On a Morelli type expression of cohomology classes of toric varieties

TL;DR

This paper presents a canonical expression for homology classes of complete $Q$-factorial toric varieties as linear combinations of orbit closure classes, generalizing Morelli's Todd class formula using rational functions on Grassmannians.

Contribution

It introduces a new canonical method to express homology classes of toric varieties in terms of orbit closures, extending Morelli's formula with rational function coefficients.

Findings

Provides a canonical linear combination representation of homology classes.

Generalizes Morelli's formula for Todd classes.

Uses rational functions on Grassmannians for coefficients.

Abstract

Let $X$ be a complete $\Q$-factorial toric variety of dimension $n$ and $\del$ the fan in a lattice $N$ associated to $X$. For each cone $\sigma$ of $\del$ there corresponds an orbit closure $V(\sigma)$ of the action of complex torus on $X$. The homology classes $\{[V(\sigma)]\mid \dim \sigma=k\}$ form a set of specified generators of $H_{n-k}(X,\Q)$. It is shown that, given $\alpha\in H_{n-k}(X,\Q)$, there is a canonical way to express $\alpha$ as a linear combination of the $[V(\sigma)]$ with coefficients in the field of rational functions of degree $0$ on the Grassmann manifold of $(n-k+1)$-planes in $N_\Q$. This generalizes Morelli's formula for $\alpha$ the $(n-k)$-th component of the Todd homology class of the variety $X$.