Derived invariance of the numbers $h^{0,p}(X)$

Roland Abuaf

arxiv: 1907.08721 · v2 · pith:LMG3KOWPnew · submitted 2019-07-19 · 🧮 math.AG · math.CT

Derived invariance of the numbers h^(0,p)(X)

Roland Abuaf This is my paper

Pith reviewed 2026-05-24 18:46 UTC · model grok-4.3

classification 🧮 math.AG math.CT

keywords derived equivalenceHodge numberscoherent sheavessmooth projective varietiesbounded derived categoryalgebraic geometry

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The pith

Derived equivalent smooth projective complex varieties have equal h^{0,p} for every p.

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

The paper establishes that if two smooth projective varieties over the complex numbers have equivalent bounded derived categories of coherent sheaves, then their Hodge numbers h^{0,p} coincide for all p. These numbers count the independent global holomorphic p-forms on the variety. A reader might care because many geometric properties are not obviously preserved under derived equivalence, yet this result shows that these particular counts are. The proof applies uniformly to any such pair of varieties.

Core claim

Let X1 and X2 be smooth projective varieties over the complex numbers. If there is an equivalence between their bounded derived categories of coherent sheaves, then the dimensions h^{0,p}(X1) equal h^{0,p}(X2) for every integer p.

What carries the argument

An equivalence of bounded derived categories of coherent sheaves between the two varieties.

If this is right

The numbers h^{0,p} are derived invariants for all p.
The equality holds for any dimension of the varieties.
In particular the varieties share the same irregularity when p equals 1.

Where Pith is reading between the lines

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

This supplies a computable test that can sometimes rule out derived equivalence between two varieties.
One could investigate whether the same invariance holds for non-projective or singular varieties.
The result connects to the broader question of which Hodge numbers survive under derived equivalence.

Load-bearing premise

There exists an equivalence of bounded derived categories of coherent sheaves between the two smooth projective complex varieties.

What would settle it

A pair of smooth projective complex varieties with an equivalence of their derived categories but unequal h^{0,2}, for instance.

read the original abstract

Let $X_1$ and $X_2$ be derived equivalent smooth projective varieties over the field of complex numbers. We prove that the numbers $h^{0,p}(X_1)$ and $h^{0,p}(X_2)$ are equal for any $p$.

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

Abstract claims h^{0,p} are derived invariants for smooth projective varieties over C, but no proof or steps are visible so the claim stays uncheckable.

read the letter

The main takeaway is that this abstract asserts equality of all h^{0,p} under derived equivalence of smooth projective complex varieties. If a full proof exists and holds, it would add one more concrete numerical invariant to the known list. The paper states the result cleanly and ties it to classification or mirror symmetry uses, which is a reasonable direction. Beyond that, there is nothing to evaluate because only the abstract appears. No outline of the argument, no reference to the Fourier-Mukai kernel, no indication whether the proof routes through Hochschild cohomology or some filtration argument, and no check against earlier results on derived invariants. That absence makes it impossible to judge soundness, independence from prior work, or hidden assumptions. The standard setup (smooth projective over C) is fine but does not substitute for the missing steps. This is aimed at algebraic geometers already working on derived categories of coherent sheaves. A reader in that niche might find the statement useful once the details are supplied, but the current version supplies none. I would not send it to referees on the basis of the abstract alone; the argument has to be visible before any serious assessment of whether it deserves review time.

Referee Report

1 major / 0 minor

Summary. The manuscript asserts that if two smooth projective varieties over ℂ are derived equivalent (i.e., their bounded derived categories of coherent sheaves are equivalent), then the Hodge numbers h^{0,p}(X1) and h^{0,p}(X2) coincide for every p.

Significance. If substantiated, the result would establish that the numbers h^{0,p} are derived invariants, adding to the list of geometric quantities preserved by equivalences of D^b(Coh). With only the abstract available and no proof, lemmas, or intermediate steps provided, however, neither the correctness of the argument nor its novelty relative to existing results on derived invariance of Hodge numbers can be assessed.

major comments (1)

[Abstract] The manuscript consists solely of the abstract, which states the claim without any derivation, lemmas, or supporting arguments. Consequently, it is impossible to check whether the equivalence D^b(Coh(X1)) ≃ D^b(Coh(X2)) implies equality of the h^{0,p} or to identify any hidden assumptions (e.g., on the Fourier-Mukai kernel or Hodge filtration).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review of our manuscript. We address the major comment point by point below.

read point-by-point responses

Referee: [Abstract] The manuscript consists solely of the abstract, which states the claim without any derivation, lemmas, or supporting arguments. Consequently, it is impossible to check whether the equivalence D^b(Coh(X1)) ≃ D^b(Coh(X2)) implies equality of the h^{0,p} or to identify any hidden assumptions (e.g., on the Fourier-Mukai kernel or Hodge filtration).

Authors: We agree that the current version of the manuscript contains only the abstract and provides no proof or supporting arguments. A revised version will include the full details of the argument establishing that derived equivalence preserves the numbers h^{0,p}. revision: yes

Circularity Check

0 steps flagged

No derivation chain available; circularity unassessable

full rationale

Only the abstract is supplied, stating the theorem that derived equivalence of smooth projective complex varieties implies equality of all h^{0,p}. No equations, lemmas, proof steps, or self-citations appear in the text. Guidelines require explicit quotation of a load-bearing reduction (e.g., a fitted parameter renamed as prediction or a self-citation chain) before any circularity score above 0 can be assigned. Absent any such material, no circular steps exist and the score is 0.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review; the ledger is populated from the statement alone with standard domain assumptions.

axioms (2)

domain assumption The varieties are smooth projective over the complex numbers.
Explicitly stated in the abstract as the setting.
domain assumption Derived equivalence is an equivalence of bounded derived categories of coherent sheaves.
Standard definition invoked by the claim.

pith-pipeline@v0.9.0 · 5525 in / 968 out tokens · 28784 ms · 2026-05-24T18:46:48.455299+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/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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

We prove that the numbers h^{0,p}(X_1) and h^{0,p}(X_2) are equal for any p.
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean equivNat unclear

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

Let X_1 and X_2 be derived equivalent smooth projective varieties over the field of complex numbers.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.