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arXiv · 1804.03934

A vector bundle version of the Monge-Ampere equation

Vamsi Pritham Pingali
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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.

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Vamsi Pritham Pingali. 2018-04-11. A vector bundle version of the Monge-Ampere equation. https://arxiv.org/abs/1804.03934

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