We define a stability condition for a polarised algebraic variety and state a conjecture relating this to the existence of a Kahler metric of constant scalar curvature. The main result of the paper goes some way towards verifying this conjecture in the case of toric surfaces. We prove that, under the stability hypothesis, the Mabuchi functional is bounded below on invariant metrics, and that minimising sequences have a certain convergence property. In the reverse direction, we give new examples of polarised surfaces which do not admit metrics of constant scalar curvature. The proofs use a general framework, developed by Guillemin and Abreu, in which invariant Kahler metrics correspond to convex functions on the moment polytope of a toric variety. This paper is a step towards the solution of the general problem of finding conditions under which a complex projective variety admits a Kahler metric of constant scalar curvature. The pattern of the answer one expects is that this differential geometric condition should be equivalent to some notion of “stability” in the sense of Geometric Invariant Theory. This expectation is probably now an item of folklore: going back to suggestions put forward by Yau in the case of KahlerEinstein metrics, and the many results of Tian and others in this case; reinforced by a detailed formal picture which makes clear the analogy with the well-established relation between the stability of vector bundles and Yang-Mills connections [5]. Here, we begin the investigation of

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Scalar Curvature and Stability of Toric Varieties

We show that the Reeb vector, and hence in particular the volume, of a Sasaki–Einstein metric on the base of a toric Calabi–Yau cone of complex dimension n may be computed by minimising a function Z on

 $$\mathbb {R}^{n}$$
 which depends only on the toric data that defines the singularity. In this way one can extract certain geometric information for a toric Sasaki–Einstein manifold without finding the metric explicitly. For complex dimension n  =  3 the Reeb vector and the volume correspond to the R–symmetry and the a central charge of the AdS/CFT dual superconformal field theory, respectively. We therefore interpret this extremal problem as the geometric dual of a–maximisation. We illustrate our results with some examples, including the Y
 p,q
 singularities and the complex cone over the second del Pezzo surface.

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The Geometric Dual of a-maximisation for Toric Sasaki-Einstein Manifolds

In this paper we study compact Sasaki manifolds in view of transverse Kahler geometry and extend some results in Kahler geometry to Sasaki manifolds. In particular we define integral invariants which obstruct the existence of transverse Kahler metric with harmonic Chern forms. The integral invariant $f1$ for the first Chern class case becomes an obstruction to the existence of transverse Kahler metric of constant scalar curvature. We prove the existence of transverse Kahler-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 Sasaki-Einstein metrics on the $U(1)$-bundles associated with the canonical line bundles of toric Fano manifolds, including as a special case an irregular toric Sasaki-Einstein metrics on the unit circle bundle associated with the canonical bundle of the two-point blow-up of the complex projective plane.

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

Gravitational entropy arises in string theory via coarse graining over an underlying space of microstates. In cases with enough supersymmetry, it has been possible to explicitly construct such microstates in spacetime and understand how coarse graining of non-singular, horizon-free objects can lead to an effective description as an extremal black hole. We discuss how these results arise in type II string theory on AdS5 × S5 and on AdS3 × S3 × T4 that preserve 16 and eight supercharges, respectively. For such a picture of black holes as effective geometries to extend to cases with a finite horizon area, the scale of quantum effects in gravity would have to extend well beyond the vicinity of the singularities in the effective theory. By studying examples in M-theory on AdS3 × S2 × CY that preserve four supersymmetries, we show how this can happen.

Black holes as effective geometries

Manifolds all of whose geodesics are closed

A (symplectic) toric variety X, of real dimension 2n, is completely determined by its moment polytope Δ ⊂ ℝn Recently Guillemin gave an explicit combinatorial way of constructing "toric" Kahler metrics on X, using only data on Δ In this paper, differential geometric properties of these metrics are investigated using Guillemin's construction In particular, a nice combinatorial formula for the scalar curvature R is given, and the Euler–Lagrange condition for such "toric" metrics being extremal (in the sense of Calabi) is proven to be R being an affine function on Δ ⊂ ℝn A construction, due to Calabi, of a 1-parameter family of extremal Kahler metrics of non-constant scalar curvature on is recast very simply and explicitly using Guillemin's approach Finally, we present a curious combinatorial identity for convex polytopes Δ ⊂ ℝn that follows from the well-known relation between the total integral of the scalar curvature of a Kahler metric and the wedge product of the first Chern class of the underlying complex manifold with a suitable power of the Kahler class

