Boundedness problems on the minimal model program and singularities

• Project/Area Number: 24684003
• Research Category: Grant-in-Aid for Young Scientists (A)
• Allocation Type: Partial Multi-year Fund
• Research Field: Algebra
• Research Institution: Kyoto University
• Principal Investigator: KAWAKITA Masayuki 京都大学, 数理解析研究所, 准教授 (10378961)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥10,660,000 (Direct Cost: ¥8,200,000、Indirect Cost: ¥2,460,000); Fiscal Year 2015: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000); Fiscal Year 2014: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000); Fiscal Year 2013: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000); Fiscal Year 2012: ¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
• Keywords: 極小対数的食違い係数 / 昇鎖律 / 生成極限 / イデアル進半連続性 / 連結性補題 / 標準特異点 / 対数的標準特異点 / 形式的べき級数環 / イデアル進位相 / Gorenstein指数

Outline of Final Research Achievements

I studied the ACC for minimal log discrepancies, using generic limits introduced by de Fernex, Mustata and Kollar. I proved the discreteness of log discrepancies over all log canonical pairs for a fixed variety and fixed exponents, and obtained the ACC for minimal log discrepancies on local complete intersection singularities. Even when the exponents are not fixed, I proved the ACC for minimal log discrepancies greater than 1 on smooth 3-folds. Also, I showed the ideal-adic semi-continuity of minimal log discrepancies on surfaces.
As a boundedness problem of singularities, I proved that the index of a 3-fold strictly canonical singularity is at most 6.

