Studies on Hodge Theory and Hypergeometric Functions

在研究代数簇结构的过程中，研究给定簇上的多正则形式往往是很有帮助的。例如，簇的Kodaira维是m-亏格的增长阶，m-标准型的向量空间的维度是整数m的函数。这是代数簇二元分类的一个基本不变量。研究了给定簇上的对数多正则形式与其子簇上的多正则形式的比较问题，设X为光滑代数簇，Y为光滑因子.例如，Y是紧致代数流形，X是它的变形族的全空间。我们考虑了将Y上的多重标准型扩张到X上的多重标准型的问题，定义了X和Y上的乘子理想层序列，将扩张问题归结为这些理想之间的包含关系问题。在一定条件下证明了一个延拓定理。作为应用程序。我们证明了任何正则奇点的变形都是正则的，并且在任何只有正则奇点的一般类型变形族上，多项式是常量。

In the course of the investigation of the structure of algebraic varieties, it is often very helpful to look at the pluricanonical forms on the given varieties. For example, the Kodaira dimension of a variety is the order of growth of the m-genus, the dimension of the vector space of m-canonical forms, as a function on the integer m. This is a fundamental invariant for the birational classification of algebraic varieties. We investigated the problem of comparing log pluricanonical forms on a given variety with pluricanonical forms on its subvariety.Let X be a smooth algebraic variety and Y a smooth divisor. For example, Y is a compact algebraic manifold and X is the total space of its deformation family. We considered the problem of extending pluricanonical forms on Y to log pluricanonical forms on X.We defined sequences of multiplier ideal sheaves on X and Y, and the extension problem is reduced to the problem of the inclusion relationships among these ideals. We proved an extension theorem under some conditions. As applications. we proved that any deformations of canonical singularities are canonical, and that the plurigenus is constant on any deformation family of varieties of general type with only canonical singularities.