Complexes on algebraic varieties and their moduli spaces

• Project/Area Number: 26287007
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Kobe University
• Principal Investigator: Yoshioka Kota 神戸大学, 理学研究科, 教授 (40274047)
• Project Period (FY): 2014-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥11,440,000 (Direct Cost: ¥8,800,000、Indirect Cost: ¥2,640,000); Fiscal Year 2017: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000); Fiscal Year 2016: ¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000); Fiscal Year 2015: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000); Fiscal Year 2014: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000)
• Keywords: モジュライ / 複体 / K3 / 安定層 / 安定性 / ベクトル束 / Bridgeland stability

Outline of Final Research Achievements

We obtained some results on moduli of semi-stable sheaves on K3 surfaces and abelian surfaces. In particular we described the local structure of the singular points and get an example of strange duality. We also proved Kawamata-Morrison's cone conjectures for two series of irreducible symplectic manifolds of Beauville.
For moduli spaces of stable sheaves on Enriques surfaces, we described the condition for the non-emptyness. We also obtained some results on the birational geometry of the moduli spaces by using Bridgeland stability conditions.

Academic Significance and Societal Importance of the Research Achievements

安定層とそのモジュライ空間は微分幾何やYang-Mills理論（インスタントン）と関係し、様々な立場から研究されてきた。本研究では多くのよい性質をもつ曲面であるK3曲面、アーベル曲面、Enriques曲面に対し、そのモジュライ空間の性質を調べた。とくにEnriques曲面上の安定層についてはこれまで研究があまり進んでいなかったが、この研究で安定層の存在性やモジュライの既約性など基本的な問題に成果を得ることができた。

Research Products

• [Int'l Joint Research] University of Illinois at Chicago(米国)
• [Int'l Joint Research] グラスゴー大学(イギリス)
• [Int'l Joint Research] ボローニャ大学(イタリア)
• [Journal Article] Derived equivalent Hilbert schemes of points on K3 surfaces which are not birational (2019)
  • Author(s): Ciaran Meachan, Giovanni Mongardi, Kota Yoshioka
  • Journal Title: Math, Z. Volume: 出版予定
  • DOI: 10.1007/s00209-019-02281-1
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] The wall-crossing behavior for Bridgeland’s stability conditions on abelian and K3 surfaces (2018)
  • Author(s): H. Minamide, K. Yoshioka, S. Yanagida
  • Journal Title: J. reine angew. Math. Volume: 0 Pages: 1-107
  • DOI: 10.1515/crelle-2015-0010
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Moduli spaces of stable sheaves on Enriques surfaces (2017)
  • Author(s): Kota Yoshioka
  • Journal Title: Kyoto J. Math. Volume: -
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Fourier-Mukai duality for K3 surfaces via Bridgeland stability condition (2017)
  • Author(s): Kota Yoshioka
  • Journal Title: J. Geometry and Physics Volume: -
  • DOI: 10.1016/j.geomphys.2016.08.011
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] A note on stable sheaves on Enriques surfaces II (2017)
  • Author(s): Kota Yoshioka
  • Journal Title: Manuscripta Math Volume: 153
  • DOI: 10.1007/s00229-016-0882-7
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] On the strange duality conjecture for abelian surfaces II (2017)
  • Author(s): B., Bolognese, A., Marian, D., Oprea, K., Yoshioka
  • Journal Title: Ｊ． Ａｌｇeｂraic Ｇｅｏｍｅｔｒｙ Volume: 未定
  • Peer Reviewed
• [Journal Article] A note on stable sheaves on Enriques (2016)
  • Author(s): Yoshioka Kota
  • Journal Title: Tohoku Math. J. (2) Volume: 未定
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Bridgeland's stability and the positive cone of the moduli spaces of stable objects on an abelian surface (2016)
  • Author(s): Kota Yoshioka
  • Journal Title: Adv. Stud. Pure Math. Volume: 未定
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] A proof of the Kawamata-Morrison Cone Conjecture for holomorphic symplectic varieties of K3 or generalized Kummer deformation type (2015)
  • Author(s): E., Markman, K., Yoshioka
  • Journal Title: Int. Math. Res. Not. (IMRN) Volume: 24
  • DOI: 10.1093/imrn/rnv119
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Presentation] Categorical entropy for Fourier-Mukai transforms on generic abelian surfaces (2017)
  • Author(s): Kota Yoshioka
  • Organizer: Derived category and birational geometry
  • Place of Presentation: 大阪大学理学部
  • Year and Date: 2017-02-20
  • Invited
• [Presentation] Vector bundles on Enriques surfaces (2017)
  • Author(s): Kota Yoshioka
  • Organizer: 代数的層のモジュライの研究とその周辺
  • Place of Presentation: 数理解析研究所
  • Year and Date: 2017-02-01
  • Invited
• [Presentation] Moduli of stable sheaves on Enriques surfaces (2017)
  • Author(s): Yoshioka Kota
  • Organizer: Japanese-Europian Symposium on Symplectic Varieties
  • Invited
• [Presentation] Moduli spaces of stable sheaves on Enriques surfaces (2016)
  • Author(s): Kota Yoshioka
  • Organizer: Derived categories and Chow groups of hyperkaehler and Calabi-Yau varieties
  • Place of Presentation: Simons center for geometry and physics, Stony Brook, NY, USA
  • Year and Date: 2016-09-19
  • Int'l Joint Research / Invited
• [Presentation] Moduli spaces of stable sheaves on Enriques surfaces (2016)
  • Author(s): Kota Yoshioka
  • Organizer: On explicit description of holomorphic symplectic varieties,
  • Place of Presentation: 大沼国際セミナーハウス
  • Year and Date: 2016-08-29
  • Int'l Joint Research / Invited
• [Presentation] Fourier-Mukai duality for K3 surfaces (2015)
  • Author(s): Ｋ． Ｙｏｓｈｉｏｋａ
  • Organizer: Vector bundles on algebraic curves 2015
  • Place of Presentation: Ｗａｒｗｉｃｋ Ｕｎｉｖｅｒｓｉｔｙ， Ｅｎｇｌａｎｄ
  • Year and Date: 2015-06-15
  • Int'l Joint Research / Invited
• [Presentation] Twisted stability and Bridgeland stability (2015)
  • Author(s): 吉岡康太
  • Organizer: Geometry from stability conditions
  • Place of Presentation: Warwick University
  • Year and Date: 2015-02-16 – 2015-02-20
  • Invited
• [Presentation] Examples of movable divisors on a generalized Kummer variety and an application (2014)
  • Author(s): 吉岡康太
  • Organizer: 代数幾何学シンポジウム
  • Place of Presentation: 城崎国際アートセンター
  • Year and Date: 2014-10-20 – 2014-10-24
  • Invited
• [Presentation] 代数曲面上のベクトル束 (2014)
  • Author(s): 吉岡康太
  • Organizer: 第59回代数学シンポジウム
  • Place of Presentation: 東大数理科学研究科
  • Year and Date: 2014-09-08 – 2014-09-11
  • Invited
• [Funded Workshop] Algebraic Geometry and Integrable Systems, Kobe 2016 (2016)
  • Place of Presentation: Faculty of Science, Kobe University
  • Year and Date: 2016-12-05