Kähler–Einstein Metrics

Abstract. In this paper we study compact Sasaki manifolds in view of transverse Kähler geometry and extend some results in Kähler geometry to Sasaki manifolds. In particular we define integral invariants which obstruct the existence of transverse Kähler metric with harmonic Chern forms. The integral invariant f 1 for the first Chern class case becomes an obstruction to the existence of transverse Kähler metric of constant scalar curvature. We prove the existence of transverse Kähler-Ricci solitons (or Sasaki-Ricci soliton) on compact toric Sasaki manifolds whose basic first Chern form of the normal bundle of the Reeb foliation is positive and the first Chern class of the contact bundle is trivial. We will further show that if S is a compact toric Sasaki manifold with the above assumption then by deforming the Reeb field we get a Sasaki-Einstein structure on S. As an application we obtain irregular toric Sasaki-Einstein metrics on the unit circle bundles of the powers of the canonical bundle of the two-point blow-up of the complex projective plane.

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“…These constructions are surveyed in the article [7] (see also [8]). This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds

Futaki¹,

Ono²,

Wang³

2009

J. Differential Geom.

211

478

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“…In particular, rich infinite classes of examples were produced in the 1990s via a quotient construction (essentially the hyperKähler quotient). A review of those developments was given in a previous article in this journal series [14], with a more recent account appearing in [19]. We note that the first non-trivial dimension for a 3-Sasakian manifold is dim S = 7, and also that 3-Sasakian manifolds are automatically regular or quasi-regular as Sasaki-Einstein manifolds (indeed, the first quasi-regular Sasaki-Einstein manifolds constructed were 3-Sasakian 7-manifolds).…”

Section: Regularitymentioning

confidence: 79%

“…Also as in the smooth manifold case, one can use the continuity method to great effect. Our discussion in the remainder of this section will closely follow [19].…”

Section: 3mentioning

confidence: 99%

Sasaki-Einstein Manifolds

Sparks¹

2010

Preprint

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A Sasaki-Einstein manifold is a Riemannian manifold (S, g) that is both Sasakian and Einstein.Sasakian geometry is the odd-dimensional cousin of Kähler geometry. Indeed, just as Kähler geometry is the natural intersection of complex, symplectic, and Riemannian geometry, so Sasakian geometry is the natural intersection of CR, contact, and Riemannian geometry. Perhaps the most straightforward definition is the following: a Riemannian manifold (S, g) is Sasakian if and only if its metric cone C(S) = R >0 × S, ḡ = dr 2 + r 2 g is Kähler. In particular, (S, g) has odd dimension 2n − 1, where n is the complex dimension of the Kähler cone.A metric g is Einstein if Ric g = λg for some constant λ. It turns out that a Sasakian manifold can be Einstein only for λ = 2(n − 1), so that g has positive Ricci curvature. Assuming, as we shall do throughout, that (S, g) is complete, it follows from Myers' Theorem that S is compact with finite fundamental group. Moreover, a simple calculation shows that a Sasakian metric g is Einstein with Ric g = 2(n − 1)g if and only if the cone metric ḡ is Ricci-flat, Ric ḡ = 0. It immediately follows that for a Sasaki-Einstein manifold the restricted holonomy group of the cone Hol 0 (ḡ) ⊂ SU (n).The canonical example of a Sasaki-Einstein manifold is the odd dimensional sphere S 2n−1 , equipped with its standard Einstein metric. In this case the Kähler cone is C n \ {0}, equipped with its flat metric.A Sasakian manifold (S, g) inherits a number of geometric structures from the Kähler structure of its cone. In particular, an important role is played by the Reeb vector field. This may be defined as ξ = J(r∂ r ), where J denotes the integrable complex structure of the Kähler cone. The restriction of ξ to S = {r = 1} = {1} × S ⊂ C(S) is a unit length Killing vector field, and its orbits thus define a one-dimensional foliation of S called the Reeb foliation. There is then a classification of Sasakian manifolds, and hence also Sasaki-Einstein manifolds, according to the global properties of this foliation. If all the orbits of ξ are compact, and hence circles, then ξ

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Kähler–Einstein Metrics

Cited by 2 publications

References 20 publications

Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds

Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds

Sasaki-Einstein Manifolds

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