Proves existence of weighted-cscK metrics on Kähler manifolds.
problem Existence of weighted-cscK metrics on Kähler manifolds.
method Proves existence via G-coercivity of weighted Mabuchi functional. result Existence of (v, w)-weighted-cscK metrics with v log-concave.
The paper proves the existence of singular cscK metrics on smoothable varieties.
problem Existence of singular cscK metrics on smoothable varieties.
method Developing a strong topology of pluripotential theory in families and uniform estimates for cscK metrics.
result Existence of singular cscK metrics on Q-Gorenstein smoothable klt varieties when the Mabuchi functional is coercive. The study finds conditions for Kaehler-Einstein and cscK metrics on certain manifold coverings.
problem Existence of Kaehler-Einstein and cscK metrics on ramified coverings.
method Cohomological conditions on Kaehler classes and branching divisors.
result Sufficient conditions for the existence of cscK metrics on ramified coverings.
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.
The paper proves conditions for Kähler-Einstein metrics to remain Kähler-Einstein under cscK perturbations.
problem Conditions for Kähler-Einstein metrics to remain Kähler-Einstein under cscK perturbations.
method Study of constant scalar curvature Kähler (cscK) metrics on complete non-compact Kähler--Einstein manifolds.
result Sufficient conditions for a cscK perturbation of a Kähler--Einstein metric to remain Kähler--Einstein.
Study on weighted cscK metrics on Kähler varieties with singularities.
problem Existence of singular weighted cscK metrics on Kähler varieties.
method Resolution of singularities, coercive weighted Mabuchi functional, construction of examples.
result Existence of singular weighted cscK metrics when the weighted Mabuchi functional is coercive.
We show that the existence of constant scalar curvature Kähler (cscK) metrics with cone singularities is equivalent to the properness of log K-energy. We also prove their equivalence to the geodesic stability. They are extensions of the solution of the properness conjecture and Donaldson's geodesic stability conjectu…
New invariants help solve existence of weighted cscK metrics.
problem Existence of weighted cscK metrics in K-stability.
method Introduced weighted analytic delta invariant and beta invariant.
result Sufficient condition for existence of weighted cscK metrics.
The paper introduces a new system of equations for Hessian-cscK metrics.
problem Finding constant scalar curvature Kähler metrics.
method Proposes a coupled system of complex Hessian equations and shows it can be variational.
result Proves a C0-estimate for the system that depends on entropy. This paper connects soliton-type metrics with weighted CSCK metrics on Fano manifolds.
problem Existence and properties of weighted constant scalar curvature Kähler metrics.
method Introducing a weight function g(v,w) and proving equivalence between (v,w)-CSCK metrics and g(v,w)-solitons. result Existence of (v,w)-CSCK metrics in the first Chern class is equivalent to existence of g(v,w)-solitons. Unique Poincaré type cscK metric with singularity at smooth divisor is unique up to holomorphic transformations.
problem Proving uniqueness of Poincaré type cscK metric with singularity at a smooth divisor.
method Holomorphic transformations, asymptotic behavior analysis, fixed point problem.
result Unique Poincaré type cscK metric with singularity at a smooth divisor is unique up to holomorphic transformations.
The paper finds conical higher cscK metrics on minimal ruled surfaces with conical singularities.
problem Existence of conical higher cscK metrics on minimal ruled surfaces.
method Develop conical singularities along at least one of the two special divisors and use the momentum construction.
result Conical higher cscK metrics exist in each Kähler class on minimal ruled surfaces.
This paper characterizes mu-cscK metrics using Perelman's W-entropy.
problem Characterizing mu-cscK metrics and understanding their properties.
method Using Perelman's W-entropy as a functional on the tangent bundle of Kähler metrics, the paper characterizes mu-cscK metrics as critical points of this functional.
result The W-entropy is monotonic along geodesics and provides a lower bound for mu-entropy.
Study on Kähler metrics with curvature constraints.
problem Existence of constant weighted scalar curvature Kähler metrics.
method Establish Ck-estimates for Kähler potentials. result Extends prior results on classical cscK metrics.
Researchers prove existence of cscK metrics on smooth minimal models.
problem Existence of constant scalar curvature Kähler metrics on compact Kähler manifolds.
method Direct proof showing existence on smooth minimal models and blowups.
result Compact Kähler manifolds with nef canonical bundle always admit cscK metrics.
