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Basic Dolbeault cohomology and Weitzenböck frmulas on transversely Kähler foliations

Seoung Dal Jung, Ken Richardson

Abstract

We study basic Dolbeault cohomology and find new Weitzenböck formulas on a transversely Kähler foliation. We investigate conditions on mean curvature and Ricci curvature that impose restrictions on basic Dolbeault cohomology. For example, we prove that on a transversely Kähler foliation with positive transversal Ricci curvature, there are no nonzero basic-harmonic forms of type $(r,0)$, among other results.

Basic Dolbeault cohomology and Weitzenböck frmulas on transversely Kähler foliations

Abstract

We study basic Dolbeault cohomology and find new Weitzenböck formulas on a transversely Kähler foliation. We investigate conditions on mean curvature and Ricci curvature that impose restrictions on basic Dolbeault cohomology. For example, we prove that on a transversely Kähler foliation with positive transversal Ricci curvature, there are no nonzero basic-harmonic forms of type , among other results.

Paper Structure

This paper contains 4 sections, 34 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The basic Dolbeault cohomology on transversely Kähler foliations
  4. The Weitzenböck formula

Key Result

Proposition 2.1

(In Al,PaRi for the compact case) Let $(M,\mathcal{F})$ be a Riemannian foliation of codimension $q$ such that the leaf closures of $\mathcal{F}$ are compact. Then the formal adjoint operators $\delta_B$ and $\delta_T$ of $d_B$ and $d_T$ with respect to $\ll\cdot,\cdot\gg$ on basic forms are given b respectively, applied to basic $r$-forms $\phi$.

Theorems & Definitions (44)

  • Proposition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Corollary 2.7
  • Theorem 2.8
  • Example 3.1
  • Definition 3.2
  • ...and 34 more