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.
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.
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 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.
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.
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.
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.
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.
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…
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.
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.
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. The paper improves the regularity and existence of pseudo Calabi flow.
problem Improving the smoothness and existence of pseudo Calabi flow.
method Analyzing the initial conditions and using volume form closeness to smooth metrics.
result The pseudo Calabi flow becomes smooth immediately and exists for all time under certain conditions.
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.
Researchers prove K-semistability for cscK manifolds with transcendental cohomology.
problem K-semistability of cscK manifolds with transcendental cohomology class.
method Utilizing a recent result by R. Berman, T. Darvas, and C. Lu, the authors establish a formula relating the Donaldson-Futaki invariant to the asymptotic slope of the K-energy.
result cscK manifolds with transcendental cohomology class are K-semistable.
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.
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.
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.
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…
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. Paper studies constant scalar curvature equation and its smooth solutions on Kähler manifolds.
problem Analyzing constant scalar curvature equation and its weak solutions on Kähler manifolds.
method Defined weak solutions for CSCK, proved smoothness with uniform L∞ bound, used W2,2 regularity for Laplacian equation. result Weak solutions of CSCK with uniform L∞ bound are smooth. 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.
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.
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.
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 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. Paper studies K-polystability of cscK manifolds with holomorphic vector fields.
problem K-polystability of cscK manifolds with transcendental cohomology class.
method Study of subgeodesic rays and asymptotics of K-energy functional.
result Existence of cscK metric implies geodesic K-polystability.
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.
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.
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.
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.
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.
Introduces Kähler metrics with constant weighted scalar curvature and explores their stability.
problem Existence and stability of Kähler metrics with constant weighted scalar curvature.
method Defines a functional and introduces a weighted Futaki invariant to study stability.
result Shows that constant weighted scalar curvature metrics are minima of a functional and implies weighted K-semistability.
Solves Yau-Tian-Donaldson conjecture for smooth projective varieties.
problem Uniform K-stability and existence of cscK metrics.
method Special Fujita approximations and regularization of entropy functional.
result Uniformly K-stable polarized smooth projective varieties admit cscK metrics.
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…
The paper introduces new equations in Kähler geometry and proves their solutions and convexity.
problem Solving equations in Kähler geometry and understanding their geometric implications.
method Using moment map pictures to motivate and prove solutions for the equations.
result The Mabuchi functional for certain equations is shown to be convex.
Study compares different stability notions in Kähler geometry.
problem Comparing various stability notions in Kähler geometry.
method Introduce and study geodesic stability on rays with specific singularity types.
result Equivalence of some stability notions under certain conditions.
Stability of weighted extremal manifolds proven through blowups.
problem Stability of weighted extremal manifolds.
method Blowup technique to analyze weighted extremal Kähler manifolds.
result Proves weighted extremal manifolds are relatively weighted K-polystable.
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.
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. 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.
Study Kähler metrics with constant scalar curvature using coupled equations.
problem Finding Kähler metrics with constant scalar curvature.
method Solving a system of elliptic equations for a Kähler metric and a closed (1,1)-form, proving higher order estimates and smooth convergence.
result Smooth convergence to a cscK metric coupled to a harmonic (1,1)-form under uniform estimates.
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.
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).
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
Faltings heights help in Kahler geometry stability.
problem Stability of polarized manifolds and csck metrics.
method Faltings heights, Igusa local zeta functions, stability conjectures.
result Stability of (X,L) implies existence of csck metric.
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 …
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.