S. K. Donaldson
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
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We study toric nearly Kähler manifolds, extending the work of Moroianu and Nagy. We give a description of the global geometry using multi-moment maps. We then investigate polynomial and radial solutions to the toric nearly Kähler equation.
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.
Weyl group of the group of holomorphic isometries of a Kähler toric manifold
本文计算了具有实解析凯勒度量的凯勒环流形全纯等距群的Weyl群（Segal意义下）。
We study compact toric strict locally conformally Kähler manifolds. We show that the Kodaira dimension of the underlying complex manifold is $-\infty$ and that the only compact complex surfaces admitting toric strict locally conformally Kähler metrics are the diagonal Hopf surfaces. We also show that every toric Vaisman manifold has lcK rank 1 and is isomorphic to the mapping torus of an automorphism of a toric compact Sasakian manifold.
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.
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.
Abreu-Sena-Dias have constructed two distinct families of scalar-flat Kähler non-compact toric metrics using Donaldson's rephrasing of Joyce's construction in action-angle coordinates. In this paper and using the same set-up, we show that these are the only J-complete scalar-flat Kähler metrics on any given strictly unbounded toric surface. We also show that the asymptotic behaviour of such a metric determines it uniquely.
Asymptotic geometry of toric Kahler instantons
本文研究完备环面Kähler流形的辛约化闭性判据，分类标量平坦瞬子的渐近几何，并给出非闭约化的反例。
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…
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 prove that a compact toric locally conformally Kähler manifold which is not Kähler admits a toric Vaisman structure, a fact which was conjectured in \cite{mmp}. This is the final step leading to the classification of compact toric locally conformally Kähler manifolds started in \cite{p} and \cite{mmp}. We also show, by constructing an example, that unlike in the symplectic case, toric locally conformally symplectic manifolds are not necessarily toric locally conformally Kähler.
We show that 6-dimensional strict nearly Kähler manifolds admitting effective $\mathbb{T}^3$ actions by automorphisms are completely characterized in the neigbourhood of each point by a function on $\mathbb{R}^3$ satisfying a certain Monge-Ampère type equation.
Let $P$ be a Delzant polytope. We show that the quantization of the corresponding toric manifold $X_{P}$ in toric Kähler polarizations and in the toric real polarization are related by analytic continuation of Hamiltonian flows evaluated at time $t = \sqrt{-1} s$. We relate the quantization of $X_{P}$ in two different toric Kähler polarizations by taking the time-$\sqrt{-1} s$ Hamiltonian "flow" of strongly convex functions on the moment polytope $P$. By taking $s$ to infinity, we obtain the quantization of $X_{P}$ in the (singular) real toric polarization. Recall that $X_{P}$ has an open dens…
Convergence of Generalized Kähler--Ricci Flow on Toric Fano Manifolds
本文在环面Fano流形上定义扩展Mabuchi泛函并证明其适定性，通过单调性证明广义Kähler-Ricci流收敛到Kähler-Ricci孤子，证实了Apostolov-Streets-Ustinovskiy猜想。
Toric separable geometries and extremal Kähler metrics
本文提出环面可分几何的统一框架，将已知显式卡拉比极值环面凯勒度量归结为两类分解结构，并通过显式计算数量曲率与求解极值性偏微分方程，系统构造了新的极值度量。
We generalise a theorem of Engman and Abreu--Freitas on the first invariant eigenvalue of non-negatively curved $S^{1}$-invariant metrics on $\mathbb{CP}^{1}$ to general toric Kähler metrics with non-negative scalar curvature. In particular, a simple upper bound of the first non-zero invariant eigenvalue for such metrics on complex projective space $\mathbb{C}P^{n}$ is exhibited. We derive an analogous bound in the case when the metric is extremal and a detailed study is made of the accuracy of the bound in the case of Calabi's extremal metrics on $\mathbb{C}P^{2}\sharp -\mathbb{C}P^{2}$.
