Project/Area Number  09440010 
Research Category 
GrantinAid for Scientific Research (B).

Section  一般 
Research Field 
Algebra

Research Institution  KYOTO UNIVERSITY 
Principal Investigator 
MORI Shigefumi RIMS, KYOTO UNIVERSITY PROFESSOR, 数理解析研究所, 教授 (00093328)

CoInvestigator(Kenkyūbuntansha) 
MUKAI Shigeru DEPT. OF MATH., NAGOYA UNIV., PROFESSOR, 大学院・多元数理科学研究科, 教授 (80115641)
SAITO Masahiko DEPT. OF MATH., KOBE UNIV., PROFESSOR, 理学部, 教授 (80183044)
NAKAYAMA Noboru RIMS, KYOTO UNIVERSITY ASSOCIATE PROFESSOR, 数理解析研究所, 助教授 (10189079)
MIYAOKA Yoichi RIMS, KYOTO UNIVERSITY PROFESSOR, 数理解析研究所, 教授 (50101077)
SAITO Kyoji RIMS, KYOTO UNIVERSITY PROFESSOR, 数理解析研究所, 教授 (20012445)

Project Fiscal Year 
1997 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥9,400,000 (Direct Cost : ¥9,400,000)
Fiscal Year 1999 : ¥3,000,000 (Direct Cost : ¥3,000,000)
Fiscal Year 1998 : ¥3,100,000 (Direct Cost : ¥3,100,000)
Fiscal Year 1997 : ¥3,300,000 (Direct Cost : ¥3,300,000)

Keywords  minimal model / abundance conjecture / mirror conjecture / K3 surface / CalabiYau manifold / Abelian variety / Bmodel / terminal singularity / 極小モデル / アバンダンス予想 / ミラー予想 / K3曲面 / カラビーヤウ多様体 / アーベル多様体 / Bモデル / 端末特異点 / モーデル・ヴェイユ格子 / アパンダンス予想 / Aモデル / Chern数 / 極小モデル理論 / flip / Mirror対称 / CalabiYau多様体 / Hilbert scheme / special Lagrangian 
Research Abstract 
Mori, together with Kollar, published an book on the birational geometry of algebraic varieties. Topics treated in the book include a simpler alternate definition of dlt singularity, a simpler proof of the rationality of dlt singularities and an alternate proof of the existence of the 3dimensional stable flips. He also published a review of his work on the existence of rational curves on algebraic varieties, in which he posed problems on the refinement of the existence theorem, the generalization of the cone theorem, etc. Together with Kollar, Miyaoka and Takagi, he finished the proof of the boundedness of the terminal QFano 3folds, which will be published shortly. (The proof of Reid's conjecture on 3dimensional flips in the reducible case is in preparation.) Miyaoka is preparing the proof for the assertion that every projective smooth nfold with an external ray of length at least n+1 is isomorphic to the projective space, which is to be published soon. Nakayama has investigated prob
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lems related to the minimal model theory. He proved that small deformation of terminal singularities are terminal (in preparation). He also proved that, assuming the abundance conjecture, every nonsingular projective manifold whose universal covering is an affine space has an abelian variety as a finite etale covering. Mukai's work are on the algebraic construction of moduli spaces and various geometries on them, including certain duality of polarized K3 surfaces. He is also investigating the Verlinde formula on the moduli spaces of the parabolic vector bundles. Masahiko Saito, together with Hosono and Takahashi, has formulated a generalzation of the holomorphic anomaly equation on the counting of higher genus curves on rational elliptic surfaces and verified that it is consistent with the Bmodel computation in the case of genus 0,1. Hayakawa has proved that, for every 3dimensional terminal singularity of index at least two, an arbitrary exceptional divisor with the minimal discrepancy can be obtained by an explicit "weighted blowup". Dr. Kenji Matsuki at Purdue University was invited for two weeks from the end of June 1999 to present his recent work on the weak factorization of the birational map in a series of lectures (the lecture notes will be published. ) Less
