Constant Scalar Curvature Kahler Metrics Research Articles

Uniform K-stability of polarized spherical varieties

We express notions of K-stability of polarized spherical varieties in terms of combinatorial data, vastly generalizing the case of toric varieties. We then provide a combinatorial sufficient condition of G-uniform K-stability by studying the corresponding convex geometric problem. Thanks to recent work of Chi Li and a remark by Yuji Odaka, this provides an explicitly checkable sufficient condition of existence of constant scalar curvature Kahler metrics. As a side effect, we show that, on several families of spherical varieties, G-uniform K-stability is equivalent to K-polystability with respect to G-equivariant test configurations for polarizations close to the anticanonical bundle.

A Moment Map Picture of Relative Balanced Metrics on Extremal Kähler Manifolds

We give a moment map interpretation of some relatively balanced metrics. As an application, we extend a result of S. K. Donaldson on constant scalar curvature Kahler metrics to the case of extremal metrics. Namely, we show that a given extremal metric is the limit of some specific relatively balanced metrics. As a corollary, we recover uniqueness and splitting results for extremal metrics in the polarized case.

Expansions of solutions to extremal metric type equations on blowups of cscK surfaces

The aim of this article is to study expansions of solutions to an extremal metric type equation on the blowup of constant scalar curvature Kahler surfaces. This is related to finding constant scalar curvature Kahler (cscK) metrics on K-stable blowups of extremal Kahler surfaces (Szekelyhidi in Duke Math J 161(8):1411–1453, 2012).

Complete cscK Metrics on the Local Models of the Conifold Transition

In this paper, we construct complete constant scalar curvature Kahler (cscK) metrics on the complement of the zero section in the total space of \({\mathcal{O}(-1)^{\oplus2}}\) over \({\mathbb{P}^{1}}\), which is biholomorphic to the smooth part of the cone C 0 in \({\mathbb{C}^{4}}\) defined by equation \({\Sigma_{i=1}^{4} w_{i}^{2}=0}\). On its small resolution and its deformation, we also consider complete cscK metrics and find that if the cscK metrics are homogeneous, then they must be Ricci-flat.

Kähler–Einstein Metrics and Stability

This is a note to announce a solution of the well-known question: which Fano manifolds admit Kahler–Einstein metrics? The idea that the appropriate condition should be in terms of “algebro-geometric stability” was proposed about 20 years ago by Yau [21, 22] (partly by analogy with the “Kobayashi–Hitchin correspondence” in the case of holomorphic bundles). For more on the history of this problem please see our forthcoming full paper. Over the years various different notions of stability have been discussed in the literature, both in the Fano/Kahler–Einstein case and in the more general situation of constant scalar curvature Kahler metrics on polarized manifolds. But the condition we end up using is essentially as proposed by Tian [20] and which we now recall (see also [14] for the equivalence with a priori stronger definitions).

ON GENERALIZED KÄHLER-RICCI SOLITONS

By definition, Kahler‐Ricci solitons are defined on Fano mani folds. In this note, we shall generalize the notion of Kahler‐Ricci solitons to the case of general polarized manifolds from the view point of K-energy, which are called “generalized Kahler‐Ricci solitons”. Moreover, “generalized Kahler‐Ricci solitons” are also one of generalizations of constant scalar curvature Kahler metrics. Furthermore, we shall give a non-trivial example of a “generalized Kahler‐Ricci soliton”.

Constant Scalar Curvature Kahler Metric Obtains the Minimum of K-energy

Based on Donaldson's method, we prove that, for an integral Kähler class, when there is a Kähler metric of constant scalar curvature, then it minimizes the K-energy. We do not assume that the automorphism group is discrete.

Blowing up Kähler manifolds with constant scalar curvature, II

In this paper we prove the existence of Kahler metrics of constant scalar curvature on the blow up at finitely many points of a compact manifold that already carries a constant scalar curvature Kahler metric. In the case where the manifold has nontrivial holomorphic vector fields with zeros, we give necessary conditions on the number and locations of the blow up points for the blow up to carry constant scalar curvature Kahler metrics.

Compact complex surfaces and constant scalar curvature Kähler metrics

In the thesis, I prove the following statement: Every compact complex surface with even first Betti number is deformation equivalent to one which admits an extremal Kahler metric. In fact, this extremal Kahler metric can even be taken to have constant scalar curvature in all but two cases: the deformation equivalence classes of the blow-up of P 2 at one or two points. The explicit construction of compact complex surfaces with constant scalar curvature Kahler metrics in different deformation equivalence classes is given. The main tool repeatedly applied here is the gluing theorem of C. Arezzo and F. Pacard which states that the blow-up/resolution of a compact manifold/orbifold of discrete type, which admits cscK metrics, still admits cscK metrics.

A REMARK ON KÄHLER METRICS OF CONSTANT SCALAR CURVATURE ON RULED COMPLEX SURFACES

In this paper we point out how some recent developments in the theory of constant scalar curvature Kahler metrics can be used to clarify the existence issue for such metrics in the special case of (geometrically) ruled complex surfaces.

Unstable products of smooth curves

We give examples of smooth surfaces with negative first Chern class which are slope unstable with respect to certain polarisations, and so have Kahler classes that do not admit any constant scalar curvature Kahler metrics. We also compare this to the work of Song-Weinkove on the J-flow.

Blowing up and desingularizing constant scalar curvature Kähler manifolds

This paper is concerned with the existence of constant scalar curvature Kahler metrics on blow-ups at finitely many points of compact manifolds which already carry constant scalar curvature Kahler metrics. We also consider the desingularization of isolated quotient singularities of compact orbifolds which carry constant scalar curvature Kahler metrics.

Constant scalar curvature Kähler metrics on fibred complex surfaces

This article finds constant scalar curvature Kahler metrics on certain compact complex surfaces. The surfaces considered are those admitting a holomorphic submersion to curve, with fibres of genus at least 2. The proof is via an adiabatic limit. An approximate solution is constructed out of the hyperbolic metrics on the fibres and a large multiple of a certain metric on the base. A parameter dependent inverse function theorem is then used to perturb the approximate solution to a genuine solution in the same cohomology class. The arguments also apply to certain higher dimensional fibred Kahler manifolds.

Scalar Curvature and Projective Embeddings, I

We prove that a metric of constant scalar curvature on a polarised Kahler manifold is the limit of metrics induced from a specific sequence of projective embeddings; satisfying a condition introduced by H. Luo. This gives, as a Corollary, the uniqueness of constant scalar curvature Kahler metrics in a given rational cohomology class. The proof uses results in the literature on the asymptotics of the Bergman kernel. The arguments are presented in a general framework involving moment maps for two different group actions.

Hamiltonian constructions of Kähler-Einstein metrics and Kähler metrics of constant scalar curvature

Assuming the existence of a real torus acting through holomorphic isometries on a Kahler manifold, we construct an ansatz for Kahler-Einstein metrics and an ansatz for Kahler metrics with constant scalar curvature. Using this Hamiltonian approach we solve the differential equations in special cases and find, in particular, a family of constant scalar curvature Kahler metrics describing a non-linear superposition of the Bergman metric, the Calabi metric and a higher dimensional generalization of the LeBrun Kahler metric. The superposition contains Kahler-Einstein metrics and all the geometries are complete on the open disk bundle of some line bundle over the complex projective spaceP n. We also build such Kahler geometries on Kahler quotients of higher cohomogeneity.