極小モデル理論に現れる特異点の理論

• Project/Area Number: 18K13384
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: The University of Tokyo
• Principal Investigator: 中村 勇哉 東京大学, 大学院数理科学研究科, 助教 (20780034)
• Project Period (FY): 2018-04-01 – 2025-03-31
• Project Status: Granted (Fiscal Year 2023)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2021: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2020: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 極小モデル理論 / ACC予想 / LSC予想 / PIA予想 / 特異点 / 極小ログ食い違い係数 / MLD / 双有理幾何 / 有理点問題 / ファノ多様体 / 特異点理論 / フリップの停止問題

Outline of Annual Research Achievements

今年度は、昨年度に引き続き、商特異点の極小ログ食い違い係数について研究した。超商特異点とは超曲面特異点の有限商となっているような特異点のクラスである。極小ログ食い違い係数に関する予想としてLSC(lower semi-continuity)予想とPIA(precise inversion of adjunction)予想があり、極小ログ食い違い係数を研究するモチベーションとなっている。

昨年度までの研究において、群作用が線形である場合に、PIA予想とLSC予想の成立を証明した。また、さらにそれを群作用が線形と限らない場合に証明している。いずれの場合も、超曲面の定義多項式が、群作用で不変な場合を扱っていた。本年度は、柴田康介氏との共同研究により、超曲面の定義多項式が群作用で不変とは限らない場合(semi-invariantの場合)を研究し、その場合のPIA予想とLSC予想の成立を証明した。このカテゴリーは、3次元端末的特異点を含んでいるため、より自然なクラスであり、応用も期待できる。また、Cartierとは限らないWeil因子へのPIA予想を証明した初めてのカテゴリーである点でも重要だと考えている。

昨年度までの研究と同様に、証明には弧空間の理論を用いている。今回の新しい点は、semi-invariantな元を扱うために、新しい不変量を導入している。この不変量が、極小ログ食い違い係数を弧空間で表示する際に重要となっている。

Current Status of Research Progress

2: Research has progressed on the whole more than it was originally planned.

Reason

本研究では、超曲面特異点の有限商となっているような特異点のクラスを扱っている。昨年度までに、一番重要なinvariantケースについて理論を完成させた。今年度は、残っていたsemi-invariantケースについて理論を完成することができた。

Strategy for Future Research Activity

超曲面特異点の有限商となっているような特異点のクラスをこれまでの研究で扱っている。超曲面特異点に限らない一般の商多様体について統一的に扱うことを一つの目標としている。また、これまでの研究では、証明に必要となる特異点の仮定をつけている。この仮定を外すことも目標となっている。

