice-Sophia Antipolis)
From Lang's Conjecture to finiteness properties of Torelli groups
[ 講演概要 ]
First we recall one of Lang's conjectures in diophantine geometry
on the interplay between subvarieties and translated subgroups in a
commutative algebraic group
(proved by M. Laurent in the case of affine tori in 1984).

Then we present the technique of resonance and characteristic varieties,
a powerful tool in the study of fundamental groups of algebraic varieties.

Finally, using the two ingredients above, we show that the Torelli
groups $T_g$
have some surprising finiteness properties for $g>3$.
In particular, we show that for any subgroup $N$ in $T_g$ containing
the Johnson kernel $K_g$, the complex vector space $N_{ab} \\otimes C$
is finite dimensional.

All the details are available in our joint preprint with S. Papadima
arXiv:1002.0673.

### 2010年02月01日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室

Extensions of two Chow stability criteria to positive characteristics
[ 講演概要 ]
I will talk about two results on Chow (semi-)stability of cycles in positive characteristics, which were originally known in characteristic 0. One is on the stability of non-singular projective hypersurfaces of degree greater than 2, and the other is the criterion by Y. Lee in terms of the log canonical threshold of Chow divisor. A couple of examples will be discussed in detail.

### 2010年01月25日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室

On weak Fano varieties with log canonical singularities
[ 講演概要 ]
We prove that the anti-canonical divisors of weak Fano
3-folds with log canonical singularities are semiample. Moreover, we consider
semiampleness of the anti-log canonical divisor of any weak log Fano pair
with log canonical singularities. We show semiampleness dose not hold in
general by constructing several examples. Based on those examples, we propose
sufficient conditions which seem to be the best possible and we prove
semiampleness under such conditions. In particular we derive semiampleness of the
anti-canonical divisors of log canonical weak Fano 4-folds whose lc centers
are at most 1-dimensional. We also investigate the Kleiman-Mori cones of
weak log Fano pairs with log canonical singularities.

### 2010年01月18日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室
Anne-Sophie Kaloghiros 氏 (RIMS)
The divisor class group of terminal Gorenstein Fano 3-folds and rationality questions
[ 講演概要 ]
Let Y be a quartic hypersurface in CP^4 with mild singularities, e.g. no worse than ordinary double points.
If Y contains a surface that is not a hyperplane section, Y is not Q-factorial and the divisor class group of Y, Cl Y, contains divisors that are not Cartier. However, the rank of Cl Y is bounded.

In this talk, I will show that in most cases, it is possible to describe explicitly the divisor class group Cl Y by running a Minimal Model Program (MMP) on X, a small Q-factorialisation of Y. In this case, the generators of Cl Y/ Pic Y are topological traces " of K-negative extremal contractions on X.
This has surprising consequences: it is possible to conclude that a number of families of non-factorial quartic 3-folds are rational.
In particular, I give some examples of rational quartic hypersurfaces Y_4\\subset CP^4 with rk Cl Y=2 and show that when the divisor class group of Y has sufficiently high rank, Y is always rational.

### 2009年12月21日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室

Ampleness of two-sided tilting complexes
[ 講演概要 ]
From the view point of noncommutative algebraic geometry (NCAG),
a two-sided tilting complex is an analog of a line bundle.
In this talk we introduce the notion of ampleness for two-sided
tilting complexes over finite dimensional algebras.
From the view point of NCAG, the Serre functors are considered to be
shifted canonical bundles. We show by examples that the property
of shifted canonical bundle captures some representation theoretic
property of algebras.

### 2009年12月14日(月)

14:40-16:10   数理科学研究科棟(駒場) 126号室
いつもと時間帯が異なります。ご注意ください。
Sergey Galkin 氏 (IPMU)
Invariants of Fano varieties via quantum D-module
[ 講演概要 ]
We will introduce and compute Apery characteristic
class and Frobenius genera - invariants of Fano variety derived from
it's Gromov-Witten invariants. Then we will show how to compute them
and relate with other invariants.

