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現代数学シリーズ連接層の導来圏と代数幾何学(電子書籍版)

出版社：	丸善出版
著者：	上原北斗(著) 戸田幸伸(著) 中村周(編) 山田澄生(編)
発行日：	2021-01-25
分野：	その他 > 一般
ISBN：	9784621305911
電子書籍版：	2021-01-25 （電子書籍版）

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「アーベル圏の導来圏」はアーベル圏の対象の複体からなる三角圏であり、1960年代にグロタンディックとヴェルディエにより導入された概念である。以来、導来圏は活発に研究されており、現在では代数幾何はもとより数論幾何学・代数解析・表現論・数理物理などで有用な道具、言語として活躍し大きな発展に寄与している。本書は代数多様体上の連接層の導来圏に関し、特に重要な発展と思われるトピックを選び、最新の諸研究結果までを概説した。できる限りself-containedに、また本質を見失わない程度に簡単な設定にした上で、各トピックについて核となるアイディアが現れるよう展開されている。

表紙
序文
目次
第1章本書の構成と記号
1.1 本書の構成
1.2 記号
第2章導来圏の定義と性質
2.1 なぜ導来圏を考えるのか ?
2.2 Abel圏の複体のホモトピー圏
2.3 三角圏
2.4 コホモロジー的関手
2.5 三角圏の局所化
2.6 Abel圏の導来圏
2.7 導来関手
2.8 代数多様体上の連接層の導来圏
2.9 導来関手の性質
2.10 Fourier - Mukai変換
第3章三角関手の同値性の判定条件
3.1 随伴関手とSerre関手
3.2 三角圏の生成元
3.3 三角関手の充満忠実性, 同値性の判定条件
第4章導来圏の半直交分解と例外生成列
4.1 代数多様体の導来圏の不変量
4.2 半直交分解
4.3 例外生成列
4.4 例外生成列を持つ代数多様体
4.5 傾斜対象
4.6 半直交分解および例外列の変異
4.7 お化けが出た !
第5章安定層のモジュライ空間とFourier - Mukai変換
5.1 代数曲線上の ( 半 ) 安定ベクトル束
5.2 高次元代数多様体上の ( 半 ) 安定層
5.3 ( 半 ) 安定層のモジュライ空間
5.4 ( 半 ) 安定層のモジュライ空間の構成
5.5 安定層の局所変形理論
5.6 普遍層による導来同値
5.7 K3曲面上の安定層のモジュライ空間
5.8 相対的安定層のモジュライ空間
5.9 Pfaffian - Grassmannian導来同値
5.10 捻れ層とFourier - Mukai変換
第6章導来McKay対応
6.1 クレパント解消 ( CCR ) とフロップ
6.2 G - Hilb ( M )
6.3 交叉定理
6.4 導来McKay対応
第7章フロップによる導来同値
7.1 t - 構造と捩れ対
7.2 一般的なひねくれ者
7.3 ひねくれた同僚
7.4 ひねくれた妻
7.5 Van den Berghによるフロップの導来同値
7.6 非可換クレパント解消 ( NCCR )
7.7 なぜNCCRと呼ぶのか ?
7.8 NCCR vs. CCR
7.9 フロップ - フロップ関手
7.10 非可換変形とフロップ
第8章連接層の導来圏と行列因子化
8.1 完全圏の導来圏
8.2 Serre商とVerdier商
8.3 次数付き加群の商圏
8.4 Gorenstein多様体の導来圏
8.5 行列因子化の圏と極大Cohen - Macaulay加群の安定圏
8.6 gauged LG模型の因子化の圏
8.7 行列因子化の圏と特異点の三角圏
8.8 行列因子化の圏とCalabi - Yau超曲面の導来圏
第9章ホモロジー的射影双対理論
9.1 Lefschetz分解
9.2 普遍超平面
9.3 ホモロジー的射影双対
9.4 2次超曲面のHPD
9.5 圏論的特異点解消
9.6 Grassmann多様体のHPD
9.7 4次元3次超曲面の導来圏
9.8 長方形型Lefschetz分解に関するHPDの証明
第10章連接層の導来圏と幾何学的不変式論
10.1 幾何学的不変式論
10.2 導来圏とGIT ( アファイン空間の場合 )
10.3 導来圏とGIT ( 一般の場合 )
10.4 トーリックDeligne - Mumfordスタック
10.5 トーリックDeligne - Mumfordスタックと例外生成列
10.6 特異点の三角圏とGIT
第11章 Bridgeland安定性条件
11.1 Abel圏の安定性条件
11.2 三角圏の安定性条件
11.3 安定性条件の空間
11.4 楕円曲線上の安定性条件の空間
11.5 ミラー対称性との関係
11.6 極大体積極限の近傍
11.7 連接層のなすAbel圏の傾斜
11.8 代数曲面上の安定性条件
11.9 代数曲面の極大体積極限
11.10 壁と部屋の構造
11.11 K3曲面上の安定性条件の空間
11.12 3次元代数多様体上の安定性条件
11.13 Bogomolov - Gieseker型不等式予想
第12章 Donaldson - Thomas不変量
12.1 3次元Calabi - Yau多様体上の安定層のモジュライ空間
12.2 完全障害理論
12.3 仮想サイクル
12.4 Behrend関数
12.5 曲線を数え上げるDT不変量
12.6 君のママは僕のママ ( GW / DT対応 )
12.7 安定対不変量
12.8 君のママもか ! ( DT / PT対応 )
12.9 ArtinスタックのGrothendieck群
12.10 モチーフ的Hall代数
12.11 道代数のモチーフ的Hall代数
12.12 Poisson代数
12.13 イプシロン関数
12.14 一般化DT不変量
12.15 Poissonトーラス
12.16 一般化DT不変量の壁越え
12.17 君のママが僕のママであるために
付録A Grothendieck双対性
A.1 双対化複体
A.2 Cohen - Macaulay加群とGorenstein環
A.3 標準層
付録B 代数上の加群
B.1 有限次元代数
B.2 箙
B.3 箙上の道代数
B.4 箙上の表現
付録C 代数群の代数多様体への作用
C.1 代数群の座標環への作用
C.2 G - 同変層
C.3 代数群のベクトル束への作用
付録D 商スタック
D.1 スタック
D.2 2 - 圏の補足
D.3 スタックとしてのスキーム
D.4 Artinスタック, Deligne - Mumfordスタック
D.5 商スタック
参考文献
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