This is a continuation of the previous articles on Kahler cone metrics. In this article, we introduce weighted function spaces and provide a self-contained treatment on cone angles in the whole interval (0,1]. We first construct geodesics in the space of Kahler cone metrics (cone geodesics). We next determine the ver…
Let (X,ω) be a compact Kähler manifold and H the space of Kähler metrics cohomologous to ω. If a cscK metric exists in H, 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 a…
Non-Archimedean balanced metrics approximate cscK metrics for totally degenerate abelian varieties
problem Non-Archimedean balanced metrics for polarized abelian varieties
method Non-Archimedean analogue of the cscK metric
result Uniform estimate for Calabi-Yau metrics on fibers
In this paper, we generalize our apriori estimates on cscK(constant scalar curvature Kähler) metric equation to more general scalar curvature type equations (e.g., twisted cscK metric equation). As applications, under the assumption that the automorphism group is discrete, we prove the celebrated Donaldson's conjecture…
Study on Kähler metrics on ruled surfaces, proving existence and non-existence.
problem Existence and non-existence of Kähler metrics on minimal ruled surfaces.
method Analysis of twisted and coupled constant scalar curvature Kähler metrics.
result Bound for Chen-Cheng invariant on ruled surfaces.
Study moduli space of cscK surfaces around toric ones, introducing foldable surfaces.
problem Moduli space of cscK surfaces around toric ones.
method Introduced foldable surfaces and classified them. Studied moduli space locally.
result Moduli space locally modeled on a finite quotient of a toric affine variety with terminal singularities.
Study shows convergence of cscK surfaces in Hilbert scheme.
problem Understanding convergence of cscK surfaces.
method Gromov--Hausdorff convergence and Hilbert scheme approach.
result Established convergence of non-collapsed polarized cscK surfaces in a Hilbert scheme.
We partially confirm an old conjecture of Donaldson that if there exists a cscK metrics in a given Kähler class, then there is no degenerated geodesic ray which is tamed by a bounded ambient geometry unless it parallels to a holomorphic line consists of cscK metrics only. We also prove that for simple test configuratio…
Let X be a compact toric surface. There exists a sequence of torus equivariant blow-ups of X such that the blown-up toric surface obtained admits a cscK metric.
We introduce a cohomological obstruction to solving the constant scalar curvature Kähler (cscK) equation twisted by a semipositive form, appearing in works of Fine and Song-Tian. Geometrically this gives an obstruction for a manifold to be the base of a holomorphic submersion carrying a cscK metric in certain ``adiabat…
The paper proves the existence of a special type of metric on complex manifolds.
problem Finding metrics with specific properties on complex manifolds.
method Proving the existence of n-complex dimensional manifolds with strictly partially regular and cscK metrics.
result For n ≥ 3, the (constant) scalar curvature of the metric can be zero, positive, or negative.
Compact metrics found near Kähler manifold's canonical class.
problem Finding unique cscK metrics near Kähler manifold's canonical class.
method Proved existence and uniqueness of cscK metrics for Kähler classes near canonical class.
result Metric spaces are pre-compact in Gromov-Hausdorff sense.
In this paper we compute the Futaki invariant of adiabatic Kaehler classes on resolutions of Kaehler orbifolds with isolated singularities. Combined with previous existence results of extremal metrics by Arezzo-Lena-Mazzieri, this gives a number of new existence and non-existence results for cscK metrics.
Analytic K-semistability connects curvature to metric existence.
problem Establishing constant scalar curvature Kähler metrics.
method Small polarized deformations and Futaki invariant computation.
result K-polystability implies existence of cscK metrics locally.
The paper defines and analyzes Kähler metrics near a compact manifold, showing their deviation from Poincaré-type metrics.
problem Understanding the behavior of Kähler metrics near a compact manifold.
method Defining and analyzing Kähler metrics on a trivial holomorphic open disk bundle, showing their deviation from Poincaré-type metrics.
result The Kähler metrics near a compact manifold deviate exponentially from Poincaré-type metrics, and they arise naturally in perturbing cscK metrics.
The aim of this article is to study expansions of solutions to an extremal metric type equation on the blow-up of constant scalar curvature Kähler surfaces. This is related to finding constant scalar curvature Kähler (cscK) metrics on K-stable blow-ups of extremal Kähler surfaces
Study of twisted Calabi flow connecting J-flow and Calabi flow on Kähler manifolds.
problem Existence and convergence of twisted Calabi flow on compact Kähler manifolds.
method Analysis of a family of twisted Calabi flows connecting J-flow and Calabi flow, showing long-time existence and convergence to cscK metrics.
result Long-time existence and convergence of twisted Calabi flow to cscK metrics, implying openness of continuity method.