### 2009年11月16日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室
Colin Ingalls 氏 (University of New Brunswick and RIMS)
Rationality of the Brauer-Severi Varieties of Skylanin algebras
[ 講演概要 ]
Iskovskih's conjecture states that a conic bundle over
a surface is rational if and only if the surface has a pencil of
rational curves which meet the discriminant in 3 or fewer points,
(with one exceptional case). We generalize Iskovskih's proof that
such conic bundles are rational, to the case of projective space
bundles of higher dimension. The proof involves maximal orders
and toric geometry. As a corollary we show that the Brauer-Severi
variety of a Sklyanin algebra is rational.

### 2009年11月02日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室
Gerard van der Geer 氏 (Universiteit van Amsterdam)
Cohomology of moduli spaces of curves and modular forms
[ 講演概要 ]
The Eichler-Shimura theorem expresses cohomology of local systems
on the moduli of elliptic curves in terms of modular forms. The
cohomology of local systems can be succesfully explored by counting
points over finite fields. We show how this can be applied to
obtain a lot of information about the cohomology of other moduli spaces
of low genera and also about Siegel modular forms of genus 2 and 3.
This is joint work with Jonas Bergstroem and Carel Faber.

### 2009年10月19日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室

ファノ多様体上の有理曲線の鎖の長さについて
[ 講演概要 ]
ピカール数1のファノ多様体に対し、一般の二点を結ぶために必要な

ファノ多様体や余指数が3以下のファノ多様体などに対し、長さを求める。

### 2009年10月05日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室

[ 講演概要 ]

その随伴束が自由になったり、基底点集合が具体的にかけることがある。しかし
、曲線の場合は簡単であるが高次元の場合は難しい。今回の講演では主に代数曲

### 2009年09月01日(火)

16:30-18:00   数理科学研究科棟(駒場) 002号室

Matthias Schuett 氏 (Leibniz University Hannover)
Arithmetic of K3 surfaces
[ 講演概要 ]
This talk aims to review recent developments in the arithmetic of K3 surfaces, with emphasis on singular K3 surfaces.
We will consider in particular modularity, Galois action on Neron-Severi groups and behaviour in families.

### 2009年07月13日(月)

16:30-18:00   数理科学研究科棟(駒場) 126号室

Seshadri constants on rational surfaces with anticanonical pencils

[ 講演概要 ]

この不変量を調べることでしばしば幾何的な情報が得られる。

が得られた。

### 2009年07月06日(月)

16:30-18:00   数理科学研究科棟(駒場) 126号室

アーベル曲面上の安定層とフーリエ向井変換について
[ 講演概要 ]

アーベル曲面上には半等質層と呼ばれる半安定層があり, その分類, 構成方法やコホモロジーが完全に知られている. アーベル曲面のフーリエ向井対は半等質層のモジュライ空間であることも知られている.

また安定層のフーリエ変換における振舞いの記述において, 算術群や整数係数2次形式が重要な役割を果たすことも分かる. この事と先に述べた表示の存在から, 安定層のモジュライとアーベル曲面上の点のヒルベルトスキームとの間の双有理変換が明示的に構成できる.
アーベル曲面のフーリエ向井変換のフォーマリズムはK3曲面の変換と共通する部分も少なくない. 講演ではそうした点にも触れつつ, 今回の結果とその証明の概要を解説したい.

### 2009年06月29日(月)

16:30-18:00   数理科学研究科棟(駒場) 126号室

Moduli on the projective plane and the wall-crossing
[ 講演概要 ]

を用いることにより、ある有限次元代数の半安定表現のモジュライ空間
として構成する。階数が2以下の場合、表現の安定性条件を変化させること
により、壁越え現象としてのflip の記述を得る。

### 2009年06月23日(火)

16:30-18:00   数理科学研究科棟(駒場) 126号室

Group actions on affine cones
[ 講演概要 ]
The action of the additive group scheme C_+ on normal affine varieties is one of main subjects in affine algebraic geometry for a long time. In this talk, we shall mainly consider the problem about the existence of C_+-actions on affine cones, more precisely, the question:

"Determine the affine cones over smooth projective varieties admitting a (non-trivial) C_+-action ".

