Study on toric varieties, vector bundles on them and their subvarieties
Project/Area Number 
11640005

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

Research Institution  Tohoku University 
Principal Investigator 
OGATA Shoetsu Tohoku University, Mathematics, Associate Professor, 大学院・理学研究科, 助教授 (90177113)

CoInvestigator(Kenkyūbuntansha) 
KAJIWARA Takeshi Tohoku University, Mathematics, Research Assistant, 大学院・理学研究科, 助手 (00250663)
NAKAGAWA Yasuhiro Tohoku University, Mathematics, Lecturer, 大学院・理学研究科, 講師 (90250662)
ODA Tdao Tohoku University, Mathematics, Professor, 大学院・理学研究科, 教授 (60022555)

Project Period (FY) 
1999 – 2000

Project Status 
Completed (Fiscal Year 2000)

Budget Amount *help 
¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 2000: ¥1,500,000 (Direct Cost: ¥1,500,000)

Keywords  toric variety / convex polytpe / abelian surface / Fano variety / Kaehler metric 
Research Abstract 
It is known that for an ample line bundle on a projective toric variety of dimension n the global sections of the line bundle twisted by itself over n1 times defines an embedding of the toric variety into a projective space. In 1999 we proved that the ideal defining the embedded toric variety is generated by quadrics when the line bundle is twisted over n times. Ample line bundles on toric varieties are heavily related with convex polytopes, we investigated the convex polytopes of dimension 3 and lattice points in them by using computors, and we showed that the ideals of the toric varieties embedded by n1 times twisted ample line bundles are generated by quadrics when the varieties are quotients of projective space by finite abelian groups. In 2000 we proved that the ideals defining the toric varieties of dimension n embedded by n1 times twisted ample line bundles are generated by quadrics in general. We have studied the Futaki invariant which is an obstraction to the existence of an Einstein Kaehler metric on the tangent bundle over toric Fano manifold. In 1999 and 2000 we generalized it as the BandoCalabiFutaki character which is an obstraction to the existance of a Kaehler metric with constant sscalar curvature, and we showed that the BandoCalabiFutaki character is also an obstraction to semistability of algebraic varieties in the geometric invariant theory. We also studied the embedding problem of abelian surfaces into nonsingular toric varieties of dimension 4. We showed that only the product of elliptic curves can be embedded into the product of toric varieties of dimension 2.

Report
(3 results)
Research Products
(9 results)