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Study of canonical divisors on higher dimensional algebraic variety

Research Project

Project/Area Number 22244002
Research Category

Grant-in-Aid for Scientific Research (A)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionThe University of Tokyo

Principal Investigator

Kawamata Yujiro  東京大学, 数理(科)学研究科(研究院), 教授 (90126037)

Project Period (FY) 2010-04-01 – 2016-03-31
Project Status Completed (Fiscal Year 2015)
Budget Amount *help
¥45,630,000 (Direct Cost: ¥35,100,000、Indirect Cost: ¥10,530,000)
Fiscal Year 2014: ¥7,930,000 (Direct Cost: ¥6,100,000、Indirect Cost: ¥1,830,000)
Fiscal Year 2013: ¥9,100,000 (Direct Cost: ¥7,000,000、Indirect Cost: ¥2,100,000)
Fiscal Year 2012: ¥9,100,000 (Direct Cost: ¥7,000,000、Indirect Cost: ¥2,100,000)
Fiscal Year 2011: ¥9,100,000 (Direct Cost: ¥7,000,000、Indirect Cost: ¥2,100,000)
Fiscal Year 2010: ¥10,400,000 (Direct Cost: ¥8,000,000、Indirect Cost: ¥2,400,000)
Keywords代数多様体 / 標準因子 / 導来圏 / 極小モデル / トーリック多様体 / DK予想 / アバンダンス予想 / 非可換変形 / 単純組 / 例外対象 / 球面対象 / 収縮写像 / 重み付き射影空間 / 商スタック / 相対的例外対象 / 半直交分解 / 森夢空間 / コックス環 / カラビヤウ / ファノ / クレパント / フロップ / 正値性定理 / 正規交差多様体 / 混合ホッジ構造 / 小平次元 / 数値的小平次元 / 半豊富 / 対数的標準因子
Outline of Final Research Achievements

I described the change of the derived categories under the minimal model program in the case of toric varieties, and proved the DK conjecture in this case. I gave similar description for the divisorial extractions, and proved the derived McKay correspondence for finite abelian groups. I also proved the derived McKay correspondence for any finite subgroup of general linear group in the case of dimension 2, and described the semi-orthogonal complements.
I proved the abundance conjecture in the case of numerical Kodaira dimension 0. I proved that a Mori dream space is of Calabi-Yau type if and only if the Cox ring has only log canonical singularities. I extended the semi-positivity theorem of the Hodge bundle in the case of reducible fibers. I described the decomposition of the space of boundary divisors according to the minimal models and the canonical models.

Report

(6 results)
  • 2015 Final Research Report ( PDF )
  • 2014 Annual Research Report
  • 2013 Annual Research Report
  • 2012 Annual Research Report
  • 2011 Annual Research Report
  • 2010 Annual Research Report
  • Research Products

    (41 results)

All 2016 2015 2014 2013 2012 2011 2010 Other

All Journal Article (13 results) (of which Peer Reviewed: 11 results,  Acknowledgement Compliant: 1 results) Presentation (19 results) (of which Invited: 13 results) Book (3 results) Remarks (6 results)

  • [Journal Article] Derived categories of toric varieties III2016

    • Author(s)
      Y. Kawamata
    • Journal Title

      European Journal of Mathematics

      Volume: 2 Issue: 1 Pages: 196-207

    • DOI

      10.1007/s40879-015-0065-1

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Mori dream spaces of Calabi-Yau type and log canonicity of Cox rings2015

    • Author(s)
      Yujiro Kawamata and Shinnosuke Okawa
    • Journal Title

      Journal für die reine und angewandte Mathematik

      Volume: 701 Issue: 701 Pages: 195-203

    • DOI

      10.1515/crelle-2013-0029

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Variation of mixed Hodge structures and the positivity for algebraic fiber spaces2015

    • Author(s)
      Y. Kawamata
    • Journal Title

      Adv. St. Pure Math.

      Volume: 65 Pages: 27-58

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Variation of mixed Hodge structures and the positivity for algebraic fiber spaces2015

    • Author(s)
      Y. Kawamata
    • Journal Title

      Adv. St. Pure Math.

      Volume: 未定

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Mori dream spaces of Calabi-Yau type and the log canonicity of the Cox rings2015

    • Author(s)
      Y. Kawamata and S. Okawa
    • Journal Title

      J. reine angew. Math.

      Volume: 未定

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Hodge theory on generalized normal crossing varieties2014

    • Author(s)
      Y. Kawamata
    • Journal Title

      Proc. Edinburgh Math. Soc.

