Birational geometry for higher-dimensional algebraic varieties

• Project/Area Number: 16H03925
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Osaka University
• Principal Investigator: Fujino Osamu 大阪大学, 理学研究科, 教授 (60324711)
• Project Period (FY): 2016-04-01 – 2021-03-31
• Project Status: Completed (Fiscal Year 2020)
• Budget Amount: ¥17,940,000 (Direct Cost: ¥13,800,000、Indirect Cost: ¥4,140,000); Fiscal Year 2020: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000); Fiscal Year 2019: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000); Fiscal Year 2018: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000); Fiscal Year 2017: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000); Fiscal Year 2016: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
• Keywords: 極小モデル理論 / 混合ホッジ構造 / トーリック多様体 / 双有理幾何学 / 小平消滅定理 / 高次元代数多様体 / 飯高予想 / 対数的曲面 / クラスシー / 藤田予想 / 飯高プログラム / 標準束公式 / 混合ホッジ構造の変動 / 対数的標準環 / 乗数イデアル層 / トーリック多様体論 / トーリック幾何学

Outline of Final Research Achievements

I am mainly interested in higher-dimensional complex projective varieties. I have already established some powerful generalizations of the Kodaira vanishing theorem by using the theory of mixed Hodge structures on cohomology with compact support. I have also studied the theory of variations of mixed Hodge structure on cohomology with compact support. Now I am interested in applications of the theory of variations of mixed Hodge structure for higher-dimensional algebraic varieties. I obtained a generalization of Kodaira's canonical bundle formula and now tries to apply this new result for some geometric problems. In 2020, I published a book on the Iitaka conjecture.

Academic Significance and Societal Importance of the Research Achievements

この5年間の主な研究成果は、混合ホッジ構造の変動の理論を高次元代数多様体の研究に組織的に持ち込んだことである。すでにコンパクト台コホモロジーに入る混合ホッジ構造が高次元代数多様体論で有益であることは知っていたが、混合ホッジ構造の変動も考えることで様々な応用が考えられることに気づいた。安定多様体のモジュライ空間の射影性の証明は一番最初の素朴な応用である。

