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特異エルミート計量とザリスキー分解及びアバンダンス予想に関する研究

Research Project

Project/Area Number 13J02869
Research Category

Grant-in-Aid for JSPS Fellows

Allocation TypeSingle-year Grants
Section国内
Research Field Geometry
Research InstitutionThe University of Tokyo

Principal Investigator

小池 貴之  東京大学, 数理科学研究科, 特別研究員(PD)

Project Period (FY) 2013-04-01 – 2016-03-31
Project Status Completed (Fiscal Year 2015)
Budget Amount *help
¥2,970,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥270,000)
Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2014: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2013: ¥900,000 (Direct Cost: ¥900,000)
Keywords極小特異エルミート計量 / 上田理論 / レヴィ平坦葉層 / ネフ直線束 / 数値的半正 / アバンダンス予想 / 最小特異計量 / 複素管状近傍
Outline of Annual Research Achievements

本年度も昨年度に引き続き, 滑らかな複素代数多様体上のネフ直線束上の半正曲率を持つ(特異)エルミート計量に関する研究, 特に極小特異エルミート計量に関する研究を行った. 本年度の研究では昨年度までに得た知見を反映し, 複素曲面に埋め込まれた滑らかなコンパクト曲線の近傍の複素構造の非自明さ反映する理論である上田理論の一般化,及びその応用についての研究を行っい, それを応用するという立場を取った.
まず, 埋め込まれた滑らかなコンパクト曲線に, ある程度の特異性を認めた設定での上田理論の類似・一般化を行った. 特に, コンパクト曲線が被約であり, 単純正規交差特異点のみを認める場合について, その近傍の複素解析的性質を調べた. さらに被約でない場合についても, いくつかの具体例については, この結果が応用できることを示した. これらの結果の応用として, 射影平面の任意の9点配置での爆発の, 反標準束の極小特異エルミート計量の研究を行った. その結果, 例えば, 爆発後の反標準束が半正でない (つまり, 滑らかで半正曲率を持つようなエルミート計量を許容しえない) ような, 射影平面上の9点配置の存在が分かった.
また, 上田理論の微分位相幾何学的な応用に関する研究も行った. まず, 複素多様体に埋め込まれている超曲面がコンパクトとは限らない設定での上田理論についての研究を行った. その応用として, ある種の(葉ごとに正則構造を持つ) 葉層構造付多様体の, レヴィ平坦超曲面としての実現可能性の問題についての研究も行った. その結果, Barrett, 稲葉両氏によって (オリジナルの上田の定理の応用として) 得られていた, Reeb葉層等に関するレヴィ平坦実現不可能性定理を, コンパクト葉を持たない葉層についても得ることができた.

Research Progress Status

27年度が最終年度であるため、記入しない。

Strategy for Future Research Activity

27年度が最終年度であるため、記入しない。

Report

(3 results)
  • 2015 Annual Research Report
  • 2014 Annual Research Report
  • 2013 Annual Research Report
  • Research Products

    (30 results)

All 2016 2015 2014 2013 Other

All Journal Article (4 results) (of which Peer Reviewed: 4 results,  Acknowledgement Compliant: 1 results) Presentation (25 results) (of which Int'l Joint Research: 5 results,  Invited: 18 results) Remarks (1 results)

  • [Journal Article] Toward a higher codimensional Ueda theory2015

    • Author(s)
      T. Koike
    • Journal Title

      Math. Z.

      Volume: 281 Issue: 3-4 Pages: 967-991

    • DOI

      10.1007/s00209-015-1516-6

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Minimal singular metrics of a line bundle admitting no Zariski-decomposition

    • Author(s)
      KOIKE, Takayuki
    • Journal Title

      Tohoku Math. J. (2)

    • Description
      (掲載確定)
    • Related Report
      2014 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On minimal singular metrics of certain class of line bundles whose section ring is not finitely generated

    • Author(s)
      KOIKE, Takayuki
    • Journal Title

      Ann. Inst. Fourier(Grenoble)

    • Description
      (掲載確定)
    • Related Report
      2014 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the minimality of canonically attached singular Hermitian metrics on certain nef line bundles

