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.
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 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.
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.
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.
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.
The paper proves a Basmajian identity for non-Archimedean local fields.
problem Proving Basmajian's identity over non-Archimedean local fields.
method Projective Anosov representations and Berkovich hyperbolic geometry.
result A signed finite sum series identity for Basmajian's identity.
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-…
Construct special Lagrangian fibrations on abelian varieties using retraction techniques.
problem Constructing special Lagrangian fibrations on abelian varieties.
method Explicit construction using special techniques in non-Archimedean geometry.
result Solved a conjecture of Kontsevich-Soibelman for finite quotients of abelian varieties.
Research connects probabilistic and variational approaches to Kahler-Einstein metrics.
problem Constructing Kahler-Einstein metrics on complex projective varieties.
method Combines probabilistic construction and variational methods.
result Non-Archimedean geometry of X emerges from probabilistic framework.
The purpose of the present paper is to set up a formalism inspired from non-Archimedean geometry to study K-stability. We first provide a detailed analysis of Duistermaat-Heckman measures in the context of test configurations, characterizing in particular the trivial case. For any normal polarized variety (or, more gen…
Consider a polarized complex manifold (X,L) and a ray of positive metrics on L defined by a positive metric on a test configuration for (X,L). For most of the common functionals in Kähler geometry, we prove that the slope at infinity along the ray is given by evaluating the non-Archimedean version of the functional (as…
New complete Calabi-Yau metrics found in complex space.
problem Finding metrics on complex spaces with specific conditions.
method Generalized Calabi ansatz, non-archimedean Monge-Ampère equation.
result Complete Calabi-Yau metrics constructed in Fano manifolds.
Derives formulas for Monge-Ampère measures and reduces complex conjectures to simpler existence problems.
problem Complex Monge-Ampère measures and their applications in algebraic geometry.
method Derives formulas and reduces conjectures to simpler existence problems.
result Reduces uniform Yau-Tian-Donaldson conjecture to existence of approximate decompositions.
Paper introduces hybrid curves moduli space and canonical measures.
problem Asymptotic geometry of Riemann surfaces and their moduli spaces.
method Definition of hybrid curves and canonical measures.
result Continuous variation of canonical measures over hybrid curves moduli space.
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.
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.
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.
Study of algebraic dynamics on Markov cubics in tropical geometry.
problem Understanding the dynamics of Markov cubics over non-archimedean fields.
method Tropicalization and (∞,∞,∞)-triangle reflection group on hyperbolic plane. result Existence of Fatou domain and finitude of orbits with rational points over prime power denominators.
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.
Two Calabi-Yau theorems for Kähler manifold degenerations.
problem Understanding degenerations of compact Kähler manifolds.
method Direct proof for big test configurations and broader class of degenerations in non-Archimedean Kähler geometry.
result Established a connection between big cohomology classes and their volumes.
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.
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.
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.
Mathematically proves SYZ conjecture for conifold transition.
problem Tackles the SYZ conjecture for conifold transition.
method Uses family Floer context and non-archimedean setting.
result Explicitly writes singular T-duality fibers and confirms missing points in mirror cluster variety.
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.
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.
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
Proves uniform K-stability is open in Kähler cone.
problem Stability of Kähler metrics in complex geometry.
method Introduced new norm on test configurations and estimates for non-archimedean energy functionals.
result Uniform K-stability is an open condition in the Kähler cone.
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.
Introduces non-Archimedean metrics for pseudoeffective classes on Kähler manifolds.
problem Characterizing and approximating non-Archimedean metrics on pseudoeffective classes.
method Extending Ross-Witt Nyström correspondence to relative case, introducing flag configurations.
result Non-Archimedean finite energy metrics are approximable by flag configurations, and very general Ding energies are continuous.
