A vector bundle version of the Monge-Ampere equation

TL;DR

This paper introduces a vector bundle version of the complex Monge-Ampere equation, explores its geometric properties on complex surfaces, and establishes links to stability conditions and inequalities related to vector bundles.

Contribution

It formulates a new vector bundle Monge-Ampere equation, provides a moment map interpretation, and proves stability and inequality results, including a Kobayashi-Hitchin correspondence.

Findings

Existence of solutions implies stability conditions.

Derived a Kobayashi-Lubke-Bogomolov-Miyaoka-Yau type inequality.

Established a Kobayashi-Hitchin correspondence for the reduced equation.

Abstract

We introduce a vector bundle version of the complex Monge-Ampere equation motivated by a desire to study stability conditions involving higher Chern forms. We then restrict ourselves to complex surfaces, provide a moment map interpretation of it, and define a positivity condition (MA positivity) which is necessary for the infinite-dimensional symplectic form to be Kahler. On rank-2 bundles on compact complex surfaces, we prove two consequences of the existence of a "positively curved" solution to this equation - Stability (involving the second Chern character) and a Kobayashi-Lubke-Bogomolov-Miyaoka-Yau type inequality. Finally, we prove a Kobayashi-Hitchin correspondence for a dimensional reduction of the aforementioned equation.