Global representation of Segre numbers by Monge–Ampère products

Abstract

AbstractOn 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 $$\mathcal {B}(X)$$ 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 $$\mathcal {B}$$ B -analogues of classical intersection theoretic constructions: For an analytic subspace $$V\subset X$$ V ⊂ X we define a $$\mathcal {B}$$ B -Segre class, which is an element of $$\mathcal {B}(X)$$ 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 $$\mathcal {B}$$ B -analogue of the Gysin morphism.