ＩＳＳＮ 18802818

数理解析研究所講究録 1605

Ｇｅｏｍｅｔｒｙ ｒｅｌａｔｅｄ ｔｏ ｔｈｅ ｔｈｅｏｒｙ ｏｆ
ｉｎｔｅｇｒａｂｌｅ ｓｙｓｔｅｍｓ
可積分系に関わる幾何学の研究

京都大学数理解析研究所

2008 年 6 月

RIMS K6kyOroku 1605

Geomet7y related to the theoiy ofintegrable systemsu]tzne, 2008

Research Znstitute for Mathematical SciencesI¡yoto Untverszty, KYoto, ,lapan
This is a report of research done at Research Institute for Mathematical
Sciences, Kyoto University The papers contained herein are m final fbrm
and will not be submitted for publication elsewherePreface

The RIMS workshop entitled ”Geometry related to the theory of inte-grable systems” was held at RIMS Kyoto University from S eptember 25 toSeptember 28 2007 with the assistance of Osaka City Umversity Advanced

Mathematical Institute, fo11owed by a ”Part II” workshop at Nara-gaku

Semmar House

This volume consists mainly of survey articles of the talks given in ”PartI” at RIMS In these articles the key aspects of various important results aredescribed, leading the reader back to the relevant onginal papers where necessary The topics cover arather wide class of subjects in geometry related

to integrable systems’ the geometry of character varieties, quantum coho-mology theory, quantum Painleve equations, non-commutative mtegrablesystems, affine Kac-Moody algebras, Einstein homogeneous manifolds, andsurface theory Two speakers have already published then results in otherpubhcations, the references are as follows

K Iwasakr andT Uehara’ Geometry ofPaznleve’ equatzons, RIMS Kokyuroku

Bessatsu B2 (2007), 73-88, and Math Ann 338 (2007), 295–345

M Micallef and J Gray The work of Jesse Douglas on Mznzmal Su7:fttces,
Bulletin of the American Mathematical Society, 45 (2008), 293-302

NMe have organized several international conferences on similar themes,

starting with the MSJ-IRI (Mathematical Society of Japan Internati onalResearch Institute) ”Integrable Systems in Differential Geometry” held at

the University of ’Tbkyo in 2000 It is our pleasure to promote research inthis area and encourage the participation of many young mathematicians,

and we intend to contmue this direction in the future
I vyouid Iike to express my sincere gratitude to all the speakers and theparticipants for their contnbutions in making the workshop successfu1

June 6, 2008

Reiko Miyaoka

Ｇｅｏｍｅｔｒｙ ｒｅｌａｔｅｄ ｔｏ ｔｈｅ ｔｈｅｏｒｙ ｏｆ ｍｔｅｇｒａｂｌｅ ｓｙｓｔｅｍｓ可積分系に関わる幾何学の研究
ＲＩＭＳ研究集会報告集

2007 年 9 月 25 日〜9 月 28 日

研究代表者 宮岡 礼子（Ｒｅｌｋｏ Ｍｌｙａｏｋａ）
目次

1 ＧＥＯＭＥＴＲＹ ＯＦ ＣＨＡＲＡＣＴＥＲ ＶＡＲＩＥＴＩＥＳ ＯＦ ＳＵＲＦＡＣＥ ＧＲＯＵＰＳ 一ｅ ｅ一一 一 一一一一一 1
ＵＣａｌｌｆ・Ｄａｖｌｓ 村瀬 元彦（Ｍｏｔｏｈｌｃｏ Ｍｕｌａｓｅ）
2 量子コホモロジーの整構造について一・・一一・一一一一……一一一一一ｄｄ一一一一一一 一…一…一一一…一一…一一一一…22

九大・数理学（Ｋｙｕｓｈｕ Ｕ） 入谷 寛（Ｈｉｒｏｓｈｉ ｌｒｉｔａｍ）
3 Ｎｏｎｃｏｍｍｕｔａｔｉｖｅ ｌｎｔｅｇｒａｂｌｅ Ｓｙｓｔｅｍｓ ａｎｄ Ｔｗｉｓｔｏｒ Ｇｅｏｍｅｔｒｙ 一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一 33

名大・多元数理科学（Ｎａｇｏｙａ Ｕ） 浜中 真志（Ｍａｓａｓｈｌ Ｈａｍａｎａｋａ）4 Ｑｕａｎｔｕｍ Ｐａｍｌｅｖｅ ｅｑｕａｔｉｏｎｓ 一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一 53

