# Coupled complex Monge-Amp\`ere equations on Fano horosymmetric manifolds

## TL;DR

This paper establishes criteria for solving coupled complex Monge-Ampère equations on Fano horosymmetric manifolds, enabling the characterization of special Kähler metrics through combinatorial data.

## Contribution

It provides necessary and sufficient conditions for the existence of coupled Kähler-Ricci solitons, Mabuchi metrics, and twisted Kähler-Einstein metrics on these manifolds.

## Key findings

- Derived combinatorial criteria for metric existence
- Characterized solutions for coupled Kähler-Ricci equations
- Connected geometric solutions with manifold data

## Abstract

We give necessary and sufficient conditions for existence of solutions to a general system of complex Monge-Amp\`ere equations on Fano horosymmetric manifolds. In particular, we get necessary and sufficient conditions for existence of coupled K\"ahler-Ricci solitons, Mabuchi metrics and twisted K\"ahler-Einstein metrics in terms of combinatorial data of the manifold.