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arxiv: 2604.14040 · v1 · submitted 2026-04-15 · 🧮 math.AG

A lower bound on the Calabi functional for a degeneration of polarized varieties

Gabriel Frey This is my paper

Pith reviewed 2026-05-10 12:20 UTC · model grok-4.3

classification 🧮 math.AG
keywords Calabi functionalCM degreesGIT heightpolarized varietiesdegenerationsChow varietynon-Archimedean geometry
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The pith

The Calabi functional on degenerations of polarized varieties is bounded below by the difference of CM degrees between generically isomorphic families.

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

The paper proves a lower bound on the Calabi functional that applies when a polarized variety degenerates within a family. The bound is expressed using the difference in CM degrees between two families that agree generically. A reader would care because the result extends Donaldson's earlier lower bound from a single variety to the setting of degenerations and supplies a discretely valued version of that bound in non-Archimedean geometry. The proof proceeds by developing the GIT height and applying it to the family GIT problem on the Chow variety, which also yields a numerical proof of separatedness for GIT quotients under general and special linear actions.

Core claim

We prove that for degenerations of polarized varieties the Calabi functional admits a lower bound given by the difference of CM degrees between generically isomorphic families. The argument develops the theory of GIT height and applies it directly to the family GIT problem on the Chow variety, thereby obtaining the bound and simultaneously furnishing a numerical proof of separatedness of GIT quotients for general and special linear actions. This generalizes Donaldson's lower bound for a single polarized variety to the discretely valued case in the sense of non-Archimedean geometry.

What carries the argument

The GIT height applied to the family GIT problem on the Chow variety, which produces the lower bound on the Calabi functional in terms of the difference of CM degrees.

If this is right

  • The bound generalizes Donaldson's result from a single polarized variety to degenerations.
  • The same GIT height machinery supplies a numerical proof of separatedness for GIT quotients under general and special linear actions.
  • The construction yields a discretely valued version of Donaldson's lower bound in non-Archimedean geometry.
  • The approach strengthens earlier work of Wang and Xu on separatedness of GIT quotients.

Where Pith is reading between the lines

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

  • The GIT height technique developed here could be tested on concrete families such as toric degenerations to check the sharpness of the bound.
  • The separatedness result for GIT quotients might extend to additional group actions beyond those treated in the paper.
  • If the bound holds, it offers a way to compare stability invariants across different models of the same generic polarized variety.

Load-bearing premise

The families under consideration are generically isomorphic polarized varieties and the GIT height applies directly to the family GIT problem on the Chow variety without further restrictions on the degeneration.

What would settle it

An explicit degeneration of polarized varieties together with two generically isomorphic families for which the Calabi functional falls below the difference of their CM degrees would falsify the stated lower bound.

read the original abstract

We prove a lower bound on the Calabi functional for degenerations of polarized varieties, involving the difference of CM degrees between generically isomorphic families. This may be viewed as a discretely valued version of Donaldson's lower bound for models, in the sense of non-Archimedean geometry. In particular, this generalizes a result of Donaldson, who considered a single polarized variety. As a main tool, we develop the theory of GIT height, introduced by Wang, and apply it to the family GIT problem of the Chow variety. Using the GIT height, we also give a numerical proof of separatedness of GIT quotients for general and special linear actions, strengthening prior work of Wang--Xu.

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 / 3 minor

Summary. The paper proves a lower bound on the Calabi functional for degenerations of polarized varieties, expressed via the difference of CM degrees between generically isomorphic families. This is presented as a discretely valued analogue of Donaldson's lower bound in non-Archimedean geometry. The main technical tool is the development of GIT height (following Wang) applied to the family GIT problem on the Chow variety; the same tool yields a numerical proof of separatedness for GIT quotients under general and special linear actions, strengthening earlier work of Wang-Xu.

Significance. If the central identities hold, the result supplies a useful generalization of Donaldson's bound to degenerations and supplies a parameter-free numerical criterion for separatedness of GIT quotients. The construction of GIT height for families and the direct reduction to the numerical criterion (without extra restrictions beyond generic isomorphism) are genuine strengths that could be cited in future work on non-Archimedean stability.

minor comments (3)
  1. [§1] §1, paragraph after the statement of the main theorem: the sentence claiming the result is 'parameter-free' should be qualified by noting that the families are assumed generically isomorphic; this hypothesis is used in the reduction and is not automatic.
  2. [§3] §3, definition of the GIT height for the family problem: the notation for the difference of CM degrees (appearing in the key identity) is introduced without an explicit cross-reference to the earlier definition in §2; adding the reference would improve readability.
  3. [§4] The proof of the numerical criterion for separatedness (used in both the Calabi bound and the GIT separatedness statement) is presented as self-contained, but a short remark comparing the argument to the original Wang-Xu approach would help readers assess the strengthening.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the paper, the assessment of its significance as a generalization of Donaldson's bound, and the recommendation for minor revision. No specific major comments are provided in the report, so there are no individual points requiring detailed rebuttal or clarification at this stage.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation develops GIT height as a new tool for the family GIT problem on the Chow variety and establishes the lower bound on the Calabi functional directly from the difference of CM degrees for generically isomorphic families. Key identities equate this difference to the GIT height difference by definition in §§2–3, with the separatedness criterion proved numerically without parameter fitting or self-referential assumptions. The argument generalizes Donaldson's prior result using external foundations and is self-contained against the stated hypotheses, with no load-bearing steps reducing to self-citation chains, ansatzes, or renamed inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard background from algebraic geometry including properties of CM degrees, Chow varieties, and GIT quotients, with GIT height developed as an application tool rather than a new postulate.

axioms (2)
  • standard math Standard properties of CM degrees and the Calabi functional for polarized varieties
    The paper invokes these as established invariants in the field.
  • standard math Existence and basic properties of GIT quotients and heights on Chow varieties
    Relies on prior developments in geometric invariant theory.

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Reference graph

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