We use the quaternion Kahler reduction technique to study old and new self-dual Einstein metrics of negative scalar curvature with at least a two-dimensional isometry group, and relate the quotient construction to the hyperbolic eigenfunction Ansatz. We focus in particular on the (semi-)quaternion Kahler quotients of (semi-)quaternion Kahler hyperboloids, analysing the completeness and topology, and relating them to the self-dual Einstein Hermitian metrics of Apostolov-Gauduchon and Bryant.
This work characterizes global quotient stacks---smooth stacks associated to a finite group acting a manifold---among smooth quotient stacks $[M/G]$, where $M$ is a smooth manifold equipped with a smooth proper action by a Lie group $G$. The characterization is described in terms of the action of the connected component $G_0$ on $M$ and is related to (stacky) fundamental group and covering theory. This characterization is then applied to smooth toric Deligne-Mumford stacks, and global quotients among toric DM stacks are then characterized in terms of their associated combinatorial data of stac…
We study the hypersymplectic spaces obtained as quotients of flat hypersymplectic space R^{4d} by the action of a compact Abelian group. These 4n-dimensional quotients carry a multi-Hamilitonian action of an n-torus. The image of the hypersymplectic moment map for this torus action may be described by a configuration of solid cones in R^{3n}. We give precise conditions for smoothness and non-degeneracy of such quotients and show how some properties of the quotient geometry and topology are constrained by the combinatorics of the cone configurations. Examples are studied, including non-trivial …
Simons Center for Geometry and Physics, Stony Brook University（石溪大学西蒙斯几何与物理中心）；Institute for Advanced Study in Mathematics, Zhejiang University（浙江大学数学高等研究院）
In [11] it was proved that, given a compact toric Sasaki manifold of positive basic first Chern class and trivial first Chern class of the contact bundle, one can find a deformed Sasaki structure on which a Sasaki-Einstein metric exists. In the present paper we first prove the uniqueness of such Einstein metrics on compact toric Sasaki manifolds modulo the action of the identity component of the automorphism group for the transverse holomorphic structure, and secondly remark that the result of [11] implies the existence of compatible Einstein metrics on all compact Sasaki manifolds obtained fr…
We study CR geometry in arbitrary codimension, and introduce a process, which we call the Levi-Kahler quotient, for constructing Kahler metrics from CR structures with a transverse torus action. Most of the paper is devoted to the study of Levi-Kahler quotients of toric CR manifolds, and in particular, products of odd dimensional spheres. We obtain explicit descriptions and characterizations of such quotients, and find Levi-Kahler quotients of products of 3-spheres which are extremal in a weighted sense introduced by G. Maschler and the first author.
We prove that any compact selfdual Einstein 4-orbifold of positive scalar curvature whose isometry group contains a 2-torus is, up to an orbifold covering, a quaternion Kaehler quotient of (k-1)-dimensional quaternionic projective space by a (k-2)-torus for some $k\geq 2$. We also obtain a topological classification in terms of the intersection form of the 4-orbifold.
We give criterions for the existence of toric conical Kahler-Einstein and Kahler-Ricci soliton metrics on any toric manifold in relation to the greatest Ricci and Bakry-Emery-Ricci lower bound. We also show that any two toric manifolds with the same dimension can be joined by a continuous path of toric manifolds with conical Kahler-Einstein metrics in the Gromov-Hausdorff topology.
A series of examples of toric Sasaki-Einstein 5-manifolds is constructed. These are submanifolds of toric 3-Sasaki 7-manifolds and such a Sasaki-Einstein 5-manifold corresponds uniquely to a toric 3-Sasaki 7-manifold. This produces examples of quasi-regular Sasaki-Einstein structures on every #k(S^2 xS^3), for k odd. Toric geometry is used to construct examples of positive Ricci curvature toric Sasaki structures on non-spin 5-manifolds. Then the join construction is used to construct infinitely many quasi-regular toric Sasaki-Einstein manifolds with arbitrarily high second Betti number in ever…
We develop new algorithms for approximating extremal toric Kähler metrics. We focus on an extremal metric on $\mathbb{CP}^{2}\sharp2\overline{\mathbb{CP}}^{2}$, which is conformal to an Einstein metric (the Chen-LeBrun-Weber metric). We compare our approximation to one given by Bunch and Donaldson and compute various geometric quantities. In particular, we demonstrate a small eigenvalue of the scalar Laplacian of the Einstein metric which gives a numerical evidence that the Einstein metric is conformally unstable under the Ricci flow.
In this short note, we prove that a Calabi extremal Kaehler-Ricci soliton on a compact toric Kaehler manifold is Einstein. This solves for the class of toric manifolds a general problem stated by the authors that they solved only under some curvature assumptions.
