On nearly smooth complex spaces

Abstract

We introduce a class of normal complex spaces having only mild singularities (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.

On proper intersections on a singular analytic space

“…We prove the global Theorem 1.3 in Section 7 and provide various examples in Section 8. In the last section we show that our product, at least for RE-cycles, coincides with the product in [7] when Y is nearly smooth.…”

“…Let G ⊂ Y × S be the graph of f and let If q : Y → Y is a local model and µ is a cycle in Y , then there is a natural pullback cycle q * µ in Y , see [7,Section 2.1]. We have the following corollary of Lemma 9.3.…”

“…By [7, Proposition 1.1.5], the inequality (1.1) holds for nearly smooth spaces. For such spaces proper intersection thus has a clear meaning, and an intersection product µ 2 ∩ Y µ 1 for any two properly intersecting cycles µ 1 and µ 2 in Y is introduced in [7]. The main result of this section is the following proposition.…”

“…The ideal (u 2 , v 2 , uv) has the same integral closure as the regular sequence (u Recently, in [7], Barlet and Magnússon introduced a class of analytic spaces called nearly smooth. An analytic space Y is nearly smooth if it is normal and if for each y ∈ Y there is a neighborhood U of y, a connected complex manifold Y , and a proper holomorphic surjective finite mapping q : Y → U .…”

“…There is a proper intersection theory on normal surfaces due to Mumford, [15]. Recently Barlet and Magnússon, [7], defined proper intersections on a so-called nearly smooth Y by analytic methods. In Section 9 we show that for RE-cycles our intersection product coincides with the intersection in [7].…”