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

主要研究了代数变量上有理曲线族的结构，特别是具有nef切束的Fano流形。本研究的主要成果之一是用反正则因子与有理曲线的交点数对二次超曲面进行了简单表征，发表为“超二次曲面的数值表征”。其中的结果不仅统一了迄今已知的各种表征（Brieskorn的一个定理，Kobayashi-Ochi的一个定理等），而且给出了许多进一步的应用。他还结合Green-Griffiths-Lang的一个猜想，研究了一般型曲面上曲线的正则度（与正则因子的交数），证明了在第一陈氏数大于第二陈氏数的条件下，曲线的正则度是由曲线的几何属和周围曲面的陈氏数的某个显式函数所限定的。直接的结果是，在这样的曲面上只有有限的有理曲线和椭圆曲线（代数朗猜想的一个特例）。这第二个结果以“关于Bogomolov定理的注解”为题提出。他研究的第三个课题是复辛流形的纤维空间结构，在这方面他没有取得太大进展。在合作研究者的工作中，我们应该提到：Y.Kawamata对衍生范畴的研究；M.Kondo关于附加结构K3曲面模空间的研究齐藤的算术分支理论tamagawa在anabel几何上的研究和ibukiyama在模形式上的研究。特别是，T.Terasoma对多重zeta值理论做出了杰出的贡献，并在2006年马德里国际数学家大会上被提名为演讲者。

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.