Abstract
On a reduced analytic space X we introduce the concept of a generalized cycle, which extends the notion of a formal sum of analytic subspaces to include also a form part. We then consider a suitable equivalence relation and corresponding quotient mathcal {B}(X) that we think of as an analogue of the Chow group and a refinement of de Rham cohomology. This group allows us to study both global and local intersection theoretic properties. We provide many mathcal {B}-analogues of classical intersection theoretic constructions: For an analytic subspace Vsubset X we define a mathcal {B}-Segre class, which is an element of mathcal {B}(X) with support in V. It satisfies a global King formula and, in particular, its multiplicities at each point coincide with the Segre numbers of V. When V is cut out by a section of a vector bundle we interpret this class as a Monge–Ampère-type product. For regular embeddings we construct a mathcal {B}-analogue of the Gysin morphism.
Highlights
Throughout this paper X is a reduced analytic space of pure dimension n and J → X is a coherent ideal sheaf with zero set Z with codimension κ
Following [13] we call these numbers Segre numbers and, we will see in Theorem 1.1 below that they are closely related to Segre classes
To find global representations we introduce an extension GZ(X ) of Z(X ) that we call the Z-module of generalized cycles
Summary
Throughout this paper X is a reduced analytic space of pure dimension n and J → X is a coherent ideal sheaf with zero set Z with codimension κ. Since our second goal concerns intersection theory we will pay special attention to such sheaves J and describe S(J , X ) in more detail In this case the normal cone NJ X is a vector bundle over Z and we let s(NJ X ) = 1 + s1(NJ X ) + s2(NJ X ) + · · · + sn−κ (NJ X ) be its associated total Segre class. As in the case with general ideal sheaves we are interested in specific representatives, so let us assume that J is defined by a section φ of a Hermitian vector bundle F → X and let F be the pull-back of F to Z. 2. Proposition 1.5 If φ is a section of the Hermitian vector bundle F defining J , we have the equality of generalized cycles.
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