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

首席研究员对代数簇上有理曲线族的结构进行了详细的研究，特别是对具有nef切丛的Fano流形。本文的主要研究成果之一是利用反正则因子与有理曲线的交数对二次超曲面进行了一个简单的刻画，发表于《超二次曲面的数值刻画》。其中的结果不仅统一了迄今为止已知的各种特征（Brieskorn定理、Kobayashi-Ochi定理等），而且还统一了已知的各种特征（Brieskorn定理、Kobayashi-Ochi定理等）。而且还提供了许多进一步的应用。他还研究了正则度本文结合Green-Griffiths-Lang的一个猜想，讨论了一般型曲面上曲线的标准度（与标准因子的交数），证明了曲线的标准度受曲线的几何亏格和周围曲面的Chern数的显函数的约束，在第一陈氏数大于第二陈氏数的条件下。作为一个直接的结果，它遵循只有1000个合理的和椭圆曲线在这样的表面（代数朗猜想的一个特殊情况）。这第二个结果是提交的标题下“一个评论定理的Bogomolov”。他的第三个研究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的进展，在合作研究者的工作中，我们应该提到：Y.Kawamata对导范畴的研究，M.Kondo对具有额外结构的K3曲面的模空间的研究，T.Saito的算术分支理论，以及Y. Kawamata对导范畴的研究。A.Tamagawa对Anabel几何的研究和T.Ibukiyama对模形式的研究。特别是T.Terasoma对多重zeta值理论做出了杰出贡献，并被提名为2006年马德里国际数学家大会的演讲者。

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.