Defines operations in non-Archimedean metrics theory.
problem No specific problem stated; focuses on theory development.
method Defines operations using non-Archimedean metrics theory.
result Establishes the envelope conjecture holds in the theory.
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48 results for “non-Archimedean pluripotential theory”
Lecture notes on using non-Archimedean geometry for complex variety degenerations.
problem Complex algebraic variety degenerations with non-Archimedean Berkovich spaces.
method Hybrid spaces and non-Archimedean pluripotential theory.
result Relation between convergence of psh metrics and Monge-Ampere measures in hybrid spaces.
Synthetic approach to pluripotential theory measures finite energy.
problem Global pluripotential theory on compact Kähler manifolds and projective Berkovich spaces.
method Definition and study of measures of finite energy, introduction of twisted and free energy functionals.
result Coercivity of energy functionals is an open condition with respect to polarization.
Survey on metric SYZ conjecture and non-archimedean geometry.
problem Existence of special Lagrangian fibrations on Calabi-Yau manifolds.
method Pluripotential theory and non-archimedean geometry.
result Subtleties and open questions in the conjectural picture.
We survey some recent developments in the direction of the Yau-Tian-Donaldson conjecture, which relates the existence of constant scalar curvature Kähler metrics to the algebro-geometric notion of K-stability. The emphasis is put on the use of pluripotential theory and the interpretation of K-stability in terms of non-…
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.
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.
Uniform proof of Kähler-Einstein metrics with arbitrary polarizations.
problem Existence of Kähler-Einstein metrics with arbitrary polarizations.
method Quantization techniques and pluripotential theory.
result Uniform Yau-Tian-Donaldson theorem for Kähler-Einstein metrics.
The paper explores maximal destabilizers for both K-stability and Chow-stability in unstable situations.
problem Exploring maximal destabilizers for K-stability and Chow-stability in unstable situations.
method Using non-Archimedean pluripotential theory and idealistic assumptions, the paper provides a route to show that maximal K-destabilizers are quantized by maximal Chow-destabilizers.
result Maximal K-destabilizers are quantized by maximal Chow-destabilizers.
We give a variational proof of a version of the Yau-Tian-Donaldson conjecture for twisted Kähler-Einstein currents, and use this to express the greatest (twisted) Ricci lower bound in terms of a purely algebro-geometric stability threshold. Our approach does not involve the continuity method or Cheeger-Colding-Tian the…
Study of non-archimedean μ-entropy and its connection to K-stability.
problem Understanding K-stability in non-archimedean settings.
method Introducing non-archimedean μ-entropy and its properties, connecting it to K-semistability.
result Established a criterion for K-semistability without vector ξ, using the non-archimedean μ-entropy.
We propose a list of open problems in pluripotential theory partially motivated by their applications to complex differential geometry. The list includes both local questions as well as issues related to the compact complex manifold setting.
We extend profound results in pluripotential theory on Kahler manifolds to Sasaki setting via its transverse Kahler structure. As in Kahler case, these results form a very important piece to solve the existence of Sasaki metrics with constant scalar curvature (cscs) in terms of properness of K-energy. One main result i…
Study complex Monge-Ampère flows on Kähler manifolds using Perron method.
problem Complex Monge-Ampère flows in big cohomology classes.
method Perron method for pluripotential subsolutions.
result Upper envelope of subsolutions is a unique pluripotential solution with regularity.
New proof for stability estimates in complex equations without pluripotential theory.
problem Stability estimates for complex Monge-Ampère and Hessian equations.
method New proof using general degenerations of background metrics.
result Uniform stability estimates for both equations under various degenerations.
We develop the first steps of a parabolic pluripotential theory in bounded strongly pseudo-convex domains of Cn. We study certain degenerate parabolic complex Monge-Amp{è}re equations, modelled on the K{ä}hler-Ricci flow evolving on complex algebraic varieties with Kawamata log-terminal singularities. Under natural ass…
Smoothness proven for conical Calabi-Yau potentials on Fano cones.
problem Smoothness of conical Calabi-Yau potentials on Fano cones.
method Pluripotential theory on degenerate Sasakian manifolds.
result Locally bounded conical Calabi-Yau potentials are smooth on the regular locus.