Research Products

• [Journal Article] On generalized minimal log discrepancy (2024)
  • Author(s): CHEN Weichung、GONGYO Yoshinori、NAKAMURA Yusuke
  • Journal Title: Journal of the Mathematical Society of Japan Volume: 76 Issue: 2 Pages: 393-449
  • DOI: 10.2969/jmsj/90119011
  • ISSN: 0025-5645, 1881-1167, 1881-2333
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Upper bounds of orders of automorphism groups of leafless metric graphs (2023)
  • Author(s): Nakamura Yusuke、Song JuAe
  • Journal Title: AKCE International Journal of Graphs and Combinatorics Volume: 21 Issue: 1 Pages: 71-76
  • DOI: 10.1080/09728600.2023.2251042
  • Peer Reviewed / Open Access
• [Journal Article] Inversion of adjunction for quotient singularities (2022)
  • Author(s): Nakamura Yusuke、Shibata Kohsuke
  • Journal Title: Algebraic Geometry Volume: 9 Pages: 214-251
  • DOI: 10.14231/ag-2022-007
  • Peer Reviewed / Open Access
• [Journal Article] Coordination sequences of crystals are of quasi-polynomial type (2021)
  • Author(s): Nakamura Yusuke、Sakamoto Ryotaro、Mase Takafumi、Nakagawa Junichi
  • Journal Title: Acta Crystallographica Section A Foundations and Advances Volume: 77 Issue: 2 Pages: 138-148
  • DOI: 10.1107/s2053273320016769
  • Peer Reviewed / Open Access
• [Journal Article] Minimal model program for log canonical threefolds in positive characteristic (2020)
  • Author(s): Hashizume Kenta、Nakamura Yusuke、Tanaka Hiromu
  • Journal Title: Mathematical Research Letters Volume: 27 Issue: 4 Pages: 1003-1054
  • DOI: 10.4310/mrl.2020.v27.n4.a3
  • Peer Reviewed
• [Journal Article] Dual complex of log Fano pairs and its application to Witt vector cohomology (2020)
  • Author(s): Yusuke Nakamura
  • Journal Title: Int. Math. Res. Not. IMRN Volume: － Issue: 13 Pages: 9802-9833
  • DOI: 10.1093/imrn/rnz356
  • Peer Reviewed
• [Journal Article] A Witt Nadel vanishing theorem for threefolds (2020)
  • Author(s): Yusuke Nakamura, Hiromu Tanaka
  • Journal Title: Compos. Math. Volume: 156 Issue: 3 Pages: 435-475
  • DOI: 10.1112/s0010437x1900770x
  • Peer Reviewed
• [Journal Article] Rational points on log Fano threefolds over a finite field (2019)
  • Author(s): Yoshinori Gongyo, Yusuke Nakamura, Hiromu Tanaka
  • Journal Title: J. Eur. Math. Soc. (JEMS) Volume: 21 Issue: 12 Pages: 3759-3795
  • DOI: 10.4171/jems/913
  • Peer Reviewed
• [Journal Article] Minimal model program for log canonical threefolds in positive characteristic (2019)
  • Author(s): Kenta Hashizume, Yusuke Nakamura, Hiromu Tanaka
  • Journal Title: Mathematical Research Letters Volume: 未定
  • Peer Reviewed
• [Journal Article] A boundedness conjecture for minimal log discrepancies on a fixed germ (2018)
  • Author(s): Mircea Mustata, Yusuke Nakamura
  • Journal Title: Contemporary Mathematics Volume: 712 Pages: 287-306
  • DOI: 10.1090/conm/712/14351
  • Peer Reviewed / Int'l Joint Research
• [Presentation] Minimal log discrepancies of quotient singularities (2023)
  • Author(s): 中村勇哉
  • Organizer: FRG Special Month in Ann Arbor
  • Int'l Joint Research / Invited
• [Presentation] Ehrhart theory of periodic graphs (2023)
  • Author(s): 中村勇哉
  • Organizer: FRG Special Month in Ann Arbor
  • Int'l Joint Research / Invited
• [Presentation] Ehrhart theory on periodic graphs (2023)
  • Author(s): 中村 勇哉
  • Organizer: 九州大学代数学セミナー
  • Invited
• [Presentation] Minimal log discrepancies of quotient singularities (2023)
  • Author(s): 中村 勇哉
  • Organizer: Recent Developments in Algebraic Geometry, Arithmetic and Dynamics Part 2
  • Int'l Joint Research / Invited
• [Presentation] Shokurov’s index conjecture and PIA conjecture for quotient singularities (2023)
  • Author(s): 中村 勇哉
  • Organizer: 城崎代数幾何学シンポジウム2023
  • Invited
• [Presentation] Ehrhart theory of periodic graphs (2023)
  • Author(s): 中村 勇哉
  • Organizer: The 1st Algebraic geometry Atami symposium
  • Int'l Joint Research
• [Presentation] Toward an Ehrhart theory of periodic graphs (2023)
  • Author(s): 中村 勇哉
  • Organizer: Order seminar in Osaka
