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A foliated Hitchin-Kobayashi correspondence
- Source :
- Adv. Math. 408 (2022), 108661
- Publication Year :
- 2018
-
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 Sasaki 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.<br />Comment: 42 pages
Details
- Database :
- arXiv
- Journal :
- Adv. Math. 408 (2022), 108661
- Publication Type :
- Report
- Accession number :
- edsarx.1802.09699
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1016/j.aim.2022.108661