cscK Metrics Research Articles

Convergence of cscK metrics on smooth minimal models of general type

Convergence of cscK metrics on smooth minimal models of general type

On the Futaki invariant of Fano threefolds

We study the zero locus of the Futaki invariant on K-polystable Fano threefolds, seen as a map from the Kähler cone to the dual of the Lie algebra of the reduced automorphism group. We show that, apart from families 3.9, 3.13, 3.19, 3.20, 4.2, 4.4, 4.7 and 5.3 of the Iskovskikh–Mori–Mukai classification of Fano threefolds, the Futaki invariant of such manifolds vanishes identically on their Kähler cone. In all cases, when the Picard rank is greater or equal to two, we exhibit explicit 2-dimensional differentiable families of Kähler classes containing the anti-canonical class and on which the Futaki invariant is identically zero. As a corollary, we deduce the existence of non Kähler–Einstein cscK metrics on all such Fano threefolds.

The balanced metrics and cscK metrics on polyball bundles

The balanced metrics and cscK metrics on polyball bundles

A product model for generalizing Poincaré-type Kähler metrics

We begin by defining a type of Kähler metric near the zero section of a trivial holomorphic open disk bundle N N over a compact Kähler manifold X X by incorporating flows generated by holomorphic vector fields on X X . These metrics are then shown to deviate exponentially from Poincaré-type metrics on N ∖ X N\setminus X in terms of the log-polar distance from X X in N N . Lastly we see that they arise naturally when perturbing classes containing Poincaré-type Kähler metrics of constant scalar curvature to obtain nearby cscK metrics even when the perturbed class on X X does not admit a cscK metric.

Constant scalar curvature Kähler metrics on ramified Galois coverings

Abstract We give sufficient conditions for the existence of Kähler–Einstein and constant scalar curvature Kähler (cscK) metrics on finite ramified Galois coverings of a cscK manifold in terms of cohomological conditions on the Kähler classes and the branching divisor. This result generalizes previous work on Kähler–Einstein metrics by Li and Sun [C. Li and S. Sun, Conical Kähler–Einstein metrics revisited, Comm. Math. Phys. 331 2014, 3, 927–973], and extends Chen–Cheng’s existence results for cscK metrics in [X. Chen and J. Cheng, On the constant scalar curvature Kähler metrics (II)—Existence results, J. Amer. Math. Soc. 34 2021, 4, 937–1009].

The Miyaoka–Yau inequality on smooth minimal models

AbstractIn this short note, we offer an observation that the Miyaoka–Yau inequality holds for any compact Kähler manifold with nef canonical bundle, that is, a smooth minimal model. It follows directly from the existence of cscK metrics in a neighborhood of the canonical class which was confirmed both by the work of Dyrefelt and Song using different approaches.

Existence of cscK metrics on smooth minimal models

Given a compact Kahler manifold $X$ it is interesting to ask whether it admits a constant scalar curvature Kahler (cscK) metric. In this short note we show that there always exist cscK metrics on compact Kahler manifolds with nef canonical bundle, thus on all smooth minimal models, and also on the blowup of any such manifold. This confirms an expectation of Jian-Shi-Song \cite{JianShiSong} and extends their main result from $K_X$ semi-ample to $K_X$ nef, with a direct proof that does not appeal to the Abundance conjecture. As a byproduct we obtain that the connected component $\mathrm{Aut}_0(X)$ of a compact Kahler manifold with $K_X$ nef is either trivial or a complex torus.

The balanced metrics and cscK metrics on certain holomorphic ball bundles

In this paper, we use the well-known Calabi ansatz, further generalized by Hwang-Singer, to study the existence of balanced metrics and constant scalar curvature Kähler (cscK for short) metrics on certain holomorphic ball bundles M which are locally expressed as M={(z,u,w)∈Ω×Cd1×Cd2:eλ1ϕ(z)‖u‖2+eλ2ϕ(z)‖w‖2<1}. Let g be the Kähler metric on M associated with the Kähler forms locally expressed as ω=−1∂∂‾(ϕ(z)+F(eλ1ϕ(z)‖u‖2+eλ2ϕ(z)‖w‖2)). Firstly, we obtain sufficient and necessary conditions for g to be cscK metrics. Secondly, using this result, we obtain necessary and sufficient conditions for mg to be balanced metrics for all sufficiently large positive integer numbers m. Finally, we obtain complete cscK metrics and balanced metrics on the ball bundles over simply connected Riemann surfaces. The main contribution of this paper is the explicit construction of complete, non-compact cscK metrics and balanced metrics.

