# Logarithmic Chow semistability of polarized toric manifolds

## TL;DR

This paper explores the relationship between logarithmic Chow semistability and log K-semistability in polarized toric manifolds, providing combinatorial criteria and examples, including non-semistable cases with conical Kähler Einstein metrics.

## Contribution

It introduces an obstruction criterion for semistability in polarized toric manifolds and links asymptotic log Chow semistability to log K-semistability using combinatorial methods.

## Key findings

- Established an obstruction criterion for semistability.
- Proved that asymptotic log Chow semistability implies log K-semistability.
- Presented a non-semistable example with a conical Kähler Einstein metric.

## Abstract

The logarithmic Chow semistability is a notion of Geometric Invariant Theory for the pair consists of varieties and its divisors. In this paper we introduce a obstruction of semistability for polarized toric manifolds and its toric divisors. As its application, we show the implication from the asymptotic log Chow semistability to the log K-semistability by combinatorial arguments. Furthermore we give a non-semistable example which has a conical Kahler Einstein metric.