The study examines discrete subgroups of PSL2 over non-archimedean fields.
problem Conditions for discrete subgroups of PSL2 over non-archimedean fields.
method Structure theorem for two-generator groups acting by isometries on a Λ-tree, practical algorithms.
result Necessary and sufficient conditions for discrete subgroups of PSL2 over non-archimedean fields.
Study bubbling Kahler metrics using algebraic geometry.
problem Analyzing the degeneration of Kahler metrics with Euclidean volume growth.
method Algebraic construction of birational modifications to simplify degenerations, comparing with analytic constructions.
result Provide a framework to compare algebraic and analytic approaches to bubbling phenomena.
The study constructs universal invariants for non-Archimedean metrics on projective varieties.
problem Understanding the singularity of non-Archimedean metrics on projective varieties.
method Constructing partial Okounkov bodies and Duistermaat--Heckman measures for non-Archimedean metrics.
result Generalization of Duistermaat--Heckman measures to finite energy metrics on Berkovich analytifications.
The paper provides a combinatorial criterion for realizing tropical pluri-canonical divisors.
problem Determining when a tropical pair corresponds to a smooth algebraic curve with a pluri-canonical divisor.
method Introducing tropical normalized covers and reducing the problem to their realizability.
result Generalizes previous work on tropical canonical divisors and incorporates recent progress on k-differentials. Proves regularity of harmonic maps into Euclidean buildings and applies to superrigidity of algebraic groups.
problem Regularity of harmonic maps into Euclidean buildings.
method Analyzes singular sets and applies geometric settings.
result Proves singular sets of Hausdorff codimension 2 for harmonic maps.
Simplified Milnor-Schwarz lemma for geometric group theory.
problem Conditions for orbit maps to be quasi-isometries.
method Succinct treatment and applications to non-Archimedean groups.
result Sharpened results on mapping class groups and quasi-isometry classification.
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 metric improves clustering in persistent homology.
problem Improving clustering accuracy in persistent homology.
method Defined a new non-archimedean cophenetic metric.
result Cophenetic metric enhances clustering quality and inter-relations.
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.
Characterizes K-semistability for log Fano cone singularities.
problem K-semistability of log Fano cone singularities.
method Non-Archimedean characterization and special test configurations.
result K-semistability agrees with Collins--Székelyhidi's definition.
Study Monge-Ampère equations on Calabi-Yau hypersurfaces, proving unique solutions and implications for special Lagrangian fibrations.
problem Existence of special Lagrangian fibrations in Calabi-Yau hypersurfaces.
method Non-Archimedean and tropical Monge-Ampère equations on Berkovich and skeleton spaces, proving uniqueness and deriving solutions.
result Unique solutions to tropical and non-Archimedean Monge-Ampère equations, leading to existence of special Lagrangian fibrations.
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.
Mathematical analysis of Riemann surfaces and their moduli spaces using hybrid Laplacians.
problem Analyzing the asymptotics of Arakelov Green functions on Riemann surfaces near boundary of moduli spaces.
method Introducing hybrid Laplacian, solving hybrid Poisson equation, and defining hybrid Green functions.
result Layered description of asymptotics of Arakelov Green functions on Riemann surfaces near boundary of their moduli 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.
Mabuchi solitons generalize Kähler-Einstein metrics on Fano manifolds, which constitute a Yau-Tian-Donaldson type correspondence with relative Ding stability. Comparing with Kähler-Ricci solitons, there is a distinct necessary condition for the existence. We show this condition can be implied by the uniformly relative …
Let K be an algebraically closed field endowed with a complete non-archimedean norm with valuation ring R. Let f:Y -> X be a map of K-affinoid varieties. In this paper we study the analytic structure of the image f(Y) in X; such an image is a typical example of a subanalytic set. We show that the subanalytic sets are p…
Lecture notes on Kähler geometry in toric varieties.
problem Existence of extremal Kähler metrics in toric manifolds.
method Using the Delzant polytope and stability condition.
result Existence of extremal Kähler metrics in terms of stability condition.