Research Products

• [Journal Article] A connectedness theorem over the spectrum of a formal power series ring (2015)
  • Author(s): Masayuki Kawakita
  • Journal Title: International Journal of Mathematics Volume: 26 Issue: 11 Pages: 1-27
  • DOI: 10.1142/s0129167x15500883
  • NAID: 120005698003
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] The index of a threefold canonical singularity (2015)
  • Author(s): Masayuki Kawakita
  • Journal Title: American Journal of Mathematics Volume: 137 Issue: 1 Pages: 271-280
  • DOI: 10.1353/ajm.2015.0006
  • NAID: 120005555505
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Discreteness of log discrepancies over log canonical triples on a fixed pair (2014)
  • Author(s): Masayuki Kawakita
  • Journal Title: Journal of Algebraic Geometry Volume: 23 Issue: 4 Pages: 765-774
  • DOI: 10.1090/s1056-3911-2014-00630-5
  • NAID: 120005530932
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Ideal-adic semi-continuity of minimal log discrepancies on surfaces (2013)
  • Author(s): Masayuki Kawakita
  • Journal Title: Michigan Mathematical Journal Volume: 62 Issue: 2 Pages: 443-447
  • DOI: 10.1307/mmj/1370870381
  • Peer Reviewed
• [Journal Article] Ideal-adic semi-continuity problem for minimal log discrepancies (2013)
  • Author(s): Masayuki Kawakita
  • Journal Title: Mathematische Annalen Volume: 356 Issue: 4 Pages: 1359-1377
  • DOI: 10.1007/s00208-012-0885-y
  • NAID: 120005474460
  • Peer Reviewed
• [Journal Article] Discreteness of log discrepancies over log canonical triples on a fixed pair (2013)
  • Author(s): Masayuki Kawakita
  • Journal Title: Journal of Algebraic Geometry Volume: (掲載決定)
  • NAID: 120005530932
  • Peer Reviewed
• [Journal Article] Supplement to classification of threefold divisorial contractions (2012)
  • Author(s): 川北 真之
  • Journal Title: Nagoya Math. J. Volume: 206巻 Pages: 67-73
  • DOI: 10.1215/00277630-1548493
  • NAID: 120004873687
  • Peer Reviewed
• [Presentation] On the ACC conjecture for minimal log discrepancies (2016)
  • Author(s): Masayuki Kawakita
  • Organizer: NCTS workshop in algebraic geometry at CCU
  • Place of Presentation: National Chung Cheng University, Taiwan
  • Year and Date: 2016-03-29
  • Int'l Joint Research / Invited
• [Presentation] Minimal log discrepancies (2016)
  • Author(s): Masayuki Kawakita
  • Organizer: 2016 NCTS mini-course in algebraic geometry
  • Place of Presentation: National Taiwan University, Taiwan
  • Year and Date: 2016-03-18
  • Int'l Joint Research / Invited
• [Presentation] On minimal log discrepancies (2015)
  • Author(s): Masayuki Kawakita
  • Organizer: Singularities and invariants of higher dimensional algebraic varieties
  • Place of Presentation: Kyoto University, Japan
  • Year and Date: 2015-12-17
  • Invited
• [Presentation] On the ACC conjecture for minimal log discrepancies (2015)
  • Author(s): Masayuki Kawakita
  • Organizer: Workshop on algebraic geometry
  • Place of Presentation: Fudan University, China
  • Year and Date: 2015-10-29
  • Int'l Joint Research / Invited
• [Presentation] Minimal log discrepancies and generic limits (2014)
  • Author(s): Masayuki Kawakita
  • Organizer: BICMR Algebraic Geometry Seminar
  • Place of Presentation: Peking University
  • Year and Date: 2014-05-27 – 2014-05-29
  • Invited
• [Presentation] A connectedness theorem over the spectrum of a formal power series ring (2014)
  • Author(s): Masayuki Kawakita
  • Organizer: 代数幾何学セミナー
  • Place of Presentation: 京都大学
  • Year and Date: 2014-05-09
  • Invited
• [Presentation] Ideal-adic semi-continuity of minimal log discrepancies (2013)
  • Author(s): Masayuki Kawakita
  • Organizer: Higher dimensional algebraic geometry
  • Place of Presentation: National Taiwan University,(招待講演)
  • Year and Date: 2013-03-25
• [Presentation] Discreteness of log discrepancies over log canonical triples (2012)
  • Author(s): Masayuki Kawakita
  • Organizer: Higher dimensional algebraic geometry 2012
  • Place of Presentation: Kyoto University, Japan(招待講演)
  • Year and Date: 2012-06-11
• [Presentation] Discreteness of log discrepancies (2012)
  • Author(s): Masayuki Kawakita
  • Organizer: ACC for minimal log discrepancies and termination of flips
  • Place of Presentation: American Institute of Mathematics, USA
  • Year and Date: 2012-05-17
• [Presentation] A connectedness theorem over the spectrum of a formal power series ring
  • Author(s): Masayuki Kawakita
  • Organizer: Geometry and arithmetic of surfaces
  • Place of Presentation: Ludwig Maximilians Universitaet und Technische Universitaet Muenchen, Germany
  • Invited
• [Presentation] A connectedness lemma on the spectrum of a formal power series ring
  • Author(s): Masayuki Kawakita
  • Organizer: Complex Geometry
  • Place of Presentation: National University of Singapore, Singapore
  • Invited
• [Presentation] The index of a threefold canonical singularity
  • Author(s): Masayuki Kawakita
  • Organizer: Workshop in algebraic geometry, London 2013
  • Place of Presentation: Imperial College London, UK
  • Invited
• [Presentation] The index of a threefold canonical singularity
  • Author(s): Masayuki Kawakita
  • Organizer: The commutative algebra of singularities in birational geometry: multiplier ideals, jets, valuations, and positive characteristic methods
  • Place of Presentation: Mathematical Sciences Research Institute, USA
  • Invited
• [Remarks] Website of Masayuki Kawakita
  • URL: http://www.kurims.kyoto-u.ac.jp/~masayuki
• [Remarks] Website of Masayuki Kawakita
  • URL: http://www.kurims.kyoto-u.ac.jp/~masayuki
• [Remarks]