Paper generalizes convexity to conic Mabuchi's functional.
problem Generalizing convexity to conic settings.
method Established frame for conic cscK metrics, introduced conic Mabuchi functional, proved convexity along geodesics.
result Proved convexity of conic Mabuchi's functional.
Investigates uniqueness of conic cscK metrics with cone angles less than π.
problem Uniqueness of conic constant scalar curvature Kähler metrics with cone singularities.
method Introduced a new Hölder space $\cC^{4,\a,\b}$ to study regularities, proved regularity of conic cscK metrics, and established reductivity through careful study of the conic Lichnerowicz operator.
result Any $\cC^{2,\a,\b}$ conic cscK metric is indeed of class $\cC^{4,\a,\b}$.
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.
Resolves singular cscK varieties to smooth ones.
problem Resolving conically singular cscK varieties to smooth cscK manifolds.
method Glueing result for crepant resolutions of cscK varieties with discrete automorphism groups.
result Shows a glueing result for (some) crepant resolutions of cscK varieties with discrete automorphism groups.
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.
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.
Proves complex projective varieties are birational to cscK models.
problem Complex projective varieties and their cscK metrics.
method Birational models and Lefschetz pencils.
result Every complex projective variety is birational to a smooth projective manifold with cscK metric.
The Liouville theorem and Cα-estimate for Calabi-Yau cones establish uniqueness and asymptotic behavior of metrics.
problem Establishing uniqueness and asymptotic behavior of metrics on Calabi-Yau cones.
method Developed a Liouville theorem and C0,α-estimate for Ricci-flat, conical Kähler manifolds. result Uniformly bounded Kähler metrics on a ball around the apex are asymptotic to the Ricci-flat cone metric with polynomial decay.
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.
Paper finds new cscK metrics on orbifold resolutions.
problem Existence of cscK metrics on orbifold resolutions.
method Computes Futaki invariant of Kaehler classes on resolutions.
result New existence and non-existence results for cscK metrics.
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.
Smooth minimizers of K-energy are cscK metrics in cohomologous Kähler classes.
problem Regularity of weak minimizers of the K-energy in Kähler manifolds.
method Analyzing the extended K-energy and using J-properness.
result Finite energy minimizers are smooth cscK metrics.
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.
The paper proves conditions for existence of constant scalar curvature Kähler metrics with cone singularities.
problem Existence of constant scalar curvature Kähler metrics with cone singularities.
method Equivalence to properness of log K-energy and geodesic stability. result Extensions of the solution of the properness conjecture and Donaldson's geodesic stability conjecture to cscK cone metrics.
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.
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.
This paper studies geodesics and uniqueness of cscK cone metrics.
problem Uniqueness of constant scalar curvature Kahler cone metrics.
method Introduction of weighted function spaces, construction of cone geodesics, detailed asymptotic analysis of cscK cone metrics, linear theory for Lichnerowicz operator.
result The cscK cone metric is unique up to automorphisms.
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.
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.
Proves openness of continuity path for new metric.
problem Continuity path for new metric.
method Introduced continuity path by X. Chen.
result Proved openness at t = 0.
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
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.
Paper proves existence of constant scalar curvature Kähler metrics under certain conditions.
problem Existence of constant scalar curvature Kähler metrics.
method Generalized apriori estimates and used automorphism group discreteness, K-energy non-increasing, and properness of K-energy.
result Proves equivalence of non-existence of cscK metric and existence of a destabilized geodesic ray with non-increasing K-energy.
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.
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 paper studies finite TYCZ expansions on Kaehler manifolds and their relation to cscK metrics.
problem Finite TYCZ expansions on Kaehler manifolds and their connection to cscK metrics.
method Analyzes finite TYCZ expansions on Kaehler manifolds and their properties.
result Finite TYCZ expansions imply polynomial behavior of certain metrics and vanishing of log-term in Szegö kernel.
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.
Study solutions to metric equations on modified surfaces.
problem Finding constant scalar curvature Kähler metrics on modified surfaces.
method Expanding solutions to extremal metric type equations.
result Developed methods to solve metric equations on modified surfaces.
New equations derived for Kähler metrics, linking stability and curvature.
problem Finding metrics with constant scalar curvature in Kähler geometry.
method Introduced coupled cscK equations and defined K-polystability.
result Proved existence of coupled cscK metrics for small perturbations.
The paper studies projective embeddings and K-stability for pairs.
problem Logarithmic K-stability of projective pairs.
method Analyzes projective completions and CSCK metrics, constructs almost balanced embeddings.
result Shows (L^,D,cA,0) is K-semistable. Geodesic rays prove key aspects of cscK metrics existence and stability.
problem Existence and stability of constant scalar curvature Kähler metrics.
method Reduction to regularization conjecture and analysis of geodesic rays.
result Uniform K-stability and JKX-stability are sufficient for cscK metrics existence. Study on constant mu-scalar curvature Kähler metrics, generalizing cscK and Kähler-Ricci solitons.
problem Existence and uniqueness of constant mu-scalar curvature Kähler metrics.
method Investigation of volume functional and study of a new K-energy.
result Fundamental constraints and existence conditions for constant mu-scalar curvature Kähler metrics.
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.
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).