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Construction of complex analytic theory for LC singularities and studies of positively curved varieties based on the extension problem

Research Project

Project/Area Number 19KK0342
Research Category

Fund for the Promotion of Joint International Research (Fostering Joint International Research (A))

Allocation TypeMulti-year Fund
Research Field Geometry
Research InstitutionTohoku University

Principal Investigator

Matsumura Shin-ichi  東北大学, 理学研究科, 教授 (90647041)

Project Period (FY) 2020 – 2023
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥15,080,000 (Direct Cost: ¥11,600,000、Indirect Cost: ¥3,480,000)
Keywords双有理幾何 / LC特異点 / 極小モデル理論 / アバンダンス予想 / 非消滅予想 / 正曲率性 / 調和積分論 / 葉層構造 / 単射性定理 / 正則凸多様体 / 変形族 / 混合Hodge構造 / Nakano正値性 / 基本群 / 自己同型群 / ネフ余接ベクトル束 / 第二チェーン類 / 単純正規交差因子 / 混合Hodge理論 / ファイバーの変動 / 数値的小平次元 / 数値的同値類 / Calabi-Yau多様体 / 強ネフ因子 / Serrano予想 / ハイパーケーラー多様体 / Bedford-Taylor積 / Siu分解 / Hodge理論 / Bergman核 / 順像層の弱正値性 / Lelong数 / 漸近的な基底点 / 相対的な反標準束 / 代数的ファイバー空間 / L2拡張定理 / dbar-方程式
Outline of Research at the Start

本研究では, 代数幾何(特に双有理幾何)における超越的な手法(複素解析/微分幾何の手法)を発展させ, 正則切断の拡張問題や正曲率の多様体への応用を与える. 具体的には, 双有理幾何に現れるLC特異点に対する複素解析的理論の構築を目指す. その応用として極小モデル理論におけるDLT拡張予想を大目標に正則切断のL2拡張問題を研究する. また, Bergman核や葉層構造の理論を発展させ, 正曲率の多様体に対する構造定理を研究する.

Outline of Final Research Achievements

Developing the theory of harmonic integrals for LC strata, I generalized the injectivity theorem (an extension of Kodaira's vanishing theorem) to the complex geometric setting for varieties with LC singularities, thus solving the Fujino conjecture. I also solved the abundance conjecture for minimal projective manifolds with vanishing second Chern class. Furthermore, I studied the non-vanishing conjecture in the framework of the generalized minimal model program and solved it for the nef anti-canonical bundles of three-dimensional varieties. Additionally, I clarified a relation between the geometry of algebraic fiber spaces and the positivity of relative (pluri-)anti-canonical bundles, aiming to establish structural theorems for varieties whose anti-canonical divisor or tangent vector bundle satisfies certain positivity.

Academic Significance and Societal Importance of the Research Achievements

代数幾何学(特に双有理幾何学)では, Birkar-Cascini-Hacon-McKernanの大結果以降, 半正値性と特異点の重要性が増している. 本研究は, 特異点と半正値性に対する超越的な手法を発展させた点で意義があり, 将来的にはさらなる進展が期待できる. 例えば, LC特異点を扱う複素解析理論を構築し, 消滅定理を一般化した. これにより, LC特異点の解析的側面が明らかになり, ホッジ構造の複素解析的側面の探求が期待できる. また, アバンダンス予想や非消滅予想といった当該分野の大問題に対しても, 部分的ではあるが新たな成果を上げた点で価値があると思われる.

Report

(5 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • Research Products

    (51 results)

All 2024 2023 2022 2021 2020

All Int'l Joint Research (2 results) Journal Article (17 results) (of which Int'l Joint Research: 5 results,  Peer Reviewed: 17 results,  Open Access: 3 results) Presentation (31 results) (of which Int'l Joint Research: 21 results,  Invited: 30 results) Funded Workshop (1 results)

  • [Int'l Joint Research] バイロイト大学(ドイツ)2022

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] バイロイト大学(ドイツ)2022

    • Related Report
      2023 Annual Research Report
  • [Journal Article] On the minimal model program for projective varieties with pseudo-effective tangent sheaf2023

    • Author(s)
      S. Matsumura
    • Journal Title

      Epijournal Geom. Algebriqu

      Volume: 7

    • DOI

      10.46298/epiga.2023.10337

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] On asymptotic base loci of relative anti-canonical divisors of algebraic fiber spaces2023

    • Author(s)
      Ejiri Sho、Iwai Masataka、Matsumura Shin-ichi
    • Journal Title

      Journal of Algebraic Geometry

      Volume: 32 Issue: 3 Pages: 477-517

    • DOI

      10.1090/jag/814

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Strictly nef divisors on K-trivial fourfolds2023

    • Author(s)
      H. Liu, Matsumura Shin-ichi
    • Journal Title

      Mathematische Annalen

      Volume: 387 Issue: 1-2 Pages: 985-1008

    • DOI

      10.1007/s00208-022-02461-1

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] The Nonvanishing problem for varieties with nef anticanonical bundle2023

