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

主要研究人员对代数簇上有理曲线族的结构进行了详细的研究，特别是在具有Nef切丛的Fano流形上。本研究的主要成果之一是利用反非因子与有理曲线的交数对二次超曲面进行了简单的刻画，该刻画被称为《超二次曲面的数值刻画》。其中的结果不仅统一了迄今已知的各种刻画(Brieskorn定理、Kobayashi-Ochi定理等)。而且还给出了许多进一步的应用。他还结合Green-Griffiths-Lang猜想研究了一般类型曲面上曲线的正则度(与正则因子的交数)，证明了在第一个陈数大于第二个陈数的条件下，曲线的正则度是由曲线的几何亏格和环境曲面的陈数的某个显式函数有界的。直接的结果是，在这样的曲面上只有有限多条有理曲线和椭圆曲线(代数Lang猜想的一个特例)。第二个结果以《关于博戈莫洛夫一个定理的注记》为题提交。他的第三个研究主题是复辛流形的纤维空间结构，但进展不大。在联合研究人员的工作中，值得一提的是：Y.Kawamata关于派生范畴的研究；M.Kondo关于具有额外结构的K3曲面的模空间的研究；T.Saito的算术分支理论；A.Tamagawa关于Anabelian几何的工作和T.Ibukiyama关于模形式的研究。特别是，T.Terasoma对多重Zeta价值观理论做出了杰出的贡献，并被提名为2006年马德里国际数学家大会的演讲人。

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.