On the nearly smooth complex spaces

TL;DR

This paper introduces a new class of nearly smooth complex spaces with mild singularities, extending fundamental classes and intersection theory to these spaces, and demonstrating their properties and behavior in analytic families.

Contribution

It generalizes fundamental classes and intersection theory to nearly smooth complex spaces with mild singularities, maintaining key properties of classical theory.

Findings

Established a generalized fundamental class for these spaces.

Extended geometric intersection theory to include rational coefficients.

Proved the intersection theory retains most classical properties.

Abstract

We introduce a class of normal complex spaces having only mild sin-gularities (close to quotient singularities) for which we generalize the notion of a (analytic) fundamental class for an analytic cycle and also the notion of a relative fundamental class for an analytic family of cycles. We also generalize to these spaces the geometric intersection theory for analytic cycles with rational positive coefficients and show that it behaves well with respect to analytic families of cycles. We prove that this intersection theory has most of the usual properties of the standard geometric intersection theory on complex manifolds, but with the exception that the intersection cycle of two cycles with positive integral coefficients that intersect properly may have rational coefficients. AMS classification. 32 C 20-32 C 25-32 C 36.