Study of gemetric properties and arithmetic properties of higher dimenional algebraic varieties.

高维代数簇的几何性质和算术性质的研究。

基本信息

批准号：
16340001
负责人：
MIYAOKA Yoichi
金额：
$ 6.59万
依托单位：
The University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

项目摘要

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.

首席研究人员对代数品种的理性曲线家族的结构进行了详细的研究，特别是在带有NEF切线束的Fano歧管上。这项研究的主要结果之一是，根据反典型的分裂和理性曲线的交点的简单表征，它被称为“超质量的数值特征”。其中的结果不仅统一了迄今已知的各种特征（Brieskorn的定理，Kobayashi-Ochi定理等），还提供了许多进一步的应用。他还研究了曲线在一般类型表面上的规范程度（与规范分隔线的相交数（相交数），这与绿色 - 长石的猜想有关，并证明，曲线的规范程度受到曲线几何属的一定明确函数的范围，而曲线的几何形状属的数量则是第一个数字，而不是第一个数字。直接的结果，因此，在这种表面上只有有限的理性和椭圆形曲线有限的（代数lang猜想的特殊情况）。第二个结果以“ Bogomolov定理的评论”为标题。他的研究的第三个主题是复杂的符号歧管的纤维空间结构，在其中他没有得到太大的进展。 M.Kondo对具有额外结构的K3表面模量空间的研究； T.Saito的算术分析理论； A.Tamagawa在Anabelian几何形状和T.Ibukiyama关于模块化形式的研究方面的工作。尤其是，T. terasoma对多个Zeta价值观的理论做出了很好的贡献，并在国际数学家大会上被提名为马德里，2006年。

项目成果

期刊论文数量（66）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Geometry of multiple zeta values

多个 zeta 值的几何形状

DOI：
发表时间：
2006
期刊：
Proc. Intl. Congress of Math. Vol.II
影响因子：
0
作者：
Zhihong Shen;Qinggen Wang;Yoshiro Higano;T. Terasoma
通讯作者：
T. Terasoma

Log crepant birational maps an derived categories.

Log crepant 双有理映射派生类别。

DOI：
发表时间：
2005
期刊：
J.Math.Sci.Univ.Tokyo 12-no.2
影响因子：
0
作者：
Tokuji ARAYA;Ryo TAKAHASHI;Yuji YOSHINO;Yuji YOSHINO;Tomoyoshi IBUKIYAMA;Y.Kawamata
通讯作者：
Y.Kawamata

Frobenius and quasi-Frobenius property for mod $underline C$.

mod $underline C$ 的弗罗贝尼乌斯和准弗罗贝尼乌斯性质。