    • Author(s)
      V. Lazic, S. Matsumura, T. Peternell, N. Tsakanikas, Z. Xie,
    • Journal Title

      Documenta Mathematica

      Volume: 28 Issue: 6 Pages: 1393-1440

    • DOI

      10.4171/dm/936

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] On asymptotic base loci of relative anti-canonical divisors of algebraic fiber spaces2023

    • Author(s)
      S. Ejiri, M. Iwai, S. Matsumura
    • Journal Title

      Journal of Algebraic Geometry

      Volume: 未定

    • Related Report
      2022 Research-status Report
    • Peer Reviewed
  • [Journal Article] Strictly nef divisors on K-trivial fourfolds2023

    • Author(s)
      H. Liu, S. Matsumura
    • Journal Title

      Mathematische Annalen

      Volume: 未定

    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Injectivity theorems with multiplier ideal sheaves for higher direct images under Kaehler morphisms2022

    • Author(s)
      S. Matsumura
    • Journal Title

      Algebraic Geometry

      Volume: 9

    • Related Report
      2022 Research-status Report 2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] On projective manifolds with semi-positive holomorphic sectional curvature2022

    • Author(s)
      S. Matsumura
    • Journal Title

      American Journal of Mathematics

      Volume: 未定

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] Open problems on structure of positively curved projective varieties2022

    • Author(s)
      S. Matsumura
    • Journal Title

      Annales de la Faculte des Sciences de Toulouse

      Volume: 未定

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] ON PROJECTIVE MANIFOLDS WITH PSEUDO-EFFECTIVE TANGENT BUNDLE2021

    • Author(s)
      Hosono Genki、Iwai Masataka、Matsumura Shin-ichi
    • Journal Title

      Journal of the Institute of Mathematics of Jussieu

      Volume: - Issue: 5 Pages: 1-30

    • DOI

      10.1017/s1474748020000754

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] Injectivity theorem for pseudo-effective line bundles and its applications2021

    • Author(s)
      O. Fujino, S. Matsumura
    • Journal Title

      Transactions of the American Mathematical Society, Series B

      Volume: 8

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Projective klt pairs with nef anti-canonical divisor2021

    • Author(s)
      F. Campana, J. Cao, S. Matsumura
    • Journal Title

      Algebraic Geometry

      Volume: 8

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] On projective manifolds with semi-positive holomorphic sectional curvature2021

    • Author(s)
      Shin-ichi Matsumura
    • Journal Title

      American Journal of Mathematics

      Volume: 印刷中

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] On projective manifolds with pseudo-effective tangent bundle2021

    • Author(s)
      Genki Hosono, Masataka Iwai, Shin-ichi Matsumura
    • Journal Title

      Journal of the Institute of Mathematics of Jussieu

      Volume: 印刷中

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Projective klt pairs with nef anti-canonical divisor2021

    • Author(s)
      Frederic Campana, Junyan Cao, Shin-ichi Matsumura
    • Journal Title

      Algebraic Geometry

      Volume: 印刷中

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Open problems on structure of positively curved projective varieties2021

    • Author(s)
      Shin-ichi Matsumura
    • Journal Title

      Annales de la Faculte des Sciences de Toulouse

      Volume: 印刷中

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the image of MRC fibrations of projective manifolds with semi-positive holomorphic sectional curvature2020

    • Author(s)
      Shin-ichi Matsumura
    • Journal Title

      Pure and Applied Mathematics Quarterly

      Volume: 16

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Presentation] The Nonvanishing problem for varieties with nef anticanonical bundle2024

    • Author(s)
      S. Matsumura
    • Organizer
      Algebraic Geometry Symposium
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Abundance theorem for minimal projective varieties with vanishing Chern class of Miyaoka-type2024

    • Author(s)
      S. Matsumura
    • Organizer
      One-day workshop on Algebraic and Complex Geometry
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] An injectivity theorem on snc compact Kaehler spaces2024

    • Author(s)
      S. Matsumura
    • Organizer
      Workshop on Complex Geometry
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On projective manifolds with semi-positive holomorphic sectional curvature2023

    • Author(s)
      S. Matsumura
    • Organizer
      The 7th Workshop on Complex Geometry and Lie Groups, Universita del Salento
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] The Nonvanishing problem for varieties with nef anticanonical bundle2023

    • Author(s)
      S. Matsumura
    • Organizer
      Seminar of Complex Geometry, Chinese Academy of Sciences
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] The Nonvanishing problem for varieties with nef anticanonical bundle2023

    • Author(s)
      S. Matsumura
    • Organizer
      Seminar of Complex Geometry, Wuhan University
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Compact Kaehler 3-folds with nef anti-canonical bundle2023

