Cohomologie de Dolbeault feuillet\'ee du feuilletage complexe affine de Reeb

TL;DR

This paper investigates the Dolbeault cohomology of a specific complex affine Reeb foliation on Hopf manifolds, establishing an explicit isomorphism in degree one.

Contribution

It provides a precise computation of the foliated Dolbeault cohomology for the complex affine Reeb foliation on Hopf manifolds, including an explicit generator.

Findings

Foliated Dolbeault cohomology in degree 1 is isomorphic to complex numbers

Explicit generator of the cohomology is constructed

Results deepen understanding of cohomological properties of Reeb foliations

Abstract

Let ${\cal F}$ be a complex affine Reeb foliation of dimension $1$ on the Hopf manifold ${\Bbb S}^{n+1}\times {\Bbb S}^1$. We prove that its foliated Dolbeault cohomology in degree $1$ is isomorphic to ${\Bbb C}$ by giving an explicit generator.