Mathematical Sciences: Geometry of Singular Spaces

一个代数簇(一些多项式方程的解的集合)的拓扑性质将用两种方法来研究，使用它们特有的结构。首先，代数族的光滑部分有大小和距离的概念，这是从它嵌入在空间中继承下来的。利用这一点，我们定义了平方可和函数，更一般地，定义了簇的光滑部分上的微分形式。要研究的第一个问题是，这些函数或形式的周期在多大程度上由簇的拓扑性质确定，特别是由称为“交同调群”的拓扑不变量确定。第二个问题是如何将簇上圈的交的定量度量与它的局部拓扑性质联系起来。这些几何研究不会局限于非奇异情况，使它们变得更加困难，但更适合应用于理论物理和其他分析理论中出现的空间。

The topological properties of an algebraic variety (the set of solutions of some polynomial equations) will be studied in two ways, using structures which are special to them. First, there are notions of size and distance on the smooth part of an algebraic variety, inherited from its imbedding in space. Using this, one defines square-summable functions and, more generally, differential forms on the smooth part of the variety. The first problem to be studied is to what extent the periods of these functions or forms are determined by the topological properties of the variety, in particular, by a topological invariant called "the intersection homology group." The second problem is to find ways to relate quantitative measures of the intersection of cycles on the variety with its local topological properties. These geometrical investigations will not be confined to the non-singular case, rendering them more difficult, yet more suitable of application to spaces which arise in theoretical physics and in other analytic theories.