Deformations of Constant Scalar Curvature Sasakian Metrics and K-Stability

Extending the work of G. Szekelyhidi and T. Bronnle to Sasakian manifolds, we prove that a small deformation of the complex structure of the cone of a constant scalar curvature Sasakian (cscS) manifold admits a constant scalar curvature structure if it is K-polystable. This also implies that a small deformation of the complex structure of the cone of a constant scalar curvature structure is K-semistable. As applications, we give examples of cscS manifolds which are deformations of toric examples, and we also show that if a 3-Sasakian manifold admits a nontrivial transversal complex deformation, then it admits a nontrivial Sasaki-Einstein deformation.