# Chow groups and pseudoeffective cones of complexity one $T$-varieties

## TL;DR

This paper studies the structure of pseudoeffective cones and Chow groups of complexity one $T$-varieties, revealing their rational polyhedral nature and providing explicit presentations based on combinatorial data.

## Contribution

It proves the rational polyhedrality of pseudoeffective cones for all $k$-cycles and offers a combinatorial presentation of Chow groups for rational $T$-varieties.

## Key findings

- Pseudoeffective cones are rational polyhedral for all $k$-cycles.
- Chow groups of rational $T$-varieties are described by generators and relations.
- The structure is explicitly linked to the combinatorial data defining the $T$-variety.

## Abstract

We show that the pseudoeffective cone of $k$-cycles on a complete complexity one $T$-variety is rational polyhedral for any $k$, generated by classes of $T$-invariant subvarieties. When $X$ is also rational, we give a presentation of the Chow groups of $X$ in terms of generators and relations, coming from the combinatorial data defining $X$ as a $T$-variety.