We prove that polarised manifolds that admit a constant scalar curvature Kähler (cscK) metric satisfy a condition we call slope semistability. That is, we define the slope μ for a projective manifold and for each of its subschemes, and show that if X is cscK then μ(Z)≤μ(X) for all subschemes Z. This gives man…
Study asymptotic behavior of second Chern forms on degenerating Kähler-Einstein surfaces.
problem Asymptotic behavior of second Chern forms on degenerating Kähler-Einstein surfaces with ADE singularities.
method Investigates a function on the unit disc defined by fiber integrals of the forms with a smooth test function, showing a lower bound of Hölder exponent at the origin for both cscK-metrics and Ricci-flat metrics.
result Shows bounds of Hölder exponent for both cscK-metrics and Ricci-flat metrics.
Establishes Yau-Tian-Donaldson conjecture for weighted metrics.
problem Constant scalar curvature Kähler metrics on polarized projective manifolds.
method Extends Chi Li's work to weighted case, uses a priori estimates and slope formulas.
result Proves Yau-Tian-Donaldson conjecture for weighted extremal Kähler metrics.
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 coni…
New system modifies constant scalar curvature Kähler condition with a 'Higgs field'.
problem Extending constant scalar curvature Kähler condition to higher-dimensional manifolds.
method Explicit construction of hyperkähler metrics, Hitchin's equations for harmonic bundles, and Hermitian Yang-Mills equation.
result Existence of solutions to the modified system on specific cases (Riemann surfaces, ruled surfaces, abelian and toric surfaces).
We consider the Kähler-Ricci flow on a Fano manifold. We show that if the curvature remains uniformly bounded along the flow, the Mabuchi energy is bounded below, and the manifold is K-polystable, then the manifold admits a Kähler-Einstein metric. The main ingredient is a result that says that a sufficiently small pert…
Establishes convexity and coercivity of K-energy functional for complex tori.
problem Convexity and coercivity of K-energy functional for complex tori.
method Geodesics in finite energy space, cone angle perturbations, stability of coercivity.
result Openness of coercivity under cone angle perturbations and existence of cscK cone metrics.
Uniform K-stability of Calabi-Yau fibrations linked to base curve stability.
problem K-stability of Calabi-Yau fibrations over curves.
method Adiabatic uniform K-stability and log-twisted K-stability of base curves.
result Uniform K-stability of Calabi-Yau fibrations if and only if base curves are K-stable.
The aim of this paper is to investigate uniqueness of conic constant scalar curvature Kaehler (cscK) metrics, when the cone angle is less than π. We introduce a new Hölder 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 metri…
The paper proves a unique cscK metric for uniformly K-stable Kähler manifolds.
problem Finding a unique cscK metric for uniformly K-stable Kähler manifolds.
method Developed non-Archimedean pluripotential theory, used valuative criterion, and extended Calabi-Yau Theorem.
result Proved existence and uniqueness of cscK metrics for uniformly K-stable Kähler manifolds.
The scalar curvature equation for rotation invariant Kähler metrics on Cn\{0} is reduced to a system of ODEs of order 2. By solving the ODEs, we obtain complete lists of rotation invariant zero or positive csck on Cn\{0} in lower dimensions. We also prove that there doe…
Miyaoka-Yau inequality proven for smooth minimal models.
problem Proving the Miyaoka-Yau inequality for smooth minimal models.
method Existence of cscK metrics in a neighborhood of the canonical class.
result Miyaoka-Yau inequality holds for compact Kähler manifolds with nef canonical bundle.
In this paper, we proved the openness at t =0 of the new continuity path recently introduced by X. Chen.
In this note we discuss the problem of resolving conically singular cscK varieties to construct smooth cscK manifolds, showing a glueing result for (some) crepant resolutions of cscK varieties with discrete automorphism groups.
Let 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 L^\D. When the corresponding \kahler form is in the cohomology class of a rational divisor A and when L has negative CSC…
This is a continuation of the work of Arezzo-Pacard-Singer and the author on blowups of extremal Kähler manifolds. We prove the conjecture stated in [32], and we relate this result to the K-stability of blown up manifolds. As an application we prove that if a Kähler manifold M of dimension greater than 2 admits a cscK …