A vector bundle version of the Monge-Ampere 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.
Cite
@article{arxiv.1804.03934,
title = {A vector bundle version of the Monge-Ampere equation},
author = {Vamsi Pritham Pingali},
journal= {arXiv preprint arXiv:1804.03934},
year = {2022}
}
Comments
Corrected the statement of Lemma 2.4 and the proof of Lemma 4.9