Integrated research of Calabi-Yau structures and generalized complex structures

• Project/Area Number: 25287011
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Osaka University
• Principal Investigator: Goto Ryushi 大阪大学, 理学研究科, 教授 (30252571)
• Co-Investigator(Kenkyū-buntansha): 小木曽 啓示 東京大学, 数理(科)学研究科(研究院), 教授 (40224133) ; 満渕 俊樹 大阪大学, その他部局等, 名誉教授 (80116102) ; 松本 佳彦 大阪大学, 理学(系)研究科(研究院), 助教 (00710625)
• Project Period (FY): 2013-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥12,740,000 (Direct Cost: ¥9,800,000、Indirect Cost: ¥2,940,000); Fiscal Year 2017: ¥2,470,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥570,000); Fiscal Year 2016: ¥2,470,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥570,000); Fiscal Year 2015: ¥2,470,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥570,000); Fiscal Year 2014: ¥2,470,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥570,000); Fiscal Year 2013: ¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
• Keywords: 一般化された複素構造 / カラビーヤオ構造 / 一般化されたケーラー構造 / ポアソン構造 / 変形理論 / ケーラーアインシュタイン計量 / エルミートアインシュタイン計量 / モーメント写像 / 一般化された複素多様体 / 一般化されたケーラー多様体 / 一般化された接続 / アインシュタインーエルミート計量 / モジュライ空間 / 一般化されたケーラー・アインシュタイン構造 / ケーラー・アインシュタイン計量 / 一般化されたケーラー・アインシュタイン計量 / 双エルミート構造 / カラビーヤオ多様体 / K 安定性 / ケーラーーアインシュタイン計量 / モーメントマップ / 対数変換 / ツイスター空間 / 超ケーラー多様体 / シンプレクティク構造 / 複素構造

Outline of Final Research Achievements

From a view point of the moment map, we introduce the notion of Einstein-Hermitian generalized connections over a generalized Kahler manifold of symplectic type. We show that moduli spaces of Einstein- Hermitian generalized connections arise as the Kahler quotients. The deformation complex of Einstein-Hermitian generalized connections is an elliptic complex and it turns out that the smooth part of the moduli space is a finite dimensional Kahler manifold. The canonical line bundle over a generalized Kahler manifold of symplectic type has the canonical generalized connection and its curvature coincides with ”the scalar curvature as the moment map”. Kahler-Ricci solitons provide examples of Einstein-Hermitian generalized connections and Einstein Hermitian co-Higgs bundles are also discussed.