  • Invited
• [Presentation] Inversion of adjunction for quotient singularities (2022)
  • Author(s): 中村 勇哉
  • Organizer: JAMI Conference 2022: Higher Dimensional Algebraic Geometry
  • Int'l Joint Research / Invited
• [Presentation] Minimal log discrepancies of quotient singularities (2022)
  • Author(s): 中村 勇哉
  • Organizer: Algebraic Geometry Seminar
  • Invited
• [Presentation] Minimal log discrepancies of quotient singularities (2022)
  • Author(s): 中村 勇哉
  • Organizer: 代数学シンポジウム2022
  • Invited
• [Presentation] Minimal log discrepancies of quotient singularities (2022)
  • Author(s): 中村 勇哉
  • Organizer: Birational Geometry Workshop at BIMSA
  • Int'l Joint Research / Invited
• [Presentation] Minimal log discrepancies of quotient singularities (2021)
  • Author(s): 中村勇哉
  • Organizer: Interactions of new trends in Algebraic Geometry and singularities
  • Int'l Joint Research / Invited
• [Presentation] Inversion of adjunction for quotient singularities (2021)
  • Author(s): 中村勇哉
  • Organizer: Algebraic Geometry and Related Topics
  • Int'l Joint Research / Invited
• [Presentation] Inversion of adjunction for quotient singularities (2021)
  • Author(s): 中村勇哉
  • Organizer: Fudan-SCMS AG Seminar
  • Int'l Joint Research / Invited
• [Presentation] Inversion of adjunction for quotient singularities (2021)
  • Author(s): 中村勇哉
  • Organizer: Zoom Algebraic Geometry Seminar
  • Int'l Joint Research / Invited
• [Presentation] Inversion of adjunction for quotient singularities (2020)
  • Author(s): 中村勇哉
  • Organizer: 日大特異点セミナー
  • Invited
• [Presentation] Rational point problem on singular Fano varieties (2020)
  • Author(s): Yusuke Nakamura
  • Organizer: Singularities and Arithmetics
  • Int'l Joint Research / Invited
• [Presentation] Rational point problem on singular Fano varieties (2019)
  • Author(s): Yusuke Nakamura
  • Organizer: Tokyo Denki University Mathematics Seminar
  • Invited
• [Presentation] Rational point problem on singular Fano varieties (2019)
  • Author(s): Yusuke Nakamura
  • Organizer: Number Theory/Algebraic Geometry Seminar at Boston College
  • Int'l Joint Research / Invited
• [Presentation] Rational point problem on singular Fano varieties (2019)
  • Author(s): Yusuke Nakamura
  • Organizer: Algebraic Geometry Seminar at Princeton University
  • Int'l Joint Research / Invited
• [Presentation] Rational point problem on singular Fano varieties (2019)
  • Author(s): Yusuke Nakamura
  • Organizer: Algebraic Geometry Seminar at Michigan University
  • Int'l Joint Research / Invited
• [Presentation] A rational point problem on Fano varieties (2019)
  • Author(s): Yusuke Nakamura
  • Organizer: Higher Dimensional Arithmetic Geometry
  • Invited
• [Presentation] A rational point problem on Fano varieties (2018)
  • Author(s): Yusuke Nakamura
  • Organizer: Tianyuan Advanced Seminar on the Moduli Spaces in Algebraic Geometry
  • Int'l Joint Research / Invited
• [Presentation] A vanishing theorem of Witt-vector cohomology of Ambro-Fujino type (2018)
  • Author(s): Yusuke Nakamura
  • Organizer: Differential, Algebraic and Topological Methods in Complex Algebraic Geometry
  • Int'l Joint Research / Invited
• [Presentation] A vanishing theorem of Witt-vector cohomology of Ambro-Fujino type (2018)
  • Author(s): Yusuke Nakamura
  • Organizer: Algebraic Geometry in East Asia
  • Int'l Joint Research / Invited
• [Presentation] Dual complex of Fano varieties and an application to vanishing of Witt vector cohomology (2018)
  • Author(s): Yusuke Nakamura
  • Organizer: Workshop on Birational Geometry and Related Topics
  • Int'l Joint Research / Invited
• [Presentation] A Rational Point Problem on Fano Varieties (2018)
  • Author(s): Yusuke Nakamura
  • Organizer: Colloquium
  • Int'l Joint Research / Invited
• [Presentation] Dual Complex of Fano Varieties and an Application to Vanishing of Witt Vector Cohomology (2018)
  • Author(s): Yusuke Nakamura
  • Organizer: Algebraic Geometry Seminar
  • Int'l Joint Research / Invited