Research Products

• [Int'l Joint Research] Xiamen University(中国)
• [Journal Article] Fujita-type freeness for quasilog canonical curves and surfaces (2020)
  • Author(s): Fujino Osamu、Liu Haidong
  • Journal Title: Kyoto Journal of Mathematics Volume: 60 Issue: 4 Pages: 1453-1467
  • DOI: 10.1215/21562261-2019-0068
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Corrigendum to “On subadditivity of the logarithmic Kodaira dimension” (2020)
  • Author(s): FUJINO Osamu
  • Journal Title: Journal of the Mathematical Society of Japan Volume: 72 Issue: 4 Pages: 1181-1187
  • DOI: 10.2969/jmsj/82568256
  • NAID: 130007928975
  • ISSN: 0025-5645, 1881-1167, 1881-2333
  • Peer Reviewed
• [Journal Article] Notes on toric varieties from Mori theoretic viewpoint, II (2020)
  • Author(s): Osamu Fujino and Hiroshi Sato
  • Journal Title: Nagoya Math. J. Volume: 239 Pages: 42-75
  • DOI: 10.1017/nmj.2018.27
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Simple connectedness of Fano log pairs with semi-log canonical singularities (2020)
  • Author(s): Fujino Osamu、Liu Wenfei
  • Journal Title: Mathematische Zeitschrift Volume: 295 Issue: 1-2 Pages: 341-348
  • DOI: 10.1007/s00209-019-02347-0
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Toric Fano contractions associated to long extremal rays (2020)
  • Author(s): Fujino Osamu、Sato Hiroshi
  • Journal Title: Tohoku Mathematical Journal Volume: 72 Issue: 1 Pages: 77-86
  • DOI: 10.2748/tmj/1585101622
  • Peer Reviewed
• [Journal Article] Vanishing and semipositivity theorems for semi-log canonical pairs (2019)
  • Author(s): Osamu Fujino
  • Journal Title: Publ. Res. Inst. Math. Sci. Volume: 56 Issue: 1 Pages: 15-32
  • DOI: 10.4171/prims/56-1-2
  • Peer Reviewed
• [Journal Article] On semipositivity theorems (2019)
  • Author(s): Osamu Fujino, Taro Fujisawa
  • Journal Title: Math. Res. Lett. Volume: 26 Issue: 5 Pages: 1359-1382
  • DOI: 10.4310/mrl.2019.v26.n5.a6
  • Peer Reviewed
• [Journal Article] Semipositivity theorems for moduli problems (2018)
  • Author(s): Fujino Osamu
  • Journal Title: Annals of Mathematics Volume: 187 Issue: 3 Pages: 639-665
  • DOI: 10.4007/annals.2018.187.3.1
  • Peer Reviewed
• [Journal Article] On normalization of quasi-log canonical pairs (2018)
  • Author(s): Osamu Fujino, Haidong Liu
  • Journal Title: Proc. Japan Acad. Ser. A Math. Sci. Volume: 94 Issue: 10
  • DOI: 10.3792/pjaa.94.97
  • NAID: 40021754644
  • Peer Reviewed / Open Access
• [Journal Article] Koll\'ar-Nadel type vanishing theorem (2018)
  • Author(s): Osamu Fujino
  • Journal Title: Southeast Asian Bull. Math. Volume: 42
  • Peer Reviewed / Open Access
• [Journal Article] Pull-back of quasi-log structures (2017)
  • Author(s): Osamu Fujino
  • Journal Title: Publ. Res. Inst. Math. Sci. Volume: 53 Issue: 2 Pages: 241-259
  • DOI: 10.4171/prims/53-2-1
  • Peer Reviewed
• [Journal Article] On semipositivity, injectivity and vanishing theorems (2017)
  • Author(s): Osamu Fujino
  • Journal Title: Advanced Lecture in Mathematics (ALM) Volume: 39
  • Peer Reviewed
• [Journal Article] Effective basepoint-free theorem for semi-log canonical surfaces (2017)
  • Author(s): Osamu Fujino
  • Journal Title: Publ. Res. Inst. Math. Sci. Volume: 53 Issue: 3 Pages: 349-370
  • DOI: 10.4171/prims/53-3-1
  • Peer Reviewed
• [Journal Article] 対談：森理論について---森理論誕生から最近の発展まで (2017)
  • Author(s): 藤野 修、森 重文
  • Journal Title: 数学 Volume: 69
  • NAID: 130007683907
  • Peer Reviewed
• [Journal Article] Injectivity theorems (2017)
  • Author(s): Osamu Fujino
  • Journal Title: Adv. Stud. Pure Math. Volume: 74
  • Peer Reviewed
• [Journal Article] On subadditivity of the logarithmic Kodaira dimension (2017)
  • Author(s): Osamu Fujino
  • Journal Title: Journal of the Mathematical Society of Japan Volume: 69 Issue: 4 Pages: 1565-1581
  • DOI: 10.2969/jmsj/06941565
  • NAID: 130006887111
  • ISSN: 0025-5645, 1881-1167, 1881-2333
  • Peer Reviewed
• [Journal Article] On log canonical rings (2017)
  • Author(s): Osamu Fujino, Yoshinori Gongyo
  • Journal Title: Adv. Stud. Pure Math. Volume: 74
  • Peer Reviewed
• [Journal Article] Notes on the weak positivity theorems (2017)
  • Author(s): Osamu Fujino
  • Journal Title: Adv. Stud. Pure Math. Volume: 75
  • Peer Reviewed
• [Journal Article] On log canonical rational singularities (2016)
  • Author(s): O. Fujino
  • Journal Title: Proc. Japan Acad. Ser. A Math. Sci. Volume: 92 Issue: 1
  • DOI: 10.3792/pjaa.92.13
  • NAID: 40020720229
  • Peer Reviewed / Open Access
• [Journal Article] Basepoint-free theorem of Reid--Fukuda type for quasi-log schemes (2016)
  • Author(s): O. Fujino
  • Journal Title: Publ. Res. Inst. Math. Sci. Volume: 52 Issue: 1 Pages: 63-81
  • DOI: 10.4171/prims/174
  • Peer Reviewed
• [Journal Article] Direct images of relative pluricanonical bundles (2016)
  • Author(s): O. Fujino
  • Journal Title: Algebraic Geometry Volume: 3 Issue: 1 Pages: 50-62
  • DOI: 10.14231/ag-2016-003
  • Peer Reviewed / Open Access