We study (transverse) scalar curvature type equation on compact Sasaki manifolds, in view of recent breakthrough of Chen-Cheng \cite{CC1, CC2, CC3} on existence of Kähler metrics with constant scalar curvature (csck) on compact Kähler manifolds. Following their strategy, we prove that given a Sasaki structure (with Ree…
The paper proves a comparison principle for complex Monge-Ampère flows and solves a uniqueness problem.
problem Proving uniqueness of weak solutions to the pluripotential Cauchy-Dirichlet problem.
method Proving a comparison principle for the pluripotential complex Monge-Ampère flows.
result Proves the uniqueness of the weak solution to the pluripotential Cauchy-Dirichlet problem.
We develop a parabolic pluripotential theory on compact K{ä}hler manifolds, defining and studying weak solutions to degenerate parabolic complex Monge-Amp{è}re equations. We provide a parabolic analogue of the celebrated Bedford-Taylor theory and apply it to the study of the K{ä}hler-Ricci flow on varieties with log te…
Study of non-Archimedean Hitchin map for SL2(F) characters.
problem Characterizing representations of SL2(F) characters.
method Equivariant harmonic maps into R-trees, Jenkins-Strebel differentials.
result The non-Archimedean Hitchin map is continuous and its image is contained in Jenkins-Strebel differentials.
Finite energy pluripotential theory accommodates the variational theory of equations of complex Monge-Ampère type arising in Kähler geometry. Recently it has been discovered that many of the potential spaces involved have a rich metric geometry, effectively turning the variational problems in question into problems of …
Theory developed for Hilbert geometry over valued fields, linking real and non-Archimedean geometries.
problem Understanding Hilbert geometry over general valued fields and their limits.
method Developed a theory of Hilbert geometry over general ordered valued fields, proving ultralimit results.
result Ultralimit of rescaled real Hilbert geometries is isometric to a non-Archimedean Hilbert metric space.
Defines and studies solutions to complex equations on Hermitian manifolds.
problem Solving complex equations on Hermitian manifolds.
method Extending recent theories, defines and studies pluripotential solutions to degenerate parabolic complex Monge-Ampère equations.
result Establishes existence and uniqueness of weak Chern-Ricci flow on complex compact varieties with log terminal singularities.
Paper proves all Lagrangians unobstructed if one is, using non-archimedean analytic structure.
problem Proving the existence of bounding cochains for unobstructed Lagrangians.
method Introducing non-archimedean analytic structure and using family Floer techniques.
result All Lagrangians in a connected family are unobstructed if one is.
Paper generalizes K-stability to non-algebraic spaces for Kähler metrics.
problem Existence of constant scalar curvature Kähler metrics on non-algebraic spaces.
method Developed a non-Archimedean theory of complex spaces and applied it to Kähler manifolds.
result Proved K-stability implies existence of unique constant scalar curvature Kähler metric.
This is the second paper in a series of investigations of the pluripotential theory on Teichmüller space. The main purpose of this paper is to establish the Poisson integral formula for pluriharmonic functions on Teichmüller space which are continuous on the Bers compactification. We also observe that the Schwarz type …
Sharp estimates proved for complex Monge-Ampère equations.
problem Proving sharp estimates for complex Monge-Ampère equations.
method PDE proof covering fixed and degenerating background metrics, extends to general fully non-linear equations.
result Sharp estimates proved for complex Monge-Ampère equations.
Study the limit of Calabi-Yau metrics with degenerate skeletons.
problem Understanding the behavior of Calabi-Yau metrics with degenerate skeletons.
method Using polarised degenerations and optimal transport problems.
result Describe the limiting behaviour of the Calabi-Yau potential.
Metric SYZ conjecture proved using non-archimedean geometry.
problem Proving the metric SYZ conjecture in large generality.
method Using non-archimedean geometry to support the conjecture.
result Metric SYZ conjecture can be proved in large generality.
Study on non-archimedean μ-entropy for toric varieties, proving existence and uniqueness.
problem Exploring non-archimedean μ-entropy for toric varieties and its thermodynamical structure.
method Established a Rellich type compactness result for convex functions on simple polytope, proving existence and uniqueness of optimizer.
result Existence and uniqueness of optimizer for toric non-archimedean μ^λ-entropy for λ ≤ 0.
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 -Gorenstein smoothable klt varieties when the Mabuchi functional is coercive.