    • Author(s)
      S. Matsumura
    • Organizer
      多変数函数論における擬凸性とその周辺
    • Related Report
      2023 Annual Research Report
  • [Presentation] The Nonvanishing problem for varieties with nef anticanonical bundle2023

    • Author(s)
      S. Matsumura
    • Organizer
      京都大学  代数幾何セミナー
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] The Nonvanishing problem for varieties with nef anticanonical bundle2023

    • Author(s)
      S. Matsumura
    • Organizer
      東京大学 複素解析幾何セミナー
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] Geometry of holomorphic sectional curvature2023

    • Author(s)
      S. Matsumura
    • Organizer
      Oberseminar, University of Bayreuth,
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Geometry of holomorphic sectional curvature I2023

    • Author(s)
      S. Matsumura
    • Organizer
      Seminar on Complex Geometry, Pusan National University
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Geometry of holomorphic sectional curvature II2023

    • Author(s)
      S. Matsumura
    • Organizer
      Seminar on Complex Geometry, Pusan National University
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The nonvanishing problem for varieties with nef anticanonical bundle2023

    • Author(s)
      S. Matsumura
    • Organizer
      Korea-Japan Conference in Algebraic Geometry, IBS center
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Abundance theorem for minimal compact Kaehler manifolds with vanishing second Chern class2022

    • Author(s)
      S. Matsumura
    • Organizer
      Oberseminar, University of Bayreuth
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Positivity of direct images and applications to algebraic fiber spaces2022

    • Author(s)
      S. Matsumura
    • Organizer
      The Postech Conference 2022 on Complex Analytic Geometry
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The Nonvanishing problem for varieties with nef anticanonical bundle2022

    • Author(s)
      S. Matsumura
    • Organizer
      Workshop on Complex Analysis and Geometry, University of Essen
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On asymptotic base loci of relative anti-canonical divisors of algebraic fiber spaces2021

    • Author(s)
      S. Matsumura
    • Organizer
      Grauert Theory and Recent Complex Geometry
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Structure theorem for projective klt pairs with nef anti-canonical divisor2021

    • Author(s)
      S. Matsumura
    • Organizer
      Analysis of Monge-Ampere, a tribute to Ahmed Zeriahi
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On projective manifolds with pseudo-effective tangent bundle2021

    • Author(s)
      S. Matsumura
    • Organizer
      The Conference on Complex Geometric Analysis;in honor of Kang-Tae Kim's 65th birthday
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On projective manifolds with pseudo-effective tangent bundle2021

    • Author(s)
      S. Matsumura
    • Organizer
      Subvarieties and foliations of complex projective varieties
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 非負曲率を持つ射影多様体の構造について2021

    • Author(s)
      S. Matsumura
    • Organizer
      大阪大学 談話会
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] On projective manifolds with non-negative holomorphic sectional curvature2021

    • Author(s)
      S. Matsumura
    • Organizer
      大阪大学 幾何学セミナー
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] 数値的に半正な反標準束を持つKLT対の構造定理について2021

    • Author(s)
      S. Matsumura
    • Organizer
      Higher- dimensional algebraic varieties;in honor of Professor Shigefumi Mori’s 70th birthday
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] Structure theorem for projective klt pairs with nef anti-canonical divisor2021

    • Author(s)
      S. Matsumura
    • Organizer
      城崎 代数幾何シンポジウム2021
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] Strictly nef divisors on K-trivial fourfolds2021

    • Author(s)
      S. Matsumura
    • Organizer
      2021 年度多変数関数論 冬セミナー
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] Projective klt pairs with nef anti-canonical divisor and rational curves2020

    • Author(s)
      Shin-ichi Matsumura
    • Organizer
      Complex Algebraic Geometry and Complex Analysis (Bochum-Essen-Koln-Wuppertal)
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On projective manifolds with pseudo-effective tangent bundle2020

    • Author(s)
      Shin-ichi Matsumura
    • Organizer
      Complex Geometry Seminar
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On asymptotic base loci of relative anti-canonical divisors of algebraic fiber spaces2020

    • Author(s)
      Shin-ichi Matsumura
    • Organizer
      Grauert theory and recent complex geometry
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 非負の曲率を持つ射影多様体の構造定理について2020

    • Author(s)
      Shin-ichi Matsumura
    • Organizer
      京都大学 談話会
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] Hacon-McKernan's problem on rational connectedness2020

    • Author(s)
      Shin-ichi Matsumura
    • Organizer
      第26回複素幾何シンポジウム
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On projective manifolds with pseudo-effective tangent bundle2020

    • Author(s)
      Shin-ichi Matsumura
    • Organizer
      東京大学 複素解析幾何セミナー
    • Related Report
      2020 Research-status Report
    • Invited
  • [Funded Workshop] 第19回 代数解析幾何セミナー2024

    • Related Report
      2023 Annual Research Report

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Published: 2020-02-06   Modified: 2025-01-30  

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