Transverse K\"ahler holonomy in Sasaki Geometry and ${\oldmathcal S}$-Stability

TL;DR

This paper investigates the stability of transverse Kähler holonomy groups on Sasaki manifolds under deformations, identifying conditions for stability and instability, and analyzing the effects of join operations.

Contribution

It establishes criteria for stability of Sasaki structures under transverse deformations and examines how join operations affect this stability.

Findings

Vanishing first Betti number and basic Hodge number imply stability.

Transverse hyperk"ahler structures are unstable.

Transverse Calabi-Yau structures are stable in dimension ≥7.

Abstract

We study the transverse K\"ahler holonomy groups on Sasaki manifolds $(M,{\oldmathcal S})$ and their stability properties under transverse holomorphic deformations of the characteristic foliation by the Reeb vector field. In particular, we prove that when the first Betti number $b_1(M)$ and the basic Hodge number $h^{0,2}_B({\oldmathcal S})$ vanish, then ${\oldmathcal S}$ is stable under deformations of the transverse K\"ahler flow. In addition we show that an irreducible transverse hyperk\"ahler Sasakian structure is ${\oldmathcal S}$-unstable, whereas, an irreducible transverse Calabi-Yau Sasakian structure is ${\oldmathcal S}$-stable when $\dim M\geq 7$. Finally, we prove that the standard Sasaki join operation (transverse holonomy $U(n_1)\times U(n_2)$) as well as the fiber join operation preserve ${\oldmathcal S}$-stability.