Explicit Study of Algebraic Varieties

(1)利用〓3的Artin不变量证明了特征为5的超奇异K3曲面的单旋性。(2)对复正则代数K3曲面上所有可能的有理双点构形进行了分类，并讨论了特征足够大的超奇异K3曲面的相同分类。(3)研究了通过基域的各种嵌入到复数域上的单个代数簇得到的复代数簇的超越格的同构类集合，并给出了许多在复数域的自同构下共轭的非同胚复代数簇的例子我们研究了单数的情况。利用虚二次场的类场理论，得到了K3曲面。作为应用，我们得到了奇异K3曲面的基场次数的下界。(4)假设定义在数域上的奇异K3曲面X的基场在有限位置P处约化，得到超奇异K3曲面X(P)。(5)证明了Grassman簇中对偶簇的补的拓扑基本群的Lefschetz超平面截口定理，并研究了该基本群与穿孔黎曼曲面的Barid群之间的Zariski-van Kampen型关系。

(1) By a joint work with Due Tai Pho, we proved the unirationality of supersingular K3 surfaces in characteristic 5 with Artin invariant 〓 3.(2) We classified all possible configurations of rational double points on complex normal algebraic K3 surfaces, and discussed the same classification for supersingular K3 surfaces in sufficiently large characteristics.(3) We investigated the set of isomorphism classes of transcendental lattices of complex algebraic varieties obained from a single algebraic variety defined over a number field by various embeddings of the base field into the complex number field, and produced many examples of non-homeomorphic complex algebraic varieties that are conjugate under the automorphism of the complex number field.In particular, we investigate the case of singular. K3 surfaces by means of the class field theory of imaginary quadratic fields. As an application, we obtained a lower bound of the degree of the base field of singular K3 surfaces.(4) Suppose that we obtain a supersingular K3 surface X (P) by reduction at a finite place P of a base field of a singular K3 surface X defined over a number field. We investigated the orthogonal complement of the Neron-Severi lattice of X in that of X (P), and proved an analogous result as the case of the transvendental lattices.(5) We proved a Lefschetz hyperplane section theorem for the topological fundamental group of the complement of the duial variety in the Grassmannian variety, and invetigate the Zariski-van Kampen type relation of that fundamental group with the barid group of a punctured Riemann surface.