| Summary: | We study the transverse Kähler holonomy groups on Sasaki manifolds (M, 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 b1(M) and the basic Hodge number h0,2B(S) vanish, then S is stable under deformations of the transverse Kähler flow. In addition we show that an irreducible transverse hyperkähler Sasakian structure is S-unstable, whereas, an irreducible transverse Calabi-Yau Sasakian structure is S-stable when dim M ≥ 7. Finally, we prove that the standard Sasaki join operation (transverse holonomy U(n1) × U(n2)) as well as the fiber join operation preserve S-stability.
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