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Top-Degree Global Solvability in CR and Locally Integrable Hypocomplex Structures

Abstract

We study the top-degree cohomology for the \({\bar{\partial }_b}\) operator defined on a generic submanifold of the complex space as well as for the differential complex associated with a locally integrable structure \({\mathcal {V}}\) over a smooth manifold. The main assumptions are that \({\mathcal {V}}\) is hypocomplex and that the differential complex is locally solvable in degree one. One of the main tools is an adaptation of a sheaf theoretical argument due to Ramis–Ruget–Verdier.

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Notes

  1. This is a consequence of the well known “five-term exact sequence” associated to a first quadrant spectral sequence.

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Correspondence to Paulo D. Cordaro.

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During the development of this work, P. D. Cordaro was partially supported by FAPESP (2012/03168-7) and CNPq. M. R. Jahnke was supported by CNPq (process 140199/2014-4) and CAPES (PDSE 88881.131905/2016-01).

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Cordaro, P.D., Jahnke, M.R. Top-Degree Global Solvability in CR and Locally Integrable Hypocomplex Structures. J Geom Anal 31, 8156–8172 (2021). https://doi.org/10.1007/s12220-020-00573-1

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  • DOI: https://doi.org/10.1007/s12220-020-00573-1

Keywords

  • Tangential Cauchy-Riemann complexes
  • Hypocomplex manifolds
  • Top-degree cohomology

Mathematics Subject Classification

  • 35A01
  • 35G05
  • 32V05
  • 32V40