arxiv: 2605.13808 · v1 · submitted 2026-05-13 · 🧮 math.AG