    • Author(s)
      KOIKE, Takayuki
    • Journal Title

      Kyoto J. Math

    • Description
      (掲載確定)
    • Related Report
      2014 Annual Research Report
    • Peer Reviewed
  • [Presentation] On some analogues of Ueda theory and their applications2016

    • Author(s)
      T. Koike
    • Organizer
      Komplex Analysis Winter School-KAWA 7
    • Place of Presentation
      the Grand Hotel d'Orleans (アルビ・フランス)
    • Year and Date
      2016-03-07
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On some analogues of Ueda theory and their applications2016

    • Author(s)
      T. Koike
    • Organizer
      複素解析幾何学のポテンシャル論的諸相
    • Place of Presentation
      東京大学 (東京都・目黒区)
    • Year and Date
      2016-02-12
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] On a higher codimensional analogue of Ueda theory and its applications2016

    • Author(s)
      T. Koike
    • Organizer
      九州産業大学数学教室セミナー
    • Place of Presentation
      九州産業大学 (福岡県・福岡市)
    • Year and Date
      2016-01-13
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] On some analogues of Ueda theory and their applications2015

    • Author(s)
      T. Koike
    • Organizer
      複素解析的ベクトル場・葉層構造とその周辺
    • Place of Presentation
      京都教育大学 (京都府・京都市)
    • Year and Date
      2015-12-11
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] On a neighborhood of a torus leaf of a certain class of holomorphic foliations on complex surfaces2015

    • Author(s)
      T. Koike
    • Organizer
      複素幾何シンポジウム
    • Place of Presentation
      金沢大学 (石川県・金沢市)
    • Year and Date
      2015-10-28
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Toward a higher codimensional Ueda theory2015

    • Author(s)
      T. Koike
    • Organizer
      日本数学会年会函数論分科会
    • Place of Presentation
      京都産業大学 (京都府・京都市)
    • Year and Date
      2015-09-13
    • Related Report
      2015 Annual Research Report
  • [Presentation] On some analogues of Ueda theory and their applications2015

    • Author(s)
      T. Koike
    • Organizer
      第62回幾何学シンポジウム
    • Place of Presentation
      東京理科大学 (東京都・新宿区)
    • Year and Date
      2015-08-27
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] On some analogues of Ueda theory and their applications2015

    • Author(s)
      T. Koike
    • Organizer
      Young Mathematicians Workshop on Several Complex Variables 2015
    • Place of Presentation
      同済大学 (上海・中国)
    • Year and Date
      2015-08-10
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On some analogues of Ueda theory and their applications2015

    • Author(s)
      T. Koike
    • Organizer
      2015 Seoul Workshop on Complex Geometry and Analysis
    • Place of Presentation
      Seoul National University (ソウル・韓国)
    • Year and Date
      2015-07-21
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On the semi-positivity of a nef line bundle and the neighborhood of the stable base loci2015

    • Author(s)
      T. Koike
    • Organizer
      2015 Beijing-Shanghai Workshop on Algebraic Geometry
    • Place of Presentation
      復旦大学 (上海・中国)
    • Year and Date
      2015-06-18
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On the minimality of canonically attached singular Hermitian metrics on certain nef line bundles2015

    • Author(s)
      小池貴之
    • Organizer
      日本数学会年会函数論分科会
    • Place of Presentation
      明治大学(東京都千代田区)
    • Year and Date
      2015-03-21 – 2015-03-24
    • Related Report
      2014 Annual Research Report
  • [Presentation] Toward a higher codimensional Ueda theory2015

    • Author(s)
      小池貴之
    • Organizer
      Analytical aspects of the dbar equation
    • Place of Presentation
      名古屋大学(愛知県名古屋市)
    • Year and Date
      2015-02-13 – 2015-02-15
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] On the semi-positivity of a nef line bundle and the neighborhood of the stable base loci2014

    • Author(s)
      小池貴之
    • Organizer
      多変数関数論冬セミナー
    • Place of Presentation
      金沢大学(石川県金沢市)
    • Year and Date
      2014-12-20 – 2014-12-22
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] On the minimality of canonically attached singular Hermitian metrics on certain nef line bundles2014

