Calabi-Yau threefolds in positive characteristic and related problems

Research Project

Project/Area Number	25400056
Research Category	Grant-in-Aid for Scientific Research (C)
Allocation Type	Multi-year Fund
Section	一般
Research Field	Algebra
Research Institution	Ritsumeikan University
Principal Investigator	Takayama Yukihide 立命館大学, 理工学部, 教授 (20247810)
Project Period (FY)	2013-04-01 – 2016-03-31
Project Status	Completed (Fiscal Year 2015)
Budget Amount *help	¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000) Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000) Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000) Fiscal Year 2013: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords	Calabi-Yau多様体 / 正標数 / 小平消滅定理 / 持ち上げ可能性 / 代数的基本群 / 代数幾何学 / Calabi-Yau３次元多様体 / 小平型消滅定理
Outline of Final Research Achievements	I proved that a Kodaira type vanishing theorem for the first cohomology holds for non-liftable Calabi-Yau threefolds in positive characteristic for the existing examples such as the Hirokado's example in 1999 and the Schroeer's example of 2003. Moreover, it is proved that this kind of vanishing holds for Calabi-Yau 3-folds with no p-torsion in the Picard varieties. Also I constructed a non-simply connected non-liftable Calabi-Yau threefold in positive characteritic. This example is produced from the Schroeer's example using a method ispired by an idea of Arnaud Beauville.