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arxiv: 2606.24481 · v1 · pith:ODULFVSJnew · submitted 2026-06-23 · 🧮 math.DG · math.CV

L^infty-estimates of K\"ahler-Einstein potentials on stable varieties

Rui Tang This is my paper

Pith reviewed 2026-06-25 22:47 UTC · model grok-4.3

classification 🧮 math.DG math.CV
keywords Kähler-Einstein potentialsstable varietiesnon-klt locuscomplex Monge-Ampère equationpluripotential theorylog resolutionL^∞ estimatessingularities
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The pith

Iterated logarithmic functions yield refined lower bounds for Kähler-Einstein potentials on stable varieties near the non-klt locus, with upper bounds under extra log resolution assumptions.

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

The paper establishes improved lower bounds for Kähler-Einstein potentials on stable varieties by constructing iterated logarithmic functions from a defining function of the non-klt locus. These functions produce explicit subsolutions and supersolutions to the degenerate complex Monge-Ampère equation, improving earlier estimates. Under further assumptions on the log resolution, matching upper bounds are obtained as well. The arguments rely on these constructions together with refined integrability estimates drawn from pluripotential theory. The result describes the asymptotic behavior of the potential more precisely near singularities.

Core claim

On stable varieties, iterated logarithmic functions associated with a defining function of the non-klt locus produce refined lower bounds for the Kähler-Einstein potential. When the log resolution satisfies additional assumptions, corresponding upper bounds hold. The proofs proceed via explicit subsolutions and supersolutions for the degenerate complex Monge-Ampère equation together with refined integrability estimates in pluripotential theory.

What carries the argument

Iterated logarithmic functions associated with a defining function of the non-klt locus, used to construct subsolutions and supersolutions for degenerate complex Monge-Ampère equations.

If this is right

  • Refined lower bounds hold for the Kähler-Einstein potential near the non-klt locus on stable varieties.
  • Upper bounds for the potential hold when the log resolution meets the extra assumptions.
  • Explicit subsolutions and supersolutions to the degenerate Monge-Ampère equation can be built from the iterated logarithmic functions.
  • The asymptotic behavior of the potential near singularities is controlled more sharply than in prior estimates.

Where Pith is reading between the lines

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

  • The bounds may allow sharper analysis of how Kähler-Einstein metrics converge in moduli spaces of stable varieties.
  • Similar logarithmic constructions could be tested on other classes of singular canonical metrics.
  • Explicit computation on examples with known non-klt loci would check whether the new bounds are sharp.

Load-bearing premise

The variety is stable and the non-klt locus admits a defining function that permits construction of iterated logarithmic subsolutions and supersolutions for the degenerate complex Monge-Ampère equation.

What would settle it

A stable variety whose Kähler-Einstein potential near the non-klt locus violates the refined lower bound obtained from the iterated logarithms, or whose potential violates the upper bound when the stated log resolution assumptions hold.

read the original abstract

We study the asymptotic behavior of K\"ahler-Einstein potentials on stable varieties near the singularities. Using iterated logarithmic functions associated with a defining function of the non-klt locus, we obtain refined lower bounds for the K\"ahler-Einstein potential, improving previous estimates of Di Nezza-Guedj-Guenancia and Datar-Fu-Song. Under additional assumptions on the log resolution, we also establish upper bounds. The proofs are based on the construction of explicit subsolutions and supersolutions for degenerate complex Monge-Amp\`ere equations together with refined integrability estimates in pluripotential theory.

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.

Referee Report

0 major / 2 minor

Summary. The paper studies the asymptotic behavior of Kähler-Einstein potentials on stable varieties near singularities. Using iterated logarithmic functions associated with a defining function of the non-klt locus, refined lower bounds are obtained, improving on Di Nezza-Guedj-Guenancia and Datar-Fu-Song. Upper bounds are established under additional assumptions on the log resolution. Proofs use explicit subsolutions and supersolutions for degenerate complex Monge-Ampère equations and refined integrability estimates.

Significance. If the results hold, this advances control of singular KE potentials near non-klt loci on stable varieties, improving prior L^∞ estimates via explicit iterated-log subsolutions. The direct pluripotential constructions (without hidden parameters or circular reductions) and the conditional upper bounds constitute a technical contribution useful for further work on degenerate CMA equations and moduli of stable varieties.

minor comments (2)
  1. [Introduction] Introduction: a brief explicit comparison (e.g., the precise improvement factor or the order of the iterated log relative to the cited works) would clarify the advance over Di Nezza-Guedj-Guenancia and Datar-Fu-Song.
  2. The dependence of the upper-bound statement on extra assumptions on the log resolution should be stated in the main theorem so that the scope is immediately visible.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its technical contributions, and recommendation of minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; direct construction of subsolutions

full rationale

The paper derives L^∞ bounds for Kähler-Einstein potentials via explicit construction of iterated logarithmic subsolutions and supersolutions to the degenerate complex Monge-Ampère equation, together with pluripotential integrability estimates. This is a self-contained analytic argument that improves on cited prior work (Di Nezza-Guedj-Guenancia, Datar-Fu-Song) without reducing any claimed bound to a fitted parameter, self-defined quantity, or load-bearing self-citation chain. The stability hypothesis supplies the defining function for the non-klt locus, but the estimates themselves follow from the explicit barrier functions and maximum principle comparisons, which are independent of the final bounds. No step equates a prediction to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be identified from the given text.

pith-pipeline@v0.9.1-grok · 5621 in / 1069 out tokens · 28644 ms · 2026-06-25T22:47:26.303127+00:00 · methodology

discussion (0)

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Reference graph

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