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https://hdl.handle.net/2440/136854
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Type: | Journal article |
Title: | A foliated Hitchin-Kobayashi correspondence |
Author: | Baraglia, D. Hekmati, P. |
Citation: | Advances in Mathematics, 2022; 408:108661-1-108661-47 |
Publisher: | Elsevier |
Issue Date: | 2022 |
ISSN: | 0001-8708 1090-2082 |
Statement of Responsibility: | David Baraglia, Pedram Hekmati |
Abstract: | We prove an analogue of the Hitchin-Kobayashi correspondence for compact, oriented, taut Riemannian foliated manifolds with transverse Hermitian structure. In particular, our Hitchin-Kobayashi theorem holds on any compact Sasakian manifold. We define the notion of stability for foliated Hermitian vector bundles with transverse holomorphic structure and prove that such bundles admit a basic Hermitian-Einstein connection if and only if they are polystable. Our proof is obtained by adapting the proof by Uhlenbeck and Yau to the foliated setting. We relate the transverse Hermitian-Einstein equations to higher dimensional instanton equations and in particular we look at the relation to higher contact instantons on Sasakian manifolds. For foliations of complex codimension 1, we obtain a transverse Narasimhan-Seshadri theorem. We also demonstrate that the weak Uhlenbeck compactness theorem fails in general for basic connections on a foliated bundle. This shows that not every result in gauge theory carries over to the foliated setting. |
Keywords: | Hitchin-Kobayashi; Foliations; Stable bundles; Hermitian-Einstein |
Rights: | © 2022 Elsevier Inc. All rights reserved. |
DOI: | 10.1016/j.aim.2022.108661 |
Grant ID: | http://purl.org/au-research/grants/arc/DP110103745 http://purl.org/au-research/grants/arc/DP170101054 |
Published version: | http://dx.doi.org/10.1016/j.aim.2022.108661 |
Appears in Collections: | Mathematical Sciences publications |
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