# Global representation of Segre numbers by Monge-Amp\`ere products

## TL;DR

This paper introduces a new framework using generalized cycles and a quotient group to extend intersection theory, linking Segre numbers with Monge-Ampère products on analytic spaces.

## Contribution

It develops a novel group $ ext{B}(X)$ extending classical intersection theory, defining $ ext{B}$-Segre classes and Gysin morphisms, and relating Segre numbers to Monge-Ampère products.

## Key findings

- Defines a $ ext{B}$-Segre class with support in $V$
- Establishes a King formula for $ ext{B}$-Segre classes
- Interprets classes as Monge-Ampère-type products

## 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 $V\subset X$ 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-Amp\`ere-type product. For regular embeddings we construct a $\mathcal{B}$-analogue of the Gysin morphism.