• [Journal Article] Corrigendum: Direct images of relative pluricanonical bundles (2016)
  • Author(s): O. Fujino
  • Journal Title: Algebraic Geometry Volume: 3 Pages: 261-263
  • DOI: 10.14231/ag-2016-012
  • Peer Reviewed / Open Access
• [Journal Article] Kollar-type effective freeness for quasi-log canonical pairs (2016)
  • Author(s): Osamu Fujino
  • Journal Title: Internat. J. Math. Volume: 27 Issue: 14 Pages: 1650114-1650114
  • DOI: 10.1142/s0129167x16501147
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Vanishing theorems (2016)
  • Author(s): Osamu Fujino
  • Journal Title: Adv. Stud. Pure Math. Volume: 70 Pages: 299-321
• [Presentation] Quasi-log schemes入門 第1回目 (2020)
  • Author(s): 藤野 修
  • Organizer: 阪大オンライン代数幾何学セミナー
  • Invited
• [Presentation] Quasi-log schemes入門 第2回目 (2020)
  • Author(s): 藤野 修
  • Organizer: 阪大オンライン代数幾何学セミナー
  • Invited
• [Presentation] On mixed-ω-sheaves (Part I) (2020)
  • Author(s): Osamu Fujino
  • Organizer: Tianyuan workshop series (online)
  • Int'l Joint Research / Invited
• [Presentation] On mixed-ω-sheaves (Part II) (2020)
  • Author(s): Osamu Fujino
  • Organizer: Tianyuan workshop series (online)
  • Int'l Joint Research / Invited
• [Presentation] On mixed-ω-sheaves (Part III) (2020)
  • Author(s): Osamu Fujino
  • Organizer: Tianyuan workshop series (online)
  • Int'l Joint Research / Invited
• [Presentation] Minimal model theory for log surfaces in Fujiki's class C (2020)
  • Author(s): 藤野 修
  • Organizer: 東大京大代数幾何セミナー
  • Invited
• [Presentation] On log canonical rings (2019)
  • Author(s): Osamu Fujino
  • Organizer: Workshop on Algebraic Geometry（中国、アモイ）
  • Int'l Joint Research / Invited
• [Presentation] Introduction to the theory of quasi-log canonical pairs, lecture 1 (2019)
  • Author(s): Osamu Fujino
  • Organizer: Short Course on Birational Geometry and Moduli Spaces（中国、アモイ）
  • Int'l Joint Research / Invited
• [Presentation] Introduction to the theory of quasi-log canonical pairs, lecture 2 (2019)
  • Author(s): Osamu Fujino
  • Organizer: Short Course on Birational Geometry and Moduli Spaces（中国、アモイ）
  • Int'l Joint Research / Invited
• [Presentation] Introduction to the theory of quasi-log canonical pairs, lecture 3 (2019)
  • Author(s): Osamu Fujino
  • Organizer: Short Course on Birational Geometry and Moduli Spaces（中国、アモイ）
  • Int'l Joint Research / Invited
• [Presentation] Introduction to the theory of quasi-log canonical pairs, lecture 4 (2019)
  • Author(s): Osamu Fujino
  • Organizer: Short Course on Birational Geometry and Moduli Spaces（中国、アモイ）
  • Int'l Joint Research / Invited
• [Presentation] On quasi-log canonical pairs (2019)
  • Author(s): 藤野 修
  • Organizer: 代数幾何学セミナー（東京大学） 2019年5月15日
  • Invited
• [Presentation] Simple connectedness of Fano log pairs with semi-log canonical singularities (2019)
  • Author(s): Osamu Fujino
  • Organizer: Algebraic Geometry and Moduli Theory
  • Invited
• [Presentation] Minimal model theory for log surfaces (2018)
  • Author(s): Osamu Fujino
  • Organizer: Algebraic Surfaces and Related Topics
  • Int'l Joint Research / Invited
• [Presentation] Simple connectedness of Fano log pairs with semi-log canonical singularities (2018)
  • Author(s): Osamu Fujino
  • Organizer: The 24th Symposium on Complex Geometry
  • Int'l Joint Research / Invited
• [Presentation] On quasi-log canonical pairs (2018)
  • Author(s): Osamu Fujino
  • Organizer: Workshop on Birational Geometry and Related Topics
  • Invited
• [Presentation] 安定多様体のモジュライ空間について (2018)
  • Author(s): 藤野 修
  • Organizer: 第4回杜の都代数幾何学研究集会
  • Invited
• [Presentation] On semi-log canonical Fano pairs (2018)
  • Author(s): Osamu Fujino
  • Organizer: 第16回アフィン代数幾何学研究集会
  • Int'l Joint Research / Invited
• [Presentation] Bertini type theorem for multiplier ideal sheaves (2017)
  • Author(s): Osamu Fujino
  • Organizer: Stability, boundedness and Fano varieties
  • Int'l Joint Research / Invited
• [Presentation] Direct images of relative pluricanonical bundles (2016)
  • Author(s): 藤野修
  • Organizer: 代数幾何・複素幾何セミナー、大阪大学大学院理学研究科
  • Invited
• [Presentation] 極小モデル理論の最近の発展について (2016)
  • Author(s): 藤野修
  • Organizer: 数学教室談話会、大阪大学大学院理学研究科
  • Invited
• [Presentation] Injectivity theorem for pseudo-effective line bundles and its applications (2016)
  • Author(s): Osamu Fujino
  • Organizer: Pacific RIM Conference on Mathematics 2016, Seoul National University, Korea
  • Int'l Joint Research / Invited
• [Presentation] Vanishing theorems for complex projective varieties (2016)
  • Author(s): Osamu Fujino
  • Organizer: The Asian Mathematical Conference 2016, Bali Nusa Dua Convention Center, Bali Indonesia
  • Int'l Joint Research / Invited
• [Book] Iitaka conjecture : an introduction (2020)
  • Author(s): Osamu Fujino
  • Total Pages: 128
  • Publisher: Springer
• [Book] Foundations of the minimal model program (2017)
  • Author(s): Osamu Fujino
  • Total Pages: 289
  • Publisher: The Mathematical Society of Japan
  • ISBN: 9784864970457
• [Funded Workshop] K3 Surfaces and Related Topics (2017)