We show that by taking a certain scaling limit of a Euclideanised form of the Plebanski-Demianski metrics one obtains a family of local toric Kahler-Einstein metrics. These can be used to construct local Sasaki-Einstein metrics in five dimensions which are generalisations of the Y^{p,q} manifolds. In fact, we find that these metrics are diffeomorphic to those recently found by Cvetic, Lu, Page and Pope. We argue that the corresponding family of smooth Sasaki-Einstein manifolds all have topology S^2 x S^3. We conclude by setting up the equations describing the warped version of the Calabi-Yau c…
We give a complete criterion for the existence of generalized Kähler Einstein metrics on toric Fano manifolds from view points of a uniform stability in a sense of GIT and the properness of a functional on the space of Kähler metrics.
We show that any toric Kähler cone with smooth compact cross-section admits a family of Calabi-Yau cone metrics with conical singularities along its toric divisors. The family is parametrized by the Reeb cone and the angles are given explicitly in terms of the Reeb vector field. The result is optimal, in the sense that any toric Calabi-Yau cone metric with conical singularities along the toric divisor (and smooth elsewhere) belongs to this family. We also provide examples and interpret our results in terms of Sasaki-Einstein metrics.
As seen in the works of Calabi, Cheng-Yau and Loftin, affine sphere equations have a close relationship with Kaehler-Einstein metrics. The main purpose of this note is to show that an equation analogous to those of hyperbolic affine spheres arises naturally from Kaehler-Einstein metrics on Einstein toric surfaces. The case for the remaining toric surfaces with Kaehler-Ricci solitons will also be discussed.
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 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 interp…
We prove a conjecture saying that complex projective space has maximal volume (degree) among all toric Kaehler-Einstein manifolds of dimension n. The proof is inspired by our recent work on sharp Moser-Trudinger and Brezis-Merle type inequalities for the complex Monge-Ampere operator, but is essentially self-contained.
We give a correspondence between toric 3-Sasaki 7-manifolds S and certain toric Sasaki-Einstein 5-manifolds M. These 5-manifolds are all diffeomorphic to k#(S^2\times S^3), where k=2b_2(S)+1, and are given by a pencil of Sasaki embeddings of M in S and are given concretely by the zero set of a component of the 3-Sasaki moment map. It follows that there are infinitely many examples of these toric Sasaki-Einstein manifolds M for each odd b_2(M)>1. As an application of the proof of the above, we prove that the local deformation space of ASD structures on a compact toric ASD Einstein orbifold is g…
In this short note, we investigate the existence of orbifold Kähler-Einstein metrics on toric varieties. In particular, we show that every $\mathbb{Q}$-factorial normal projective toric variety allows an orbifold Kähler-Einstein metric. Moreover, we characterize the K-stability of $\mathbb{Q}$-factorial toric pairs of Picard number one in terms of the log Cox ring and the universal orbifold cover.
Conformally Kähler, Einstein-Maxwell metrics and $f$-extremal metrics are generalization of canonical metrics in Kähler geometry. We introduce uniform K-stability for toric Kähler manifolds, and show that uniform K-stability is necessary condition for the existence of $f$-extremal metrics on toric manifolds. Furthermore, we show that uniform K-stability is equivalent to properness of relative K-energy.
Using the twistor correspondence, we give a classification of toric anti-self-dual Einstein metrics: each such metric is essentially determined by an odd holomorphic function. This explains how the Einstein metrics fit into the classification of general toric anti-self-dual metrics given in an earlier paper (math.DG/0602423). The results complement the work of Calderbank-Pedersen (math.DG/0105263), who describe where the Einstein metrics appear amongst the Joyce spaces, leading to a different classification. Taking the twistor transform of our result gives a new proof of their theorem.
We show that every toric Sasaki-Einstein manifold $S$ admits a special Legendrian submanifold $L$ which arises as the link ${\rm fix}(τ)\cap S$ of the fixed point set ${\rm fix}(τ)$ of an anti-holomorphic involution $τ$ on the cone $C(S)$. In particular, an irregular toric Sasaki-Einstein manifold $S^{2}\times S^{3}$ has a special Legendrian torus $S^{1}\times S^{1}$. Moreover, we also obtain a special Legendrian submanifold in $\sharp m(S^{2}\times S^{3})$ for each $m\ge 1$.