Research Products

• [Journal Article] Unobstructed deformations of generalized complex structures induced by C∞ logarithmic symplectic structures and logarithmic Poisson structures (2016)
  • Author(s): Ryushi Goto
  • Journal Title: Geometry and topology of manifolds Volume: 154 Pages: 159-183
  • Peer Reviewed
• [Journal Article] C∞-logarithmic transformations and generalized complex structures (2016)
  • Author(s): Ryushi Goto and Kenta Hayano
  • Journal Title: J. Symplectic Geom Volume: 14 Pages: 341-357
  • Peer Reviewed
• [Journal Article] C^\infty logarithmic transformations and generalized complex structures (2016)
  • Author(s): R. Goto and K. Hayano
  • Journal Title: The Journal of Symplectic Geometry Volume: -
  • Peer Reviewed
• [Journal Article] Unobstructed deformations of generalized complex structures induced by C∞ logarithmic symplectic structures and logarithmic Poisson structures (2016)
  • Author(s): R. Goto
  • Journal Title: Proceedings of China-Japan Geometry Conference: Geometry and Topology of Manifold ---The 10th Geometry Conference for the Friendship of China and Japan 2014 Volume: -
  • Peer Reviewed
• [Journal Article] C^\infty logarithmic transformations and generalized complex strucutres (2015)
  • Author(s): R. Goto and K. Hayano
  • Journal Title: J. Symplectic Geom, Volume: 未定
  • Peer Reviewed
• [Journal Article] On the stability of locally conformal Kähler structures (2014)
  • Author(s): Goto, Ryushi
  • Journal Title: Journal of the Mathematical Society of Japan Volume: 66 Issue: 4 Pages: 1375-1401
  • DOI: 10.2969/jmsj/06641375
  • NAID: 130004706004
  • ISSN: 0025-5645, 1881-1167, 1881-2333
  • Peer Reviewed
• [Journal Article] The Donaldson-Futaki invariant for sequences of test configurations (2014)
  • Author(s): T. Mabuchi
  • Journal Title: Progress in Math. Volume: 308 Pages: 395-403
  • Peer Reviewed
• [Journal Article] Some aspects of explicit birational geometry inspired by complex d ynamics (2014)
  • Author(s): K. Oguiso
  • Journal Title: Proceedings of the International Congress of Mathematicians, Volume: 2 Pages: 695-721
  • Peer Reviewed
• [Journal Article] On the stability of locally conformal Kaehler structures (2014)
  • Author(s): R. Goto
  • Journal Title: Journal of the Mathematical Society of Japan Volume: 未定
  • Peer Reviewed
• [Journal Article] New examples of Sasaki-Einstein manifolds (2013)
  • Author(s): T.Mabuchi and Y.Nakagawa
  • Journal Title: Tohoku Math. J Volume: 65 Pages: 243-252
  • NAID: 130005562143
  • Peer Reviewed
• [Journal Article] Quartic K3 surfaces and Cremona transformations (2013)
  • Author(s): K. Oguiso
  • Journal Title: in Arithmetic and Geometry of K3 Surfaces and Calabi-Yau Threefolds, Fields Institute Communications Volume: 67 Pages: 455-460
  • Peer Reviewed
• [Presentation] Scalar curvature is moment map in generalized Kahler geometry. (2017)
  • Author(s): R. Goto
  • Organizer: AMS Sectional Meeting AMS Special Session,Fall Western Sectional Meeting University of California, Riverside, Riverside, CA
  • Int'l Joint Research / Invited
• [Presentation] Holomorphic Poisson structures and generalized Kahler geometry (2017)
  • Author(s): R. Goto
  • Organizer: University of California, Riverside, Department of Mathematics, Colloquium
  • Int'l Joint Research / Invited
• [Presentation] Moduli spaces of Einstein-Hermitian generalized connections over generalized Kahler manifolds of symplectic type (2017)
  • Author(s): R. Goto
  • Organizer: The 23rd Symposium on Complex geometry (Kanazawa)
  • Int'l Joint Research / Invited
• [Presentation] Stratified flat structure on the moduli space of generalized complex surfaces (2016)
  • Author(s): Ryushi Goto
  • Organizer: HAYAMA Symposium on Complex Analysis in Several Variables XVIII
  • Place of Presentation: Hayama, Japan
  • Int'l Joint Research / Invited
• [Presentation] Scalar curvature as moment map in generalized K\"ahler geometry, (2016)
  • Author(s): Ryushi Goto
  • Organizer: The 22nd Symposium on Complex Geometry in Kanazawa
  • Place of Presentation: Kanazawa, Japan
  • Int'l Joint Research / Invited
• [Presentation] Scalar curvature as moment map in generalized Kahler geometry (2016)
  • Author(s): Ryushi Goto
  • Organizer: Generalized Geometry and Noncommutative Algebra, CMI Workshop,
