Minimal model program and singularity theory

• Project/Area Number: 16H06710
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Single-year Grants
• Research Field: Algebra
• Research Institution: The University of Tokyo
• Principal Investigator: Nakamura Yusuke 東京大学, 大学院数理科学研究科, 助教 (20780034)
• Project Period (FY): 2016-08-26 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000); Fiscal Year 2017: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2016: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 極小モデル理論 / ACC予想 / 有理点問題 / LSC予想 / フリップの停止問題 / 特異点 / 極小ログ食い違い係数

Outline of Final Research Achievements

I studied the following two topics: (1)Vanishing theorem of Witt vector cohomology. (2) The ACC conjecture.
(1) As joint work with H. Tanaka, we prove vanishing theorems of
Witt vector coholomogy of Nadel type for 3-dimensional Fano varieties (characteristic > 5). We have two applications. One is the connectedness lemma. Another one is a rational point formula. (2) I study the paper by Birkar on the BAB conjecture, and try to apply the ACC conjecture.

Research Products

• [Journal Article] A boundedness conjecture for minimal log discrepancies on a fixed germ (2018)
  • Author(s): M. Mustata, Y. Nakamura
  • Journal Title: Contemporary Mathematicsに掲載決定 Volume: 印刷中
  • Peer Reviewed / Int'l Joint Research
• [Presentation] Vanishing theorems of Witt-vector cohomology for Fano threefolds (2018)
  • Author(s): Yusuke Nakamura
  • Organizer: Higher dimensional algebraic geometry
  • Int'l Joint Research / Invited
• [Presentation] A vanishing theorem of Witt-vector cohomology for Fano threefolds (2018)
  • Author(s): Yusuke Nakamura
  • Organizer: Algebraic geometry in positive characteristic and related topics
  • Invited
• [Presentation] A vanishing theorem of Witt-vector cohomology for Fano threefolds (2017)
  • Author(s): Yusuke Nakamura
  • Organizer: Classification and Moduli Theory of Algebraic Varieties
  • Int'l Joint Research / Invited
• [Presentation] An application of the ACC for LCT'S to the ACC for MLD's (2017)
  • Author(s): Yusuke Nakamura
  • Organizer: NCTS Workshop in Higher Dimensional Algebraic Geometry
  • Int'l Joint Research / Invited
• [Presentation] A rational point formula on singular Fano varieties over a finite field (2017)
  • Author(s): Yusuke Nakamura
  • Organizer: BICMR-Tokyo Algebraic Geometry Workshop
  • Int'l Joint Research / Invited
• [Presentation] Birkar's paper "Singularities of linear systems and boundedness of Fano varieties" explanation (2017)
  • Author(s): Yusuke Nakamura
  • Organizer: NCTS Workshop on Singularities, Linear Systems, and Fano Varieties
  • Int'l Joint Research / Invited
• [Presentation] Rational points on log Fano threefolds over a finite field (2017)
  • Author(s): Yusuke Nakamura
  • Organizer: Workshop on Higher Dimensional Algebraic Geometry, Holomorphic Dynamics and Their Interactions
  • Int'l Joint Research / Invited
• [Presentation] New formulation of the number of rational points on singular Fano varieties over a finite field (2017)
  • Author(s): Yusuke Nakamura
  • Organizer: The 2nd Higher dimensional algebraic geometry Echigo Yuzawa symposium
  • Invited
• [Presentation] Rational points on log Fano threefolds over a finite field (2017)
  • Author(s): Yusuke Nakamura
  • Organizer: Cambridge--Tokyo Algebraic Geometry workshop 2017
  • Int'l Joint Research / Invited
• [Presentation] Rational points on log Fano threefolds over a finite field (2016)
  • Author(s): Yusuke Nakamura
  • Organizer: Tokyo-Princeton algebraic geometry conference, Princeton University
  • Int'l Joint Research / Invited
• [Presentation] Birkar's paper "Anti-pluricanonical systems on Fano varieties" explanation (2016)
  • Author(s): Yusuke Nakamura
  • Organizer: Summer School on Algebraic Geometry
  • Invited