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

By A. Futaki, Hajime Ono, Guofang Wang. 2006

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In this paper we study compact Sasaki manifolds in view of transverse K\"ahler geometry and extend some results in K\"ahler geometry to Sasaki manifolds. In particular we define integral invariants which obstruct the existence of transverse K\"ahler metric with harmonic Chern forms. The integral invariant $f_1$ for the first Chern class case becomes an obstruction to the existence of transverse K\"ahler metric of constant scalar curvature. We prove the existence of transverse K\"ahler-Ricci solitons (or {\it 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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/eca9ce6e0a3a5eef0fd6f68beeecb2af43cf54ad

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Frankel conjecture and Sasaki geometry (2012, methodology): “By [25] a Sasaki manifold is homothetic to a normalized one if and only if the basic first Chern class c1 is positive definite and the contact subbundle D has vanishing first Chern class.”
Sasaki-Einstein Manifolds (2010, methodology): “10) The equivalence of (1) and (2) follows immediately from the long exact sequence [104] relating the basic cohomology of the foliation Fξ to the cohomology of S (see [48]).”
Calabi-Yau structures and Einstein-Sasakian structures on crepant resolutions of isolated singularities (2009, methodology): “s a K¨ahler class which does not belong to the compactly supported cohomology group. Thus in the cases, we obtain a new family of Ricci-ﬂat K¨ahler metrics on KZ. Example 6.4 Recently Futaki-Ono-Wang [10] obtained an Einstein-Sasakian metric on the sphere bundle of the canonical line bundle of every”
Maximally homogeneous para-CR manifolds of semisimple type (2006, unknown): “This existence of a Sasaki-Einstein metric is proved in [ FOW06 ] although the authors do not draw all the conclusions regarding possible toric Sasaki-Einstein manifolds that can be obtained.”
Examples of asymptotically conical Ricci-flat Kähler manifolds (2008, background): “r cones. But a Kahler cone is a cone over a Sasaki manifold and is Ricci-ﬂat precisely when the Sasaki manifold is Einstein. And there has been much research recently on SasakiEinstein manifolds (cf. [7, 9, 24]). See [9] for more details on Sasaki manifolds. Deﬁnition 2.1. A Riemannian manifold (S,g”
The large N limit of quiver matrix models and Sasaki-Einstein manifolds (2011, result): “Note that although the Sasaki-Einstein metric is not known in explicit form in this case, its existence is guaranteed by the results of [38].”
Sasaki–Einstein Manifolds and Volume Minimisation (2006, background): “However, the real Monge–Ampère equation derived in [22] has recently been shown to always admit a solution [34], thus solving the existence problem for toric Sasaki–Einstein manifolds.”
Sasaki–Einstein metrics and K–stability (2015, result): “Note that it was previously shown by Futaki-Ono-Wang [44] that toric Fano cone singularities with an isolated singularity admit Ricci flat cone metrics, and we expect the same to hold when there are singularities away from the cone point.”
Ricci-flat Kähler metrics on crepant resolutions of Kähler cones (2008, background): “One can show [15] that if”
Asymptotically conical Calabi-Yau manifolds, III (2012, background): “nsider the Fano 3-fold X= Bl pP3, with D2j 1 2 K Xjthe strict transform of a smooth quadric passing through p, and we x one of the irregular Calabi-Yau cone structures (g0; 0) on N Dn0 constructed in [20]. Our goal is to prove Theorem C: the existence of AC CalabiYau structures (g; ) on the ane vari”