Deformation of Sasakian metrics

Abstract

Abstract. Deformations of the Reeb flow of a Sasakian manifold as transversely Kähler flows may not admit compatible Sasakian metrics. We show that the triviality of the (0, 2)-component of the basic Euler class characterizes the existence of compatible Sasakian metrics for given small deformations of the Reeb flow as transversely holomorphic Riemannian flows. We also prove a Kodaira-Akizuki-Nakano type vanishing theorem for basic Dolbeault cohomology of homologically orientable transversely Kähler foliations. As a consequence of these results, we show that any small deformations of the Reeb flow of a positive Sasakian manifold admit compatible Sasakian metrics.

Stability of Sasaki-Extremal Metrics under Complex Deformations

“…But the existence of a compatible Sasaki structure may be obstructed. The obstruction, due to H. Nozawa [22], is reviewed in Section 3.1. An unobstructed deformation (F ξ ,J t ) t∈B is said to be of (1, 1)-type.…”

“…An application of a transversal Kodaira-Nakano vanishing theorem gives the following which is basically Corollary 1.4 of [22].…”

“…Note that (22) implies that X ∈ fol(M, F ξ ); that is, it is automatically foliate. Also, condition (22) is equivalent to the (1, 0) vector field…”

“…By applying the familiar averaging argument using a Haar measure on G ′ to (g, η, ξ, Φ) we get a Sasaki structure (g,η,ξ,Φ) with G ′ = Aut(g,η,ξ,Φ) 0 . It is proved in [22] that there exist an f ∈ Fol(M, F ξ ,J) so that f * η (ξ) = c ∈ R >0 . This follows from a leaf wise version of Moser's argument which can be used to prove f * η | L ξ = cη| L ξ .…”

“…Although the transversely Kähler property is stable, there is a further obstruction to the existence of a Sasaki structure compatible with (F ξ ,J t ) for t ∈ B. The necessary and sufficient condition for the existence of a Sasaki structure were obtained in [22] for more general deformations, not necessarily preserving the smooth structure of F . But for our purposes, we will only consider deformations of (F ξ ,J) preserving the smooth foliation structure.…”