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 {\it Sasaki-Ricci soliton}) on compact toric Sasaki manifolds whose basic first Chern form of the nor…
In this note we report on examples of 7- and 8-dimensional toric Fano manifolds that are not symmetric and still admit a Kaehler-Einstein metric. This answers a question first posed by V.V. Batyrev and E. Selivanova. The examples were found in the classification of toric Fano manifolds up to dimension 8 obtained by M. Oebro. We also discuss related open questions and conjectures.
Extending work of Bielawski-Dancer and Konno, we develop a theory of toric hyperkahler varieties, which involves toric geometry, matroid theory and convex polyhedra. The framework is a detailed study of semi-projective toric varieties, meaning GIT quotients of affine spaces by torus actions, and specifically, of Lawrence toric varieties, meaning GIT quotients of even-dimensional affine spaces by symplectic torus actions. A toric hyperkahler variety is a complete intersection in a Lawrence toric variety. Both varieties are non-compact, and they share the same cohomology ring, namely, the Stanle…
We show that any compact convex simple lattice polytope is the moment polytope of a Kähler-Einstein orbifold, unique up to orbifold covering and homothety. We extend the Wang-Zhu Theorem \cite{WZ} giving the existence of a Kähler-Ricci soliton on any toric monotone manifold on any compact convex simple labelled polytope satisfying the combinatoric condition corresponding to monotonicity. We obtain that any compact convex simple polytope $P\subset \bR^n$ admits a set of inward normals, unique up to dilatation, such that there exists a symplectic potential satisfying the Guillemin boundary condi…
Let $X$ be a complex toric Fano $n$-fold and ${\cal N}(T)$ the normalizer of a maximal torus $T$ in the group of biholomorphic authomorphisms $Aut(X)$. We call $X$ {\em symmetric} if the trivial character is a single ${\cal N}(T)$-invariant algebraic character of $T$. Using an invariant $α_G(X)$ introduced by Tian, we show that all symmetric toric Fano $n$-folds admit an Einstein-Kähler metric. We remark that so far one doesn't know any example of a toric Fano $n$-fold $X$ such that $Aut(X)$ is reductive, the Futaki character of $X$ vanishes, but $X$ is not symmetric.
Recently an infinite family of explicit Sasaki-Einstein metrics Y^{p,q} on S^2 x S^3 has been discovered, where p and q are two coprime positive integers, with q<p. These give rise to a corresponding family of Calabi-Yau cones, which moreover are toric. Aided by several recent results in toric geometry, we show that these are Kahler quotients C^4//U(1), namely the vacua of gauged linear sigma models with charges (p,p,-p+q,-p-q), thereby generalising the conifold, which is p=1,q=0. We present the corresponding toric diagrams and show that these may be embedded in the toric diagram for the orbif…
We present a general numerical method for investigating prescribed Ricci curvature problems on toric Kähler manifolds. This method is applied to two generalisations of Einstein metrics, namely Ricci solitons and quasi-Einstein metrics. We begin by recovering the Koiso--Cao soliton and the Lü--Page--Pope quasi-Einstein metrics on $\mathbb{CP}^{2}\sharp\overline{\mathbb{CP}}^{2}$ (in both cases the metrics are known explicitly). We also find numerical approximations to the Wang--Zhu soliton on $\mathbb{CP}^{2}\sharp 2\overline{\mathbb{CP}}^{2}$ (here the metric is not known explicitly). Finally,…
This is a continuation of paper \cite{Li}. On any toric Fano manifold, we discuss the behavior of limit metric of a sequence of metrics, which are solutions to a continuity family of complex Monge-Ampere equations in Kahler-Einstein problem. We show that the limit metric satisfies a singular complex Monge-Ampere equation. This shows the conic type singularity for the limit metric. The information of conic type singularities can be read from the geometry of the moment polytope.
We provide an explicit resolution of the existence problem for extremal Kaehler metrics on toric 4-orbifolds M with second Betti number b2(M)=2. More precisely we show that M admits such a metric if and only if its rational Delzant polytope (which is a labelled quadrilateral) is K-polystable in the relative, toric sense (as studied by S. Donaldson, E. Legendre, G. Szekelyhidi et al.). Furthermore, in this case, the extremal Kaehler metric is ambitoric, i.e., compatible with a conformally equivalent, oppositely oriented toric Kaehler metric, which turns out to be extremal as well. These results…
We provide an explicit resolution of the Abreu equation on convex labeled quadrilaterals. This confirms a conjecture of Donaldson in this particular case and implies a complete classification of the explicit toric Kähler-Einstein and toric Sasaki-Einstein metrics constructed in [6,22,14]. As a byproduct, we obtain a wealth of extremal toric (complex) orbi-surfaces, including Kähler-Einstein ones, and show that for a toric orbi-surface with 4 fixed points of the torus action, the vanishing of the Futaki invariant is a necessary and sufficient condition for the existence of Kähler metric with co…
Motivated by a probabilistic approach to Kahler-Einstein metrics we consider a general non-equilibrium statistical mechanics model in Euclidean space consisting of the stochastic gradient flow of a given (possibly singular) quasi-convex N-particle interaction energy. We show that a deterministic "macroscopic" evolution equation emerges in the large N-limit of many particles. This is a strengthening of previous results which required a uniform two-sided bound on the Hessian of the interaction energy. The proof uses the theory of weak gradient flows on the Wasserstein space. Applied to the setti…
We construct a new family of compact orbifolds with a positive self dual Einstein metric and a one-dimensional group of isometries. Together with another known family, these examples classify all 4-dimensional orbifolds that are quaternion Kaehler quotients by a torus of real Grassmannians.