On the constant scalar curvature Kähler metrics (II)—Existence results

In this paper, we apply our previous estimates in Chen and Cheng [On the constant scalar curvature Kähler metrics (I): a priori estimates, Preprint] to study the existence of cscK metrics on compact Kähler manifolds. First we prove that the properness ofKK-energy in terms ofL1L^1geodesic distanced1d_1in the space of Kähler potentials implies the existence of cscK metrics. We also show that the weak minimizers of theKK-energy in(E1,d1)(\mathcal {E}^1, d_1)are smooth cscK potentials. Finally we show that the non-existence of cscK metric implies the existence of a destabilizedL1L^1geodesic ray where theKK-energy is non-increasing, which is a weak version of a conjecture by Donaldson. The continuity path proposed by Xiuxiong Chen [Ann. Math. Qué. 42 (2018), pp. 69–189] is instrumental in the above proofs.

Partially regular and cscK metrics

A Kähler metric [Formula: see text] with integral Kähler form is said to be partially regular if the partial Bergman kernel associated to [Formula: see text] is a positive constant for all integer [Formula: see text] sufficiently large. The aim of this paper is to prove that for all [Formula: see text] there exists an [Formula: see text]-dimensional complex manifold equipped with strictly partially regular and cscK metric [Formula: see text]. Further, for [Formula: see text], the (constant) scalar curvature of [Formula: see text] can be chosen to be zero, positive or negative.

Optimal lower bounds for Donaldson's J-functional

In this paper we provide an explicit formula for the optimal lower bound of Donaldson's J-functional, in the sense of finding explicitly the optimal constant in the definition of coercivity, which always exists and takes negative values in general. This constant is positive precisely if the J-equation admits a solution, and the explicit formula has a number of applications. First, this leads to new existence criteria for constant scalar curvature Kähler (cscK) metrics in terms of Tian's alpha invariant. Moreover, we use the above formula to discuss Calabi dream manifolds and an analogous notion for the J-equation, and show that for surfaces the optimal bound is an explicitly computable rational function which typically tends to minus infinity as the underlying class approaches the boundary of the Kähler cone, even when the underlying Kähler classes admit cscK metrics. As a final application we show that if the Lejmi-Székelyhidi conjecture holds, then the optimal bound coincides with its algebraic counterpart, the set of J-semistable classes equals the closure of the set of uniformly J-stable classes in the Kähler cone, and there exists an optimal degeneration for uniform J-stability.

A scalar Calabi-type flow in Hermitian geometry: short-time existence and stability

We introduce a new geometric flow of Hermitian metrics which evolves an initial metric along the second derivative of the Chern scalar curvature. The flow depends on the choice of a background metric, it always reduces to a scalar equation and preserves some special classes of Hermitian structures, such as balanced and Gauduchon metrics. We show that the flow has always a unique short-time solution and we provide a stability result when the background metric is Ka¨hler with constant scalar curvature (cscK). The main theorem is obtained by proving a general result about stability of parabolic flows on Riemannian manifolds which is interesting in its own right and in particular implies the stability of the classical Calabi flow near cscK metrics.

Regularity of weak minimizers of the K-energy and applications to properness and K-stability

Let (X,ω)(X,ω) be a compact Kahler manifold and HH the space of Kahler metrics cohomologous to ωω. If a csck metric exists in HH, we show that all finite energy minimizers of the extended K-energy are smooth csck metrics, partially confirming a conjecture of Y.A. Rubinstein and the second author. As an immediate application, we obtain that the existence of a csck metric in HH implies J-properness of the K-energy, thus confirming one direction of a conjecture of Tian. Exploiting this properness result we prove that an ample line bundle (X,L)(X,L) admitting a csck metric in c1(L)c1(L) is KK-polystable. When the automorphism group is finite, the properness result, combined with a result of Boucksom-Hisamoto-Jonsson, also implies that (X,L)(X,L) is uniformly K-stable.

