From complex to non-Archimedean geometry: an approach to the YTD conjecture

Mattias Jonsson; S\'ebastien Boucksom

arxiv: 2602.10800 · v2 · pith:LMXJMUZInew · submitted 2026-02-11 · 🧮 math.AG · math.DG

From complex to non-Archimedean geometry: an approach to the YTD conjecture

S\'ebastien Boucksom , Mattias Jonsson This is my paper

Pith reviewed 2026-05-21 13:41 UTC · model grok-4.3

classification 🧮 math.AG math.DG

keywords Yau-Tian-Donaldson conjectureconstant scalar curvature Kähler metricsnon-Archimedean geometryalgebraic stabilityKähler geometrycomplex manifoldsanalytic geometry

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The pith

Relations between complex and non-Archimedean geometry sketch a proof of a version of the Yau-Tian-Donaldson conjecture for constant scalar curvature Kähler metrics.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines connections among algebraic geometry, complex analytic geometry, and non-Archimedean geometry over the complex numbers. These connections are used to sketch one direction of the Yau-Tian-Donaldson conjecture, showing that algebraic stability implies the existence of constant scalar curvature Kähler metrics. A sympathetic reader would care because the conjecture supplies an algebraic test for when these special metrics exist on a complex manifold, unifying analytic existence questions with algebraic stability notions. The non-Archimedean viewpoint introduces discrete structures that may simplify the analysis of limits and degenerations that arise in the metric problem. The sketch therefore offers a new route to parts of a central open question in Kähler geometry.

Core claim

The authors discuss the relation between algebraic, analytic, and non-Archimedean geometry over the complex numbers and sketch a proof of a version of the Yau-Tian-Donaldson conjecture for constant scalar curvature Kähler metrics by showing that the sketched correspondences suffice to deduce metric existence from stability.

What carries the argument

The relations linking algebraic, analytic, and non-Archimedean geometry over the complex numbers, which translate stability data into statements about metric existence.

If this is right

Algebraic stability conditions imply the existence of constant scalar curvature Kähler metrics for the version of the conjecture treated here.
Non-Archimedean techniques supply an alternative to purely analytic methods for establishing metric existence.
The same geometric relations may extend to related questions about canonical metrics on complex manifolds.
Stability checks in algebraic geometry become sufficient criteria for metric existence under the sketched correspondences.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The non-Archimedean structures could allow discrete or combinatorial checks for stability that avoid direct analytic construction of the metric.
The approach may link the YTD conjecture to tropical or valuation-theoretic methods already used in algebraic geometry.
Explicit verification on toric or Fano manifolds where both stability and metrics are known would test whether the relations hold in practice.

Load-bearing premise

The sketched relations between algebraic, analytic, and non-Archimedean geometries over the complex numbers are sufficient to establish the stability-to-metric existence implication in the YTD conjecture.

What would settle it

A concrete complex manifold that meets the algebraic stability condition yet admits no constant scalar curvature Kähler metric would show that the sketched relations do not suffice for the implication.

read the original abstract

These notes expand on talks given by the authors at the 2025 Summer Research Institute in Algebraic Geometry in Fort Collins, Colorado. We discuss the relation between algebraic, analytic, and non-Archimedean geometry over the complex numbers, and sketch a proof of a version of the Yau--Tian--Donaldson conjecture for constant scalar curvature K\"ahler metrics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Boucksom and Jonsson sketch a non-Archimedean bridge from algebraic stability to cscK metrics, but the notes stay high-level.

read the letter

Hi, the main takeaway is that these notes sketch a version of the YTD conjecture for constant scalar curvature Kähler metrics by routing through non-Archimedean geometry to link algebraic stability notions with analytic existence. Boucksom and Jonsson have long worked at the intersection of these areas, so the high-level relations they draw between algebraic, analytic, and non-Archimedean pictures over the complex numbers feel grounded in their earlier results on valuations and pluripotential theory. That connection is the part that stands out as potentially useful for organizing the stability-to-metric direction. The soft spot is exactly what the stress-test note flags: this is expanded talk notes, so the central implication rests on sketched correspondences rather than spelled-out estimates, convergence arguments, or identification of a limiting current with a genuine smooth Kähler metric. Without those steps written out, it is difficult to judge whether the non-Archimedean data actually produces the required analytic object on the original manifold. The argument is not circular on its face, but the bridge looks like the load-bearing piece that still needs filling in. This is for readers already working on the YTD problem or non-Archimedean methods in Kähler geometry; a general audience would find it too compressed. I would send it to peer review because the topic is central and the authors are serious, even though the current version would need substantial expansion to stand as a complete paper.