  • Place of Presentation: Oxford. UK
  • Int'l Joint Research / Invited
• [Presentation] Scalar curvature as moment map in generalized Kahler geometry, (2016)
  • Author(s): Ryushi Goto
  • Organizer: The second China-Japan geometry conference at Fujian Normal University
  • Place of Presentation: Fujian, China
  • Int'l Joint Research / Invited
• [Presentation] Stratified flat structures on moduli spaces of generalized complex surfaces (2015)
  • Author(s): R. Goto
  • Organizer: The 21st Symposium on Complex Geometry
  • Place of Presentation: 金沢大学サテライトプラザ
  • Year and Date: 2015-10-30
  • Int'l Joint Research / Invited
• [Presentation] Flat structures on moduli spaces of generalized complex surfaces (2015)
  • Author(s): R. Goto
  • Organizer: 東京大学数理科学研究科複素解析幾何セミナー
  • Place of Presentation: 東京大学数理科学研究科
  • Year and Date: 2015-09-28
  • Invited
• [Presentation] 対数的ポアソン構造と一般化された複素多様体の非障害的な変形について (2015)
  • Author(s): 後藤竜司
  • Organizer: 東京工業大学大学院理工学研究科数学専攻大岡山談話会
  • Place of Presentation: 東京工業大学
  • Year and Date: 2015-07-01
  • Invited
• [Presentation] 一般化された複素多様体と変形理論 (2015)
  • Author(s): 後藤竜司
  • Organizer: 東京工業大学 集中講義
  • Place of Presentation: 東京工業大学
  • Year and Date: 2015-06-29
  • Invited
• [Presentation] Unobstructed deformations of generalized complex structures induced by logarithmic Poisson structures (2015)
  • Author(s): R. Goto
  • Organizer: 満渕俊樹教授退職記念小研究会
  • Place of Presentation: 大阪大学
  • Year and Date: 2015-03-12
• [Presentation] Unobstructed deformations of generalized complex structures induced by logarithmic Poisson structures (2014)
  • Author(s): R. Goto
  • Organizer: The 10th Geometry Conference for the Friendship between Chaina and Japan
  • Place of Presentation: Futan University, China
  • Year and Date: 2014-09-07 – 2014-09-11
  • Invited
• [Presentation] Deformations of generalized complex manifolds and locally conformally generalized kaehker structures (2014)
  • Author(s): R. Goto
  • Organizer: Complex Geometry and Lie Groups
  • Place of Presentation: Torino, Italy
  • Year and Date: 2014-06-18
  • Invited
• [Presentation] Deformations of generalized complex manifolds and locally conformally generalized Kaeher structures (2014)
  • Author(s): R. Goto
  • Organizer: Complex Geometry and Lie Groups
  • Place of Presentation: Torino 大学 イタリア
  • Invited
• [Presentation] Generalized complex structures on 4-manifolds and generalized hyperkaehler structures (2013)
  • Author(s): R. Goto
  • Organizer: The Asian MAthematical Conference 2013,
  • Place of Presentation: BEXCO, BUsan, Korea
  • Invited
• [Presentation] Generalized complex 4-manifolds and generalized Calabi-Yau metrical structure (2013)
  • Author(s): R. Goto
  • Organizer: The 8th Pacific Rim Complex Geometry Conference
  • Place of Presentation: Dalian, Liaoning, China
  • Invited
• [Presentation] Generalized complex structures and generalized Calabi-Yau metrics (2013)
  • Author(s): R. Goto
  • Organizer: Symplectic Algebraic geometry
  • Place of Presentation: Kansai seminar house, Kyoto
  • Invited
• [Presentation] Generalized contact structures and generalized Sasakian structures (2013)
  • Author(s): R. Goto
  • Organizer: Tokyo-Seoul Conference in Mathematics -Differential Geometry-
  • Place of Presentation: 東京大学数理科学研究科
  • Invited
• [Book] 倉西数学への誘い (2013)
  • Author(s): 藤木明, 倉西正武, 都丸正, 石井志保子, V. Gullemin, 倉西茂, 難波誠, 森本徹, 宮島公夫, 伊藤充弘,後藤竜司
  • Total Pages: 21
  • Publisher: 岩波書店
• [Funded Workshop] 第３回日中幾何学研究集会 (2017)
• [Funded Workshop] The23rd Symposium on Complex geometry (Kanazawa) (2017)
• [Funded Workshop] The 22nd Symposium on Complex Geometry in Kanazawa (2016)
  • Place of Presentation: しいのき迎賓館 金沢
  • Year and Date: 2016-11-01
• [Funded Workshop] The second China-Japan geometry conference at Fujian Normal University, (2016)
  • Place of Presentation: Fujian noram UNiversity, Fujian China
  • Year and Date: 2016-09-06
• [Funded Workshop] The 4th Workshop “Complex geometry and Lie Groups” (2016)
  • Place of Presentation: 奈良女子大学（奈良）
  • Year and Date: 2016-03-22
• [Funded Workshop] The 21st Symposium on Complex Geometry (2015)
  • Place of Presentation: 金沢大学サテライトプラザ（金沢）
  • Year and Date: 2015-10-28
• [Funded Workshop] Japan-China Geometry Conference, (2015)
  • Place of Presentation: 京都大学（京都）、奈良女子大学（奈良）
  • Year and Date: 2015-09-07