Vanishing theorem for transverse Dirac operators on Riemannian foliations

TL;DR

This paper proves a vanishing theorem for the half-kernel of a transverse Spin^c Dirac operator on a compact Riemannian foliation with specific curvature conditions, extending understanding of geometric analysis in foliated manifolds.

Contribution

It introduces a vanishing theorem for transverse Dirac operators on Riemannian foliations with particular curvature properties, advancing the theory of geometric operators in foliated spaces.

Findings

Vanishing of the half-kernel under large twisting line bundle

Conditions on curvature along leaves and transversely

Extension of vanishing theorems to foliated manifolds

Abstract

We obtain a vanishing theorem for the half-kernel of a transverse ${\rm Spin}\sp c$ Dirac operator on a compact manifold endowed with a transversely almost complex Riemannian foliation twisted by a sufficiently large power of a line bundle, whose curvature vanishes along the leaves and is transversely non-degenerate at any point of the ambient manifold.