Referee Report

2 major / 2 minor

Summary. The manuscript consists of notes expanding talks from the 2025 Summer Research Institute in Algebraic Geometry. It discusses relations among algebraic, analytic, and non-Archimedean geometries over the complex numbers and sketches a proof of a version of the Yau-Tian-Donaldson conjecture asserting that algebraic stability implies existence of a constant scalar curvature Kähler metric.

Significance. If the sketched correspondences can be made rigorous, the work would supply a new non-Archimedean route to the stability-to-existence direction of the YTD conjecture, potentially unifying techniques from Berkovich spaces, valuation theory, and Kähler geometry. The interdisciplinary framing is a clear strength, even at the level of a high-level outline.

major comments (2)

[Sketch of the proof (main body)] The central sketch (described in the abstract and expanded in the notes) asserts that non-Archimedean stability produces a limiting object whose analytic properties yield a smooth cscK metric, yet no convergence arguments, error estimates, or identification of the limiting current with a genuine Kähler form are supplied. This step is load-bearing for the stability-to-existence implication.
[Discussion of relations between algebraic, analytic, and non-Archimedean geometry] The relations between the three geometries are presented at a high level without explicit statements of the continuity or approximation properties required to pass from non-Archimedean data to the complex-analytic setting; without these, the bridge from algebraic stability to metric existence remains formally incomplete.

minor comments (2)

[Introduction] The manuscript would benefit from a short roadmap paragraph at the end of the introduction that lists the precise steps of the sketched argument and indicates where each is treated.
Notation for non-Archimedean objects (e.g., Berkovich spaces or valuations) should be introduced with a brief reminder of the relevant definitions to aid readers coming from the analytic side.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting both the potential strengths and the current limitations of these notes. We respond to the major comments point by point below, clarifying the intended scope of the manuscript as an outline of ideas from talks rather than a fully rigorous proof.

read point-by-point responses

Referee: [Sketch of the proof (main body)] The central sketch (described in the abstract and expanded in the notes) asserts that non-Archimedean stability produces a limiting object whose analytic properties yield a smooth cscK metric, yet no convergence arguments, error estimates, or identification of the limiting current with a genuine Kähler form are supplied. This step is load-bearing for the stability-to-existence implication.

Authors: We agree that the notes supply only a high-level conceptual sketch of the limiting process and do not include detailed convergence arguments, error estimates, or a precise identification of the limiting current. These notes expand talks given at the 2025 Summer Research Institute and are meant to outline a strategy rather than deliver complete technical proofs. We will revise the manuscript to state this scope explicitly in the introduction and to indicate the specific technical results (in non-Archimedean pluripotential theory and related areas) that would be needed to make the step rigorous. revision: yes
Referee: [Discussion of relations between algebraic, analytic, and non-Archimedean geometry] The relations between the three geometries are presented at a high level without explicit statements of the continuity or approximation properties required to pass from non-Archimedean data to the complex-analytic setting; without these, the bridge from algebraic stability to metric existence remains formally incomplete.

Authors: The referee correctly observes that the correspondences are described at a high level without explicit continuity or approximation statements. The manuscript's purpose is to motivate the interdisciplinary connections and sketch a possible route to the YTD conjecture. We will add a dedicated paragraph or subsection that recalls the relevant continuity and approximation results from the literature on Berkovich spaces and Kähler geometry, while noting that a complete rigorous bridge remains the subject of ongoing work. revision: yes

Circularity Check

0 steps flagged

Sketch of YTD implication remains self-contained without definitional or fitted-input reduction

full rationale

The manuscript is explicitly a sketch of relations between algebraic, analytic, and non-Archimedean geometries to support a version of the YTD conjecture. No equations, fitted parameters, or closed derivation chains are exhibited in the provided text that would allow a prediction or uniqueness claim to reduce to its own inputs by construction. Self-citations to prior work by the authors or collaborators are present in the broader literature but do not function as load-bearing justifications here; the central implication is framed as an outline relying on external correspondences rather than an internal re-derivation. No self-definitional, ansatz-smuggling, or renaming patterns are identifiable from the given material.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no information on free parameters, axioms, or invented entities; ledger left empty.

pith-pipeline@v0.9.0 · 5587 in / 1095 out tokens · 35290 ms · 2026-05-21T13:41:08.529336+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

the radial limit space N_rad(V) admits a nice description in terms of non-Archimedean norms … (N_rad(V), d_τ,rad) ≅ (N_na(V), d_τ,na)
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

E^1 is uniquely geodesic … Busemann convex … lim t→∞ t^{-1} M(ϕ_t) = M_na(φ) if φ ∈ E^1_na
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative refines

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refines
Relation between the paper passage and the cited Recognition theorem.

Mabuchi functional … geodesically convex … minimizers lie in H

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