Vaisman manifolds are strongly related to Kähler and Sasaki geometry. In this paper we introduce toric Vaisman structures and show that this relationship still holds in the toric context. It is known that the so-called minimal covering of a Vaisman manifold is the Riemannian cone over a Sasaki manifold. We show that if a complete Vaisman manifold is toric, then the associated Sasaki manifold is also toric. Conversely, a toric complete Sasaki manifold, whose Kähler cone is equipped with an appropriate compatible action, gives rise to a toric Vaisman manifold. In the special case of a strongly r…
We show that on a Kahler manifold whether the J-flow converges or not is independent of the chosen background metric in its Kahler class. On toric manifolds we give a numerical characterization of when the J-flow converges, verifying a conjecture of Lejmi and the second author in this case. We also strengthen existing results on more general inverse $σ_{k}$ equations on Kahler manifolds.
Examples of Toric Scalar-flat Kähler Surfaces with Mixed-type Ends
该研究针对严格无界环面辛4-流形，在环面除子补集上显式构造完备环面数量平坦Kähler度量，并借鉴Abreu与Sena-Dias方法给出带锥奇点的同类度量的另一种构造。
In this note, we prove that on an $n$-dimensional compact toric manifold with positive first Chern class, the Kähler-Ricci flow with any initial $(S^1)^n$-invariant Kähler metric converges to a Kähler-Ricci soliton. In particular, we give another proof for the existence of Kähler-Ricci solitons on a compact toric manifold with positive first Chern class by using the Kähler-Ricci flow.
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.
Curvature on some Kähler toric manifolds
将Guillemin-Abreu形式推广到全纯截面曲率和双截面曲率，应用于C^n、O(-ℓ)和Hirzebruch流形M_{n,ℓ}，并证明当斜率接近1时极值度量具有正全纯截面曲率，构造了全纯向量丛上的标量平坦Kähler度量。
We show that the classical Szasz analytic function $S_N(f)(x)$ is obtained by applying the pseudo-differential operator $f(N^{-1}D_θ)$ to the Bergman kernels for the Bargmann-Fock space. The expression generalizes immediately to any smooth polarized noncompact complete toric \kahler manifold, defining the generalized Szasz analytic function $S_{h^N}(f)(x)$. About $S_{h^N}(f)(x)$, we prove that it admits complete asymptotics and there exists a universal scaling limit. % We also consider some dilation operator composed with $S_{h^N}(f)(x)$ and we give an estimate about this composition. As an ex…
An almost Kähler structure is {\it extremal} if the Hermitian scalar curvature is a Killing potential [29]. When the almost complex structure is integrable it coincides with extremal Kähler metric in the sense of Calabi [8]. We observe that the existence of an extremal {\it toric} almost Kähler structure of involutive type implies uniform K-stability and we point out the existence of a formal solution of the Abreu equation for any angle along the invariant divisor. Applying the recent result of Chen--Cheng [10] and He [27], we conclude that the existence of a compatible extremal toric almost K…
In [LS], it is shown shown that the first eigenvalue of the Laplacian restricted to the space of invariant functions on a toric Kähler manifold (i.e. $λ_1^\mathbb{T}$, the invariant first eigenvalue) is an unbounded function of the toric Kähler metric. In this note we show that, seen as a function on the space of toric Kähler metrics on a fixed toric manifold, $λ_1^\mathbb{T}$ admits no analytic critical points. We also show that on $S^2$, the first eigenvalue of the Laplacian restricted to the space of $S^1$-equivariant functions of any given integer weight admits no critical points.
Moment polytopes of toric exponential families
本文证明有限样本空间上指数族环化得到的Kähler环面流形的矩多面体是高维单形的投影。
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.
The aim of this paper is to solve a uniform version of the Yau-Tian-Donaldson conjecture for polarized toric manifolds. Also, we show a combinatorial sufficient condition for uniform relative K-polystability.
In this note we prove that toric Kähler metrics on complex projective space which are also $U(n)$-invariant are determined by their equivariant spectrum i.e. the list of eigenvalues of the Laplacian together with weights of the torus representation on the eigenspaces.
The purpose of this paper is to calculate the support of the multiplier ideal sheaves derived from the Kähler-Ricci flow on certain toric Fano manifolds with large symmetry. The early idea of this paper has already been in Appendix of \cite{futaki-sano0711}.
In these notes, after an introduction to toric Kahler geometry, we present Calabi's family of U(n)-invariant extremal Kahler metrics in symplectic action-angle coordinates and show that it actually contains, as particular cases, many interesting cohomogeneity one examples of constant scalar curvature.
