THEORY OF ALGEBRAIC VARIETIES AND APPLICATIONS TO RELATED TOPICS

Miyaoka（首席研究员）主要研究高维变量上的有理曲线。主要结果如下：（1）极大有理连通纤维化的构造与应用;（2）有理连通簇的性质研究，包括有理连通3-折叠的几何刻画;（3）有理连通性与Fano n-折叠有界性的证明;（4）态射的相对变形理论及其在相对反正则除的直接象中的应用;射影空间和超二次曲面的数值特征Aoki研究了Fermat曲线和Abel L-函数，Shioda建立了Mordell-Weil格理论，得到了非常丰富的应用，如构造椭圆形，高秩的刻，Galois群同构于Weyl群的方程，高密度球填充的发现。

Miyaoka (head investigator) primarily investigated rational curves on Varieties of higher dimension. Main results are as follows: (1) Construction and applications of maximal rationally connected fibrations ; (2) Study of the properties of rationally connected varieties, including geometric characterization of rationally connected 3-folds ;(3) Proof of the rational connectedness and boundedness of Fano n-folds ;(4) Theory of relative deformation of morphisms and application to the direct images of relative anti-canonical division as ;Numerical charactcrizations of projective spaces and hyperguadrics ;Proof of the boundedness of carves of given genees on a fixed surface of * type.Decisive results to the above topics were given.Aoki studied Fermat curves and abelian L-functions.Shioda established the theory of Mordell-Weil lattices, to get extraordinarily rich applications such as the construction of elliptic, carves of high rank, equations with Galois group isomorphic to the Weyl groups, and a discovery of sphere packings with high density.