On the deformation of K？hler metrics in the presence of closed conformal vector fields

In this paper we show that if a K\"ahler manifold is furnished with a closed conformal vector field, then, in suitable open subsets, its K\"ahler structure can be deformed into a different one. This deformation can be applied to the whole class of K\"ahler manifolds that arise as Riemannian cones over Sasaki manifolds, and gives an infinite family of new examples of K\"ahler manifolds, in each complex dimension greater than $2$. Moreover, if we deform the K\"ahler metric of the cone over an Einstein-Sasaki manifold, then we get an Einstein-K\"ahler manifold, and a particular class of such deformations presents the same phenomenon of sectional holomorphic curvature decay that takes place when we deform the metric of the complex Euclidean space into that of the complex hyperbolic space. Two infinite families of such deformations arise by taking any of the two infinite families of Einstein-Sasaki manifolds constructed by Cvetic, Lu, Page and Pope.