      Volume: 57-1 Pages: 175-189

    • Related Report
      2014 Annual Research Report 2013 Annual Research Report 2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Derived categories of toric varieties II2013

    • Author(s)
      Y. Kawamata
    • Journal Title

      Michigan Math. J.

      Volume: 62 Pages: 353-363

    • Related Report
      2013 Annual Research Report 2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the abundance theorem in the case of numerical Kodaira dimension zero2013

    • Author(s)
      Y. Kawamata
    • Journal Title

      Amer. J. Math.

      Volume: 135 Pages: 115-124

    • Related Report
      2013 Annual Research Report 2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Semipositivity theorem for reducible algebraic fiber spaces2011

    • Author(s)
      Yujiro Kawamata
    • Journal Title

      Pure App. Math. Quarterly

      Volume: 7 Pages: 1427-1447

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Remarks on the cone of divisors2011

    • Author(s)
      Yujiro Kawamata
    • Journal Title

      EMS Series of Congress Reports

      Volume: 3 Pages: 317-325

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Remarks on the cone of divisors2011

    • Author(s)
      Yujiro Kawamata
    • Journal Title

      Classification of Algebraic Varieties

      Pages: 317-325

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Kodaira dimension and vanishing, a hommage to Eckart Viehweg (1948-2010)2010

    • Author(s)
      Yujiro Kawamata
    • Journal Title

      La Gazette des mathematician, Societe mathematique de France

      Volume: 125 Pages: 119-123

    • Related Report
      2010 Annual Research Report
  • [Journal Article] Derived categories and minimal models2010

    • Author(s)
      Yujiro Kawamata
    • Journal Title

      Sugaku Expositions

      Volume: 23 Pages: 235-259

    • Related Report
      2010 Annual Research Report
  • [Presentation] On some derived McKay correspondences2015

    • Author(s)
      Y. Kawamata
    • Organizer
      Categorical and Analytic Invariants in Algebraic Geometry II
    • Place of Presentation
      Kavli IPMU, University of Tokyo
    • Year and Date
      2015-11-16
    • Related Report
      2014 Annual Research Report
  • [Presentation] Multi-pointed NC deformations and CY 3folds2015

    • Author(s)
      Y. Kawamata
    • Organizer
      Categorical and Analytic Invariants in Algebraic Geometry I.
    • Place of Presentation
      Steklov Mathematical Institute, Moscow
    • Year and Date
      2015-09-14
    • Related Report
      2014 Annual Research Report
  • [Presentation] Derived McKay correspondences for finite abelian group quotients2015

    • Author(s)
      Y. Kawamata
    • Organizer
      Derived Categories and Birational Geometry
    • Place of Presentation
      University of Warwick, UK
    • Year and Date
      2015-06-01
    • Related Report
      2014 Annual Research Report
  • [Presentation] Derived categories of toric varieties2015

    • Author(s)
      Y. Kawamata
    • Organizer
      Algebraic Geometry and Applications to Physics and Dynamics
    • Place of Presentation
      Euler International Mathematical Institute, Saint Petersburg
    • Year and Date
      2015-05-25
    • Related Report
      2014 Annual Research Report
  • [Presentation] Remarks on non-commutative deformations of sheaves2015

    • Author(s)
      Y. Kawamata
    • Organizer
      Algebraic Geometry
    • Place of Presentation
      University of Bayreuth, Germany
    • Year and Date
      2015-03-12
    • Related Report
      2014 Annual Research Report
  • [Presentation] Derived category of a weighted projective space2013

    • Author(s)
      Y. Kawamata
    • Organizer
      Birational Geometry of Complex Algebraic Varieties dedicated to F. Campana
    • Place of Presentation
      Luminy, France
    • Year and Date
      2013-11-07 – 2013-11-11
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] The derived category of toric varieties2013

    • Author(s)
      Y. Kawamata
    • Organizer
      University of Milano, Algebraic Geometry Seminar
    • Place of Presentation
      University of Milano
    • Year and Date
      2013-10-01
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] Toric MMP and SOD2013

    • Author(s)
      Y. Kawamata
    • Organizer
      Toric MMP and SOD
    • Place of Presentation
      University of Catania, Italy
    • Year and Date
      2013-09-16 – 2013-09-27
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] On tilting generators2013

    • Author(s)
      Y. Kawamata
    • Organizer
      Classification of Algebraic Varieties and Related Topics
    • Place of Presentation
      Cetraro, Italy
    • Year and Date
      2013-09-08 – 2013-09-15
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] Derived categories in algebraic geometry2013

