The convex-set algebra and the toric b-Chow group

TL;DR

This paper extends the intersection theory of toric b-divisors to a convex-set algebra, providing a ring structure that captures intersection products and satisfies Hodge-type inequalities.

Contribution

It introduces the convex-set algebra extending McMullen's polytope algebra, embedding it into the toric b-Chow group for a generalized intersection theory.

Findings

Defines the convex-set algebra and embeds it into the toric b-Chow group.

Extends intersection product to arbitrary codimension for positive toric b-classes.

Shows Hodge type inequalities hold within the convex-set algebra.

Abstract

In \cite{botero}, a top intersection product of toric b-divisors on a smooth complete toric variety is defined. It is shown that a nef toric b-divisor corresponds to a convex set and that its top intersection number equals the volume of this convex set. The goal of this article is to extend this result and define an intersection product of sufficiently positive toric b-classes of arbitrary codimension. For this, we extend the polytope algebra of McMullen (\cite{McM1}, \cite{McM2}, \cite{McM3}) to the so called \emph{convex-set algebra} and we show that it embeds in the toric b-Chow group. In this way, the convex-set algebra can be viewed as a ring for an intersection theory for sufficiently positive toric b-classes. As an application, we show that some Hodge type inequalities are satisfied for the convex-set algebra.