慶磨大・理工（Ｋｅｌｏ Ｕ） 名古屋 創（ＨａＪｉｍｅ Ｎａｇｏｙａ）
5 Ｒｅａｌ ｆｏｒｍｓ ａｎｄ ｆｉｎｉｔｅ ｏｒｄｅｒ ａｕｔｏｍｏｒｐｈｉｓｍｓ ｏｆ ａｆｆｉｎｅ Ｋａｃ−Ｍｏｏｄｙ ａｌｇｅｂｒａｓ
一 ａｎ ｏｕｔｈｎｅ ｏｆ ａ ｎｅｗ ａｐｐｒｏａｃｈ 一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一 68
ＵＡｕｇｓｂｕｒｇ Ｅｒｎｓｔ Ｈｅｍｔｚｅ

6 Ｄｒａｗｍｇ ｔｈｅ ｃｏｍｐｌｅｘ ｐｒｏＪ ｅｃｔｉｖｅ ｓｔｒｕｃｔｕｒｅｓ ｏｎ ｏｎｃｅ−ｐｕｎｃｔｕｒｅｄ ｔｏｎ 一一一一一一一一一一一一一一一一一一一一一一一一一一一 81大阪市大・理学（Ｏｓａｋａ Ｃｌｔｙ Ｕ） 小森 洋平（Ｙｏｈｅｌ Ｋｏｍｏｒｌ）
7 ＥＩＮＳＴＥＩＮ ＨＯＭＯＧＥＮＥＯＵＳ ＭＡＮＩＦＯＬＤＳ ＡＮＤ

ＧＥＯＭＥＴＲＩＣ】ＮＶＡＲＩＡＮＴ ＴＨＥＯＲＹ 一・・一一一一一一一一一一一一一一一一一一一一一一一一一一・一一一一 90

ＵＮａｃｉｏｎａｌ Ｃ 6 ｒｄｏｂａ Ｊｏｒｇｅ Ｌａｕｒｅｔ
8 ＣＭＣ−ｔｎｎｏｉｄｓ ｗｉｔｈ ｅｍｂｅｄｄｅｄ ｅｎｄｓ ａ ｃｌｏｓｅｒ ｌｏｏｋ 一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一 99
ＴＵ ＭＵｎｃｈｅｎ Ｊ Ｄｏｒｆｍｅｉｓｔｅｒｉ, Ｐｈ Ｌａｎｇ

9 Ｒｅａｌ ｆｏｒｍｓ ｏｆ ｃｏｍｐｌｅｘ ｓｕｒｆａｃｅｓ ｏｆ ｃｏｎｓｔａｎｔ ｍｅａｎ ｃｕｒｖａｔｕｒｅ 一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一 120

東京電機大・情報環境（Ｔｏｋｙｏ Ｄｅｎｋｌ Ｕ） 小林 真平（Ｓｈｌｍｐｅｌ Ｋｏｂａｙａｓｈ 1）
1 Ｏ Ｓｕｒｆａｃｅｓ ｍ ｔｈｒｅｅ−ｄｉｍｅｎｓｉｏｎａｌ Ｌｉｅ ｇｒｏｕｐｓ ｍ ｔｅｒｍｓ ｏｆ ｓｐｍｏｒｓ 一一一一一一一一一一一一一一一一一一一一一一一一一一一一 133
ＵＮｏｖｏｓｉｂｉｒｓｋ ｌｓｋａｎｄｅｒ Ａ Ｔａｉｍａｎｏｖ

1 1 Ｓｏｍｅ Ｒｅｃｅｎｔ Ｄｅｖｅｌｏｐｍｅｎｔｓ ｉｎ ｔｈｅ ｓｔｕｄｙ ｏｆｍｍｉｍａｌ 2 −ｓｐｈｅｒｅｓ ｍ ｓｐｈｅｒｅｓ 一一一一一一一一一一一一一一 151
ＵＤｕｒｈａｍ Ｊ Ｂｏｌｔｏｎ

Ｂｒｏｎｘ Ｃｏｍｍｕｎｉｔｙ Ｃｏｌｌ Ｌ Ｆｅｒｎａｎｄｅｚ
ＵＬｅｅｄｓ Ｊ Ｃ Ｗｏｏｄ

1 2 Ｋｎｏｔｓ ａｎｄ ｍｍｉｍａｌ ｓｕｒｆａｃｅｓ 一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一 一一一一 170

大阪市大・数学研（Ｏｓａｋａ Ｃｌｔｙ Ｕ） 田中 利史（Ｔｏｓｈ血ｍｌ Ｔａｎａｋａ）