    • Author(s)
      Y. Kawamata
    • Organizer
      The 6th Pacific Rim Conference on Mathematics
    • Place of Presentation
      Sapporo
    • Year and Date
      2013-07-03 – 2013-07-06
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] Some remarks on tilting generators2013

    • Author(s)
      Y. Kawamata
    • Organizer
      Birational Geometry and GIT
    • Place of Presentation
      Wien
    • Year and Date
      2013-05-21 – 2013-05-24
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] Minimal models and Fourier-Mukai partners2012

    • Author(s)
      Yujiro Kawamata
    • Organizer
      Conference on complex geometry in memory of Hans Grauert
    • Place of Presentation
      Goettingen大学(ドイツ)
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] Derived categories from the viewpoint of the minimal model program2012

    • Author(s)
      Yujiro Kawamata
    • Organizer
      Algebraic and Complex Geometry
    • Place of Presentation
      Hannover大学(ドイツ)
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] Birational geometry and derived categories2012

    • Author(s)
      Yujiro Kawamata
    • Organizer
      Interactions of birational geometry with other fields
    • Place of Presentation
      Durham大学(イギリス)
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] Triangulated Categories in Algebraic Geometry2011

    • Author(s)
      Yujiro Kawamata
    • Organizer
      Triangulated categories and applications
    • Place of Presentation
      Banff International Research Station
    • Related Report
      2011 Annual Research Report
    • Invited
  • [Presentation] A remark on the abundance conjecture2011

    • Author(s)
      Yujiro Kawamata
    • Organizer
      Birational Geometry Day
    • Place of Presentation
      University of Cambridge
    • Related Report
      2011 Annual Research Report
    • Invited
  • [Presentation] On derived categories of toric varieties2011

    • Author(s)
      Yujiro Kawamata
    • Organizer
      Algebraic Geometry in East Asia
    • Place of Presentation
      National Taiwan University
    • Related Report
      2011 Annual Research Report
    • Invited
  • [Presentation] Derived categories in algebraic geometry2011

    • Author(s)
      Yujiro Kawamata
    • Organizer
      Algebraic Geometry Conference
    • Place of Presentation
      Chulalongkorn University
    • Related Report
      2011 Annual Research Report
    • Invited
  • [Presentation] Abundance theorem in the case of numerical Kodaira dimension O2010

    • Author(s)
      Yujiro Kawamata
    • Organizer
      Conference on complex geometry, group actions and moduli spaces
    • Place of Presentation
      Hyderabad大学(インド)
    • Year and Date
      2010-08-14
    • Related Report
      2010 Annual Research Report
  • [Book] 高次元代数多様体論2014

    • Author(s)
      川又雄二郎
    • Total Pages
      304
    • Publisher
      岩波書店
    • Related Report
      2014 Annual Research Report 2013 Annual Research Report
  • [Book] Derived Categories in Algebraic Geometry2013

    • Author(s)
      Y. Kawamata, ed.
    • Total Pages
      354
    • Publisher
      European Mathematical Society
    • Related Report
      2013 Annual Research Report
  • [Book] Derived Categories in Algebraic Geometry2013

    • Author(s)
      Yujiro Kawamata (editor)
    • Total Pages
      354
    • Publisher
      European Mathematical Society
    • Related Report
      2012 Annual Research Report
  • [Remarks] Homepage of Kawamata Laboratory

    • URL

      http://faculty.ms.u-tokyo.ac.jp/~kawamata_lab/

    • Related Report
      2014 Annual Research Report
  • [Remarks] 川 又 雄 二 郎 ( KAWAMATA Yujiro )

    • URL

      http://www.ms.u-tokyo.ac.jp/teacher/kawamata.html

    • Related Report
      2014 Annual Research Report
  • [Remarks] homepage of kawamata laboratory

    • URL

      http://faculty.ms.u-tokyo.ac.jp/~kawamata_lab/

    • Related Report
      2013 Annual Research Report
  • [Remarks] Homepage of Kawamata laboratory

    • URL

      http://faculty.ms.u-tokyo.ac.jp/~kawamata_lab/

    • Related Report
      2012 Annual Research Report
  • [Remarks] Homepage of Kawamata laboratory

    • URL

      http://faculty.ms.u-tokyo.ac.jp/~kawamata_lab/

    • Related Report
      2011 Annual Research Report
  • [Remarks]

    • URL

      http://faculty.ms.u-tokyo.ac.jp/~kawamata_lab/

    • Related Report
      2010 Annual Research Report

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Published: 2010-08-23   Modified: 2019-07-29  

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