Explicit Study of Algebraic Varieties

(1)与Due Tai Pho合作，证明了具有Artin不变量= 3的特征为5的超奇异K3曲面的唯一性。(2)对复正规代数K3曲面上所有可能的有理双点构型进行了分类，并对具有足够大特征的超奇异K3曲面进行了同样的分类。(3)研究了复代数变体的超越格的同构类集，这些超越格是由复数域上定义的单个代数变体通过基域的不同嵌入而得到的，并给出了在复数域自同构下共轭的非同胚复代数变体的例子。特别地，我们研究了奇异的情况。利用虚二次场的类场论研究K3曲面。作为应用，我们得到了奇异K3曲面基场度的下界。(4)假设定义在数域上的奇异K3曲面X的基域在有限点P处约简，得到一个超奇异K3曲面X (P)。我们研究了X的Neron-Severi格在X (P)格中的正交补，并证明了与跨格情况类似的结果。(5)证明了Grassmannian变体中对偶变体的补的拓扑基本群的一个Lefschetz超平面截面定理，并研究了该基本群与穿孔Riemann曲面的重子群的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.