Monge-Amp\`ere type equation for the Nakano positive curvature tensor of holomorphic vector bundles

TL;DR

This paper solves a Monge-Ampère type equation associated with Nakano positive curvature tensors in Hermitian holomorphic vector bundles, extending classical results to higher rank bundles.

Contribution

It introduces a method to solve the Monge-Ampère type equation within the conformal class of Nakano positive Hermitian metrics for higher rank bundles.

Findings

Established existence of solutions in the conformal class

Extended classical Monge-Ampère results to vector bundles of higher rank

Provided new tools for studying Nakano positivity in complex geometry

Abstract

For any Hermitian holomorphic vector bundle with Nakano positive curvature tensor, Demailly introduced a Monge-Amp\`ere type equation. When the rank of the bundle is $1$, it becomes the usual Monge-Amp\`ere equation. In this paper, we solve this equation in the conformal class of a Nakano positive Hermitian metric.