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Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds. (English) Zbl 1188.53042

Sasakian manifolds are known to have various interesting structures. They have a distinguished one-dimensional foliation, the so-called Reeb foliation, which determines a transverse Kähler structure. The cone over a Sasakian manifold has a Kähler structure. Every Sasakian manifold has a contact structure and an associated contact bundle. In this article the authors study deformations of Sasakian structures fixing the Reeb foliation, its transverse holomorphic structure and the holomorphic structure of the cone, and varying the Kähler metric on the transverse holomorphic structure. Such deformations correspond to deformations of Kähler forms in a fixed Kähler class on a Kähler manifold, and the aim of the paper is to extend results related to Calabi’s extremal problem in Kähler geometry to the above setting in Sasakian geometry.
The authors define integral invariants which obstruct the existence of transverse Kähler metrics with harmonic Chern forms, and the integral invariant \(f_1\) for the first Chern class case becomes an obstruction to the existence of a transverse Kähler metric with constant scalar curvature. Let \(S\) be a compact toric Sasakian manifold, and assume that the basic first Chern form of the normal bundle of the Reeb foliation is positive and that the first Chern class of the contact bundle is trivial. The authors prove that there exists a Sasaki metric on \(S\) which is a Sasaki-Ricci soliton. This implies that \(S\) admits a Sasaki-Einstein metric if and only if the first integral invariant \(f_1\) vanishes. As an application they prove that one can get a Sasaki-Einstein metric on \(S\) by deforming the Sasaki structure varying the Reeb vector field.
The authors then discuss some applications. One of them states that there exist Sasaki-Einstein metrics on the total space of the \(U(1)\)-bundles associated with the powers of canonical line bundles of toric Fano manifolds. As a particular consequence of this one gets the existence of an irregular toric Sasaki-Einstein metric on the unit circle bundle associated with the canonical line bundle of the two-point blow up of the complex projective plane.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53D10 Contact manifolds (general theory)
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