    • Author(s)
      小池貴之
    • Organizer
      第20回複素幾何シンポジウム
    • Place of Presentation
      プチホテル ゾンタック (長野県上田市)
    • Year and Date
      2014-11-05 – 2014-11-08
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] On the minimality of canonically attached singular Hermitian metrics on certain nef line bundles2014

    • Author(s)
      小池貴之
    • Organizer
      複素解析幾何セミナー
    • Place of Presentation
      東京大学(東京都目黒区)
    • Year and Date
      2014-10-27
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] On the minimality of canonically attached singular Hermitian metrics on certain nef line bundles2014

    • Author(s)
      小池貴之
    • Organizer
      代数幾何学セミナー
    • Place of Presentation
      京都大学(京都府京都市)
    • Year and Date
      2014-10-10
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] On the semi-positivity of a nef line bundle and the neighborhood of the stable base loci2014

    • Author(s)
      小池貴之
    • Organizer
      MAGIC seminor
    • Place of Presentation
      インペリアルカレッジ・ロンドン(英国・ロンドン)
    • Year and Date
      2014-09-15
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] On the semi-positivity of a nef line bundle and the neighborhood of the stable base loci2014

    • Author(s)
      小池貴之
    • Organizer
      Young Mathematician Workshop on Several Complex Variables 2014
    • Place of Presentation
      浦項工科大学(大韓民国慶尚北道浦項市)
    • Year and Date
      2014-08-02 – 2014-08-05
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] On the semi-positivity of a nef line bundle and the neighborhood of the stable base loci2014

    • Author(s)
      小池貴之
    • Organizer
      Asymptotic aspects of complex and algebraic geometry
    • Place of Presentation
      ミラノ・ビコッカ大学(イタリア・ミラノ)
    • Year and Date
      2014-06-23 – 2014-06-27
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] On a relationship between semi-positivity of a nef line bundle and neighborhoods of the stable base locus2014

    • Author(s)
      小池貴之
    • Organizer
      解析幾何学セミナー
    • Place of Presentation
      名古屋大学(愛知県名古屋市)
    • Year and Date
      2014-04-21
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] On minimal singular metrics of certain class of line bundles whosesection ring is not finitely generated2014

    • Author(s)
      小池 貴之
    • Organizer
      第三回若手代数複素幾何研究集会
    • Place of Presentation
      松藤プラザ「えきまえ」いきいきひろば(長崎)
    • Related Report
      2013 Annual Research Report
  • [Presentation] On minimal singular metrics of certain class of line bundles whosesection ring is not finitely generated2014

    • Author(s)
      小池 貴之
    • Organizer
      日本数学会年会函数論分科会
    • Place of Presentation
      学習院大学(東京都豊島区)
    • Related Report
      2013 Annual Research Report
  • [Presentation] Minimal singular metrics of a line bundle admitting no Zariski decomosition2013

    • Author(s)
      小池 貴之
    • Organizer
      東大代数幾何学セミナー
    • Place of Presentation
      東京大学(東京都目黒区)
    • Year and Date
      2013-11-25
    • Related Report
      2013 Annual Research Report
  • [Presentation] Minimal singular metrics of a line bundle admitting no Zariski decomosition2013

    • Author(s)
      小池 貴之
    • Organizer
      東大数理・複素解析幾何セミナー
    • Place of Presentation
      東京大学(東京都目黒区)
    • Year and Date
      2013-10-28
    • Related Report
      2013 Annual Research Report
  • [Presentation] Singular holomorphic foliations by curves whose canonical rings have infinitely generated section rings2013

    • Author(s)
      小池 貴之
    • Organizer
      Foliation theory in algebraic geometry
    • Place of Presentation
      Simons Foundation's Gerald D. Fischbach Auditorium(アメリカ合衆国・ニューヨーク)
    • Related Report
      2013 Annual Research Report
  • [Remarks] 小池貴之のwebサイト

    • URL

      http://ktakayuki.github.io/

    • Related Report
      2015 Annual Research Report

URL: 

Published: 2014-01-29   Modified: 2024-03-26  

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