On the stability of Einstein metrics carrying a special twisted spinor

TL;DR

This paper proves linear semi-stability for a broad class of Einstein manifolds with non-positive scalar curvature that admit a special twisted pure spinor, extending previous stability results to new geometric contexts.

Contribution

It establishes linear semi-stability for Einstein manifolds with twisted pure spinors, generalizing known results for parallel spin and spin^c structures.

Findings

Linear semi-stability proven for Einstein manifolds with non-positive scalar curvature and twisted pure spinors.

Extension of stability results to negative quaternion-Kähler manifolds of dimension >8.

Applicable under mild restrictions on dimension and spinor twisting.

Abstract

We prove linear semi-stability for a large class of Einstein metrics of non-positive scalar curvature. More precisely, we show that any Einstein $n$-manifold with non-positive scalar curvature carrying a parallel twisted pure spin$^r$ spinor is linearly semi-stable, under mild restrictions on $n$ and $r$. We thus extend the parallel spin and spin$^c$ stability results of Dai--Wang--Wei. As an application, our result implies linear semi-stability for all negative quaternion-K{\"a}hler manifolds of dimension greater than $8$.