math.DG

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# Title: Transverse Kähler holonomy in Sasaki Geometry and ${\oldmathcal S}$-Stability

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.
 Comments: 33 pages. An incorrect lemma was removed, and appropriate revisions made. Paper shortened by removing categorical details. Comments welcome Subjects: Differential Geometry (math.DG); Symplectic Geometry (math.SG) Journal reference: Complex Manifolds 8 (2021), no. 1, 336-353 Cite as: arXiv:2103.01112 [math.DG] (or arXiv:2103.01112v3 [math.DG] for this version)

## Submission history

From: Charles P. Boyer [view email]
[v1] Mon, 1 Mar 2021 16:24:50 GMT (32kb)
[v2] Fri, 7 May 2021 15:24:31 GMT (32kb)
[v3] Tue, 5 Oct 2021 16:10:28 GMT (27kb)

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