Log-concavity of volume and complex Monge–Ampère equations with prescribed singularity

Let $$(X,\omega )$$ be a compact Kahler manifold. We prove the existence and uniqueness of solutions to complex Monge–Ampere equations with prescribed singularity type. Compared to previous work, the assumption of small unbounded locus is dropped, and we work with general model type singularities. We state and prove our theorems in the context of big cohomology classes, however our results are new in the Kahler case as well. As an application we confirm a conjecture by Boucksom–Eyssidieux–Guedj–Zeriahi concerning log-concavity of the volume of closed positive (1, 1)-currents. Finally, we show that log-concavity of the volume in complex geometry corresponds to the Brunn–Minkowski inequality in convex geometry, pointing out a dictionary between our relative pluripotential theory and P-relative convex geometry. Applications related to stability and existence of csck metrics are treated elsewhere.

Kähler geometry of horosymmetric varieties, and application to Mabuchi’s K-energy functional

Abstract We introduce a class of almost homogeneous varieties contained in the class of spherical varieties and containing horospherical varieties as well as complete symmetric varieties. We develop Kähler geometry on these varieties, with applications to canonical metrics in mind, as a generalization of the Guillemin–Abreu–Donaldson geometry of toric varieties. Namely we associate convex functions with Hermitian metrics on line bundles, and express the curvature form in terms of this function, as well as the corresponding Monge–Ampère volume form and scalar curvature. We provide an expression for the Mabuchi functional and derive as an application a combinatorial sufficient condition of properness similar to one obtained by Li, Zhou and Zhu on group compactifications. This finally translates to a sufficient criterion of existence of constant scalar curvature Kähler metrics thanks to the recent work of Chen and Cheng. It yields infinitely many new examples of explicit Kähler classes admitting cscK metrics.

Projective embedding of pairs and logarithmic K-stability

Let $\hat{L}$ be the projective completion of an ample line bundle $L$ over $D$, a smooth projective manifold. Hwang-Singer \cite{HwangS} have constructed complete CSCK metric on $\hat{L}\backslash D$. When the corresponding \kahler form is in the cohomology class of a rational divisor $A$ and when $L$ has negative CSCK metric on $D$, we show that the Kodaira embedding induced by orthonormal basis of the Bergman space of $kA$ is almost balanced. As a corollary, $(\hat{L},D,cA,0)$ is K-semistable.

Subharmonicity of conic Mabuchi’s functional, I

The purpose of this paper is to generalize the convexity of Mabuchi's functional to the conic setting. We first established a frame to study conic cscK metrics, and then the conic Mabuchi functional was introduced in such a way that conic cscK metrics are its critical points. Finally we proved the convexity of the conic Mabuchi functional along any conic geodesics.

Uniqueness of constant scalar curvature Kähler metrics with cone singularities. I: reductivity

The aim of this paper is to investigate uniqueness of conic constant scalar curvature Kaehler (cscK) metrics, when the cone angle is less than $\pi$. We introduce a new H\"older space called $\cC^{4,\a,\b}$ to study the regularities of this fourth order elliptic equation, and prove that any $\cC^{2,\a,\b}$ conic cscK metric is indeed of class $\cC^{4,\a,\b}$. Finally, the reductivity is established by a careful study of the conic Lichnerowicz operator.

On the existence of constant scalar curvature Kähler metric: a new perspective

In this note, we introduce a new continuity path of fourth order nonlinear equations connecting the cscK equation to a second order elliptic equation, which is the critical point equation of the J-flow introduced by Donaldson (Asian J Math 3(1):1–16, 1999) and the author (Commun Anal Geom 12(4):837–852, 2004). This is a generalization of the classical Aubin continuity path for Kahler–Einstein metrics. The aim of this new path is to attack the existence problem of cscK metric. The “openness” along this continuity path is proved and a set of open problems associated with this new path is proposed.

Removable Singularities of the cscK Metric

In this paper, we consider a CscK metric defined away from divisor and with metric upper bound and lower bound going to zero in certain rate. And we'll prove that this nicely behaved metric is a smooth CscK metric across the divisor.