Cellular $\mathbb{A}^1$-Homology of Smooth Toric Varieties

TL;DR

This paper computes cellular e1^1-homology for smooth toric varieties, providing explicit descriptions for shellable cases and deriving motivic decompositions and Chow group bases.

Contribution

It offers the first explicit calculations of cellular e1^1-homology and motivic decompositions for smooth toric varieties, including shellable cases.

Findings

Explicit cellular e1^1-homology calculations for smooth toric varieties

Motivic decompositions for pure shellable cases

An additive basis for Chow groups of smooth toric varieties

Abstract

In this paper, we present the calculations of cellular $\mathbb{A}^1$-homology for smooth toric varieties, along with an explicit description of pure shellable cases. Consequently, we derive the (Milnor-Witt) motivic decomposition for these pure shellable cases. Furthermore, we obtain an additive basis for the Chow groups of general smooth toric varieties.