2006 Fiscal Year Final Research Report Summary
Study of gemetric properties and arithmetic properties of higher dimenional algebraic varieties.
| Project/Area Number |
16340001
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (B)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Algebra
|
|---|
| Research Institution | The University of Tokyo |
|---|
Principal Investigator |
MIYAOKA Yoichi University of Tokyo, Graduate school of mathematical sciences, Professor, 大学院数理科学研究科, 教授 (50101077)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
KATSURA Toshiyuki University of Tokyo, Graduate school of mathematical sciences, Professor, 大学院数理科学研究科, 教授 (40108444)
KAWAMATA Yujiro University of Tokyo, Graduate school of mathematical sciences, Professor, 大学院数理科学研究科, 教授 (90126037)
SAITO Tsuyoshi University of Tokyo, Graduate school of mathematical sciences, Professor, 大学院数理科学研究科, 教授 (70201506)
TERASOMA Tomohide University of Tokyo, Graduate school of mathematical sciences, Professor, 大学院数理科学研究科, 教授 (50192654)
中村 郁 北海道大学, 大学院理学研究科, 教授 (50022687)
|
|---|
| Project Period (FY) |
2004 – 2006
|
| Keywords | Fano manifolds / Family of rational curves / Quadric hypersurfaces / Surfaces of general type / Conjecture of Green-Griffiths-Lang / Canonical degrees |
|---|
| Research Abstract |
The principal researcher conducted a detailed study on the structure of families of rational curves on algebraic varieties, specifically on Fano manifolds with nef tangent bundles. One of the main outcome of this research is a simple characterization of quadric hypersurfaces in terms of the intersection number of anticanonical divisor and rational curves, which was published as "Numerical characterizations of hyperquadrics". The result therein not only unifies the various characterizations known to date (a theorem of Brieskorn, a theorem of Kobayashi-Ochi etc.) but also gives many further applications. He also studied the canonical degree (the intersection number with the canonical divisor) of curves on surfaces of general type in connection with a conjecture of Green-Griffiths-Lang, and proved that the canonical degree of a curve is bounded by a certain explicit function of the geometric genus of the curve and of the Chern numbers of the ambient surface, under the condition that the first Chern number is greater than the second Chern number. As a direct consequence, it follows that there are only finitely many rational and elliptic curves on such a surface (a special case of algebraic Lang conjecture). This second result is submitted under the title "A remark on a theorem of Bogomolov". The third subject of his research is the fibre space structure of complex symplectic manifolds, in which he did not get much progress.
Among the works of the joint researchers, we should mention: Y.Kawamata's study on derived categories; M.Kondo's research on the moduli spaces of K3 surfaces with extra structure; T.Saito's theory of arithmetic ramifications; A.Tamagawa's work on anabelian geometry and T.Ibukiyama's research on modular forms. In particular, T.Terasoma made excellent contributions to the theory of multiple zeta values and was nominated as a speaker at the International Congress of Mathematicians, Madrid, 2006.
|