This question has an interest from a point of view of singularities. Indeed, a normal Cohen-Macaulay affine variety admitting an action by C_+ has at most rational singularities due to the result of H. Flenner and M. Zaidenberg. In the case of dimension 2, any affine cone over the projective line P^1 has a cyclic quotient singularity, and we can see that it admits, in fact, a C_+-action. Meanwhile, in case of dimension 3, i.e., affine cones over rational surfaces, the situation becomes more subtle.

One of the main results is concerned with a criterion for the existence of a C_+-action on affine cones (of any dimension) in terms of a cylinderlike open subset on the base variety. By making use of it, it is shown that, for any rational surface Y, we can take a suitable embedding of Y in such a way that the associated affine cone admits an action of C_+. Furthermore we are able to confirm that an affine cone over an anticanonically embedded del Pezzo surface of degree greater than or equal to 4 also admits such an action.

Nevertheless, our final purpose to decide whether or not there does exist a C_+-action on the fermat cubic: x^3+y^3+z^3+u^3 =0 in C^4, which is the affine cone over an anticanonically embedded cubic surface, say Y_3, is not yet accomplished. But, we can obtain certain informations about a linear pencil of rational curves on Y_3 arising from a C_+-action which seem to be useful in order to deny an existence of an action of C_+.

### 2009年06月15日(月)

16:30-18:00   数理科学研究科棟(駒場) 128号室

アーベル曲面の分解と2次形式

[ 講演概要 ]

ありうる。いくつかの場合に分解の個数公式が求められてきた(林田、塩田-三谷
)。本講演では、すべての分解可能な複素Abel曲面に対して、2次形式論の技法
を用いて分解数の公式を与える。関連して次のことも話す:合同モジュラー曲線

の等長群の関係。

### 2009年05月22日(金)

15:00-16:30   数理科学研究科棟(駒場) 128号室
Prof. Steven Zucker 氏 (Johns Hopkins University)
The RBS compactification: a real stratified space in
algebraic geometry

### 2009年04月27日(月)

15:30-18:00   数理科学研究科棟(駒場) 122号室
Prof. Alessandra Sarti 氏 (Universite de Poitier) 15:30-16:30
Automorphism groups of K3 surfaces
[ 講演概要 ]
I will present recent progress in the study of prime order automorphisms of K3 surfaces.
An automorphism is called (non-) symplectic if the induced
operation on the global nowhere vanishing holomorphic two form
is (non-) trivial. After a short survey on the topic, I will
describe the topological structure of the fixed locus, the
geometry of these K3 surfaces and their moduli spaces.

Prof. Samuel Boissier 氏 (Universite de Nice
) 17:00-18:00
The cohomological crepant resolution conjecture

[ 講演概要 ]
The cohomological crepant resolution conjecture is one
form of Ruan's conjecture concerning the relation between the
geometry of a quotient singularity X/G - where X is a smooth
complex variety and G a finite group of automorphisms - and the
geometry of a crepant resolution of singularities of X/G ; it
generalizes the classical McKay correspondence. Following the
examples of the Hilbert schemes of points on surfaces and the
weighted projective spaces, I will present some of the recents
developments of the subject.

### 2009年02月19日(木)

15:50-18:00   数理科学研究科棟(駒場) 128号室
O. F. Pasarescu 氏 (Romanian Academy)
・Linear Systems on Rational Surfaces; Applications (15:50--16: 50)

・Some Applications of Model Theory in Algebraic Geometry (17:00 --18:00)

### 2008年11月26日(水)

16:30-18:00   数理科学研究科棟(駒場) 122号室
Piotr Pragacz
(Banach Institute)
Diagonal subschemes and vector bundles

### 2008年11月25日(火)

16:30-18:00   数理科学研究科棟(駒場) 118号室
Xavier Roulleau 氏 (東大)
Cotangent maps of surfaces of general type
[ 講演概要 ]
Surfaces are usualy studied and classified via the properties of the pluricanonical maps. For surfaces of general type whose cotangent sheaf is generated by global sections, we propose to study an other map, called the cotangent map, in order to obtain geometric informations on the surface. In this way, we obtain informations on the ampleness of the cotangent sheaf of such a surface. We will illustate this talk with the example of the Fano surface of lines of cubic threefolds.