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A vector bundle version of the Monge-Ampère equation

Pingali, VP (2020) A vector bundle version of the Monge-Ampère equation. In: Advances in Mathematics, 360 .

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Official URL: https://doi.org/10.1016/j.aim.2019.106921

Abstract

We introduce a vector bundle version of the complex Monge-Ampère 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 Kähler. 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-Lübke-Bogomolov-Miyaoka-Yau type inequality. Finally, we prove a Kobayashi-Hitchin correspondence for a dimensional reduction of the aforementioned equation.

Item Type: Journal Article
Publication: Advances in Mathematics
Publisher: Academic Press Inc.
Additional Information: The copyright for this article belongs to the Authors.
Keywords: Kazdan-Warner; Kobayashi-Hitchin correspondence; Moment-map; Vector bundle Monge-Ampère equation; Vortex bundle
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Date Deposited: 24 Jan 2023 12:00
Last Modified: 24 Jan 2023 12:00
URI: https://eprints.iisc.ac.in/id/eprint/79467

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