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Kähler geometry of toric varieties and extremal metrics

Recently Guillemin gave an explicit combinatorial way of constructing "toric" Kahler metrics on (symplectic) toric varieties, using only data on the moment polytope. In this paper, differential geometric properties of these metrics are investigated using Guillemin's construction. In particular, a nice combinatorial formula for the scalar curvature is given, and the Euler-Lagrange condition for such "toric" metrics being extremal (in the sense of Calabi) is derived. A construction, due to Calabi, of a 1-parameter family of extremal metrics of non-constant scalar curvature is recast very simply and explicitly. Finally, a curious combinatorial formula for convex polytopes, that follows from the relation between the total integral of the scalar curvature and the wedge product of the first Chern class with a suitable power of the Kahler class, is presented.

Kahler geometry of toric varieties and extremal metrics

A theorem of Delzant states that any symplectic manifold $(M,\om)$ of dimension $2n$, equipped with an effective Hamiltonian action of the standard $n$-torus $\T^n = \R^{n}/2\pi\Z^n$, is a smooth projective toric variety completely determined (as a Hamiltonian $\T^n$-space) by the image of the moment map $\phi:M\to\R^n$, a convex polytope $P=\phi(M)\subset\R^n$ In this paper we show, using symplectic (action-angle) coordinates on $P\times \T^n$, how all $\om$-compatible toric complex structures on $M$ can be effectively parametrized by smooth functions on $P$ We also discuss some topics suited for application of this symplectic coordinates approach to K\"ahler toric geometry, namely: explicit construction of extremal K\"ahler metrics, spectral properties of toric manifolds and combinatorics of polytopes

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Kahler geometry of toric manifolds in symplectic coordinates

Let M be either S 2 S 2 or the one point blow-up CP 2 # CP 2 of CP 2 . In both cases M carries a family of symplectic forms ! , where > 1 determines the cohomology class [! ]. This paper calculates the rational (co)homology of the group G of symplectomorphisms of (M; ! ) as well as the rational homotopy type of its classifying space BG . It turns out that each group G contains a nite collection Kk; k = 0; : : : ; ‘ = ‘( ), of nite dimensional Lie subgroups that generate its homotopy. We show that these subgroups \asymptotically commute", i.e. all the higher Whitehead products that they generate vanish as ! 1. However, for each xed there is essentially one nonvanishing product that gives rise to a \jumping generator" w in H (G ) and to a single relation in the rational cohomology ring H (BG ). An analog of this generator w was also seen by Kronheimer in his study of families of symplectic forms on 4-manifolds using Seiberg{Witten theory. Our methods involve a close study of the space of ! -compatible almost complex structures on M.

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Topology of symplectomorphism groups of rational ruled surfaces

A theorem of E. Lerman and S. Tolman, generalizing a result of T. Delzant, states that compact symplectic toric orbifolds are classified by their moment polytopes, together with a positive integer label attached to each of their facets. In this paper we use this result, and the existence of "global" actionangle coordinates, to give an effective parametrization of all compatible toric complex structures on a compact symplectic toric orbifold, by means of smooth functions on the corresponding moment polytope. This is equivalent to parametrizing all toric Kahler metrics and generalizes an analogous result for toric manifolds. A simple explicit description of interesting families of extremal Kahler metrics, arising from recent work of R. Bryant, is given as an application of the approach in this paper. The fact that in dimension four these metrics are self-dual and conformally Einstein is also discussed. This gives rise in particular to a one parameter family of self-dual Einstein metrics connecting the well known Eguchi-Hanson and Taub-NUT metrics.

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Miguel Abreu

Papers

Kähler geometry of toric varieties and extremal metrics

Kahler geometry of toric varieties and extremal metrics

Kahler geometry of toric manifolds in symplectic coordinates

Topology of symplectomorphism groups of rational ruled surfaces

Kähler Metrics on Toric Orbifolds