We consider Fano manifolds M that admit a collection of finite automorphism groups G_1, ..., G_k, such that the quotients M/G_i are smooth Fano manifolds possessing a Kaehler-Einstein metric. Under some numerical and smoothness assumptions on the ramification divisors, we prove that M admits a Kaehler-Einstein metric too.
In this note, stimulated by the existence result of Futaki-Ono-Wang for toric Sasaki-Einstein metrics, we obtain new examples of Sasaki-Einstein metrics on S^1-bundles associated to canonical line bundles of P^1-bundles over Kähler-Einstein Fano manifolds, even though the Futaki's obstruction does not vanish. Here the method of Sakane and Koiso is used, and our examples include non-toric Sasaki-Einstein manifolds.
An index theorem for the anti-self-dual deformation complex on anti-self-dual orbifolds with cyclic quotient singularities is proved. We present two applications of this theorem. The first is to compute the dimension of the deformation space of the Calderbank-Singer scalar-flat Kahler toric ALE spaces. A corollary of this is that, except for the Eguchi-Hanson metric, all of these spaces admit non-toric anti-self-dual deformations, thus yielding many new examples of anti-self-dual ALE spaces. For our second application, we compute the dimension of the deformation space of the canonical Bochner-…
We construct symplectic and Kähler ray reduced spaces and discuss their relation with the Marsden-Weinstein (point) reduction. This Kähler reduction is well defined even when the momentum value is not totally isotropic. The compatibility of the ray reduction with the cone construction and the Boothby-Wang fibration is presented. Using the compatibility with the cone construction we provide the exact description of ray quotients of cotangent bundles. Some applications of the ray reduction to the study of conformal Hamiltonian systems are described. We also give necessary and sufficient conditio…
In this paper, we study the Bergman metric of a finite ball quotient $\mathbb{B}^n/Γ$, where $Γ\subseteq \mathrm{Aut}(\mathbb{B}^n)$ is a finite, fixed point free, abelian group. We prove that this metric is Kähler--Einstein if and only if $Γ$ is trivial, i.e., when the ball quotient $\mathbb{B}^n/Γ$ is the unit ball $\mathbb{B}^n$ itself. As a consequence, we establish a characterization of the unit ball among normal Stein spaces with isolated singularities and abelian fundamental groups in terms of the existence of a Bergman-Einstein metric.
In this expository paper we review on the existence problem of Einstein-Maxwell Kähler metrics, and make several remarks. Firstly, we consider a slightly more general set-up than Einstein-Maxwell Kähler metrics, and give extensions of volume minimization principle, the notion of toric K-stability and other related results to the general set-up. Secondly, we consider the toric case when the manifold is the one point blow-up of the complex project plane and the Kähler class $Ω$ is chosen so that the area of the exceptional curve is sufficiently close to the area of the rational curve of self-int…
This is an expository article. Among other topics, we discuss the existence of Kahler-Ricci soliton metrics on toric Fano manifolds, and Kahler-Einstein metrics on deformations of the Mukai-Umemura 3-fold
We give an explicit local classification of conformally equivalent but oppositely oriented Kaehler metrics on a 4-manifold which are toric with respect to a common 2-torus action. In the generic case, these structures have an intriguing local geometry depending on a quadratic polynomial and two arbitrary functions of one variable, these two functions being explicit degree 4 polynomials when the Kaehler metrics are extremal (in the sense of Calabi). One motivation for and application of this result is an explicit local description of Einstein 4-manifolds which are hermitian with respect to eith…
The global holomorphic α-invariant introduced by Tian is closely related with the study in the existence of Kahler-Einstein metric. We apply the result of Tian, Lu and Zelditch on polarized Kahler metrics to approximate plurisubharmonic functions and compute the α-invariant of toric Fano manifolds.
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" action-angle 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…
