# The complex Monge-Amp\`ere equation with a gradient term

## TL;DR

This paper studies a modified complex Monge-Ampère equation that includes a gradient term, establishing existence and uniqueness of solutions on compact Hermitian manifolds, thus extending classical results to a more general setting.

## Contribution

It introduces and analyzes a complex Monge-Ampère equation with a gradient term, proving fundamental existence and uniqueness results in the Hermitian manifold context.

## Key findings

- Existence of solutions on compact Hermitian manifolds
- Uniqueness of solutions under certain conditions
- Extension of classical Monge-Ampère theory to gradient-including equations

## Abstract

We consider the complex Monge-Amp\`ere equation with an additional linear gradient term inside the determinant. We prove existence and uniqueness of solutions to this equation on compact Hermitian manifolds.