Solves non-Archimedean Calabi-Yau equation on complex log pairs.
problem Non-Archimedean Monge-Ampère equation on Berkovich analytification.
method Solves complex Monge-Ampère equation, then adapts to non-Archimedean setting.
result Non-Archimedean analog of Ricci-flat metric potentials on complex affine varieties.
This is the first paper in a series of investigation of the pluripotential theory on Teichmüller space. The main purpose of this paper is to give an alternative approach to the Krushkal formula of the pluricomplex Green function on Teichmüller space. We also show that Teichmüller space carries a natural stratified stru…
Study Kähler-Einstein potentials on stable varieties near singularities
problem Asymptotic behavior of Kähler-Einstein potentials on stable varieties near singularities
method Using iterated logarithmic functions and refined lower bounds
result Improved estimates for Kähler-Einstein potentials
This paper constructs a non-Archimedean Teichmüller space using tropical geometry.
problem Constructing a non-Archimedean analogue of Teichmüller space.
method Using techniques from tropical and logarithmic geometry.
result The skeleton of non-Archimedean Teichmüller space is the tropical Teichmüller space.
Let (X,L) be a (semi-) polarized complex projective variety and T a real torus acting holomorphically on X with moment polytope P. Given a probability density g on P we introduce a new type of Monge-Ampere measure on X, defined for singular T-invariant metrics on the line bundle L, generalizing the ordinary Monge-Amper…
We give a version of the comparison principle from pluripotential theory where the Monge-Ampère measure is replaced by the Bergman kernel and use it to derive a maximum principle
Geodesics in non-Archimedean metrics are continuous.
problem Understanding geodesics in spaces of non-Archimedean metrics.
method Maximal psh segments are geodesics, and continuity of these segments is proven.
result Maximal psh segments joining continuous psh metrics are continuous.
Non-archimedean SYZ fibration constructed for Calabi-Yau hypersurfaces.
problem Analyzing Calabi-Yau hypersurfaces using non-archimedean geometry.
method Yamamoto's tropical contractions and Li's Fermat degeneration, with toric plurisubharmonic metrics.
result Constant potential along fibers of retraction under discrete symmetry assumption.
Proves existence and uniqueness of weak solutions for specific equations.
problem Existence and uniqueness of solutions for generalized Monge-Ampère and deformed Hermitian-Yang-Mills equations.
method Combines viscosity-theoretic and pluripotential-theoretic techniques.
result Existence and uniqueness of weak solutions in boundary cases.
Richberg technique adapted for nonlinear subequations.
problem Approximating strictly subharmonic functions in -potential theory.
method Adapting Richberg technique to -potential theory.
result Local approximation to global approximation for subharmonic functions.
This note explores norms beyond ultrametric inequalities in non-Archimedean analysis.
problem Analyzing norms beyond ultrametric inequalities in non-Archimedean analysis.
method Characterization of isometries between finite-dimensional spaces with a specific norm.
result Characterization of isometries between finite-dimensional linear spaces over a valued field.
Authors discuss complex and non-Archimedean geometry, proving a conjecture.
problem Proving a version of the Yau--Tian--Donaldson conjecture for Kähler metrics.
method Relation between complex, analytic, and non-Archimedean geometry.
result Sketch of proof for Yau--Tian--Donaldson conjecture.
This is a survey of some of the recent developments in the theory of complex Monge-Ampere equations. The topics discussed include refinements and simplifications of classical a priori estimates, methods from pluripotential theory, variational methods for big cohomology classes, semiclassical constructions of solutions …
Proves existence of Kähler-Einstein metrics in big cohomology classes.
problem Existence of Kähler-Einstein metrics in big cohomology classes.
method Using a divisorial stability condition and Fujita-Odaka type delta invariants, building up from scratch the theory of pluripotential theory.
result Uniform Yau-Tian-Donaldson existence theorem for Kähler-Einstein metrics in the big cohomology class setting.
In this work, we study Monge-Ampere equations over closed Kähler manifolds with degenerated cohomology classes. Classic results and arguments in pluripotential theory are generalized a little bit to be applied to our situation.
New space for valuations in non-Archimedean setting with duality properties.
problem Developing a non-Archimedean analogue of classical valuation spaces.
method Construction of a new space of valuations with similar structures to classical spaces.
result The new space satisfies Poincaré duality and hard Lefschetz theorem.