Toric geometry of generalized Kähler-Ricci solitons
该研究建立环面稳态Kähler-Ricci孤立子与A型环面GKRS的局部及全局等价关系，给出高维新例构造方法，并完成四维GKRS的局部分类与完备辛型例子的推测性穷尽构造。
In the same way that a contact manifold determines and is determined by a symplectic cone, a Sasaki manifold determines and is determined by a suitable Kahler cone. Kahler-Sasaki geometry is the geometry of these cones. This paper presents a symplectic action-angle coordinates approach to toric Kahler geometry and how it was recently generalized, by Burns-Guillemin-Lerman and Martelli-Sparks-Yau, to toric Kahler-Sasaki geometry. It also describes, as an application, how this approach can be used to relate a recent new family of Sasaki-Einstein metrics constructed by Gauntlett-Martelli-Sparks-W…
We established a Yau--Tian--Donaldson type correspondence, expressed in terms of a single Delzant polytope, concerning the existence of extremal Kähler metrics on a large class of toric fibrations, introduced by Apostolov--Calderbank--Gauduchon--Tonnesen-Friedman and called semi-simple principal toric fibrations. We use that an extremal metric on the total space corresponds to a weighted constant scalar curvature Kähler metric (in the sense of Lahdili) on the corresponding toric fiber in order to obtain an equivalence between the existence of extremal Kähler metrics on the total space and a su…
In a recent paper Donaldson explains how to use an older construction of Joyce to obtain four dimensional local models for scalar-flat Kahler metrics with a 2-torus symmetry. Using this idea, he recovers and generalizes the Taub-NUT metric by including it in a new family of complete scalar-flat toric Kahler metrics. In this paper we generalize Donaldson's method and construct complete scalar-flat toric Kahler metrics on any symplectic toric 4-manifold with "strictly unbounded" moment polygon. These include the asymptotic locally Euclidean scalar-flat Kahler metrics previously constructed by Ca…
In this paper we construct a family of complex structures on a complex flag manifold that converge to the real polarization coming from the Gelfand-Cetlin integrable system, in the sense that holomorphic sections of a prequantum line bundle converge to delta-function sections supported on the Bohr-Sommerfeld fibers. Our construction is based on a toric degeneration of flag varieties and a deformation of Kähler structure on toric varieties by symplectic potentials.
In this paper we study the gradient steady Kähler-Ricci soliton metrics on non-compact toric manifolds. We show that the orbit space of the free locus of such a manifold carries a natural Hessian structure with a nonnegative Bakry-Émery tensor. We generalize Calabi's classical rigidity result and use this to prove that any complete $\mathbf T^n$-invariant gradient steady Kähler-Ricci soliton with a free torus action must be a flat $(\mathbb C^*)^n$.
We prove a necessary and sufficient condition in terms of the barycenters of a collection of polytopes for existence of coupled Kähler-Einstein metrics on toric Fano manifolds. This confirms the toric case of a coupled version of the Yau-Tian-Donaldson conjecture. We also obtain a necessary and sufficient condition for existence of torus-invariant solutions to a system of soliton type equations on toric Fano manifolds. Some of these solutions provide natural candidates for the large time limits of a certain geometric flow generalizing the Kähler-Ricci flow.
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.
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 paper completes a programme to determine which toric surfaces admit Kahler metrics of constant scalar curvature/
Toric Fano manifolds that do not admit extremal Kähler metrics
本文构造了维数为 10 的环面 Fano 流形，使其第一陈类中无极值 Kähler 度量，并通过乘积推广到所有 n≥11，回答了 Mabuchi 的问题。
In this paper, we show that the complete scalar-flat Kahler metrics constructed by Abreu and the author on strictly unbounded toric 4-dimensional orbifolds have finite $L^2$ norm of the full Riemannian tensor. In particular, this answers a question of Donaldon's on the corresponding Generalized Taub-NUT metric on $R^4$. This norm is explicitly determined when the underlying toric manifold is the minimal resolution of a cyclic singularity. In the Ricci-flat case corresponding to gravitational instantons, this recovers a recent result of Atiyah-Lebrun.
One has believed that low energy effective theories of the Higgs branch of gauged linear sigma models correspond to supersymmetric nonlinear sigma models, which have been already investigated by many works. In this paper we discuss a explicit derivation of supersymmetric nonlinear sigma models from gauged linear sigma models. In this process we construct Kahler potentials of some two-dimensional toric varieties explicitly. Thus we will be able to study some algebraic varieties in the language of differential geometry.
We develop a general theory for the existence of extremal Kähler metrics of Poincaré type in the sense of Auvray, defined on the complement of a toric divisor of a polarized toric variety. In the case when the divisor is smooth, we obtain a list of necessary conditions which must be satisfied for such a metric to exist. Using the explicit methods of Apostolov-Calderbank-Gauduchon together with the computational approach of Sektnan, we show that on a Hirzebruch complex surface the necessary conditions are also sufficient. In particular, on such a complex surface the complement of the infinity s…
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 …
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…
Using the twistor correspondence, this article gives a one-to-one correspondence between germs of toric anti-self-dual conformal classes and certain holomorphic data determined by the induced action on twistor space. Recovering the metric from the holomorphic data leads to the classical problem of prescribing the Cech coboundary of 0-cochains on an elliptic curve covered by two annuli. The classes admitting Kahler representatives are described; each such class contains a circle of Kahler metrics. This gives new local examples of scalar-flat Kahler surfaces and generalises work of Joyce who con…
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.
We generalize the results of Song-Zelditch on geodesics in spaces of Kahler metrics on toric varieties to harmonic maps of any compact Riemannian manifold with boundary into the space of Kahler metrics on a toric variety. We show that the harmonic map equation can always be solved and that such maps may be approximated in the C^2 topology by harmonic maps into the spaces of Bergman metrics. In particular, WZW maps, or equivalently solutions of a homogeneous Monge-Ampere equation on the product of the manifold with a Riemann surface with S^1 boundary admit such approximations. We also show that…
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…
We show the properties of the blowup limits of \KRf solutions on Fano surfaces if Riemannian curvature is unbounded. As an application, on every toric Fano surface, we prove that \KRf converges to a Kähler Ricci soliton metric if the initial metric has toric symmetry. Therefore we give a new Ricci flow proof of existence of Kähler Ricci soliton metrics on toric surfaces.
Various curvature conditions are studied on metrics admitting a symmetry group. We begin by examining a method of diagonalizing cohomogeneity-one Einstein manifolds and determine when this method can and cannot be used. Examples, including the well-known Stenzel metrics, are discussed. Next, we present a simplification of the Einstein condition on a compact four manifold with $T^{2}$-isometry to a system of second-order elliptic equations in two-variables with well-defined boundary conditions. We then study the Einstein and extremal Kahler conditions on Kahler toric manifolds. After constructi…
Strong formality of toric and homogeneous compact Kähler manifolds
本文研究紧 Kähler 流形的强形式性，证明完备光滑复环面簇与齐性紧 Kähler 流形能以相容方式同时满足理性形式性和强形式性。
Hermitian, Ricci-flat toric metrics on non-compact surfaces à la Biquard-Gauduchon
本文反转 Biquard-Gauduchon 的统一构造，证明在矩多面体温和假设下，环面除子余集上所有共形 Kähler、Ricci 平坦的环面度量均属于已知族且均为 ALF。
Kähler metrics and toric Lagrangian fibrations
论文扩展了Abreu-Guillemin理论，将不变Kähler度量从toric辛流形推广到具有toric作用的辛流形，揭示了这些流形恰好是具有椭圆奇点的拉格朗日纤维化，并建立了不变Kähler度量与椭圆连接和混合b度量的对应关系。
In this paper, we extend the result about the existence of Kähler-Ricci soliton on toric manifold (proved by Wang and Zhy) by proving this existence on horospherical varieties using the continuity method.
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.
On a given compact complex manifold or orbifold $(M,J)$, we study the existence of Hermitian metrics $\tilde g$ in the conformal classes of Kähler metrics on $(M,J)$, such that the Ricci tensor of $\tilde g$ is of type $(1,1)$ with respect to the complex structure, and the scalar curvature of $\tilde g$ is constant. In real dimension $4$, such Hermitian metrics provide a Riemannian counter-part of the Einstein--Maxwell (EM) equations in general relativity, and have been recently studied in \cite{ambitoric1, LeB0, LeB, KTF}. We show how the existence problem of such Hermitian metrics (which we …
Kähler-Einstein toric submanifolds of the projective space
该研究证明了某些托卡蒂芬诺流形无法通过射影浸入诱导出凯勒-爱因斯坦度量，仅当其为射影线乘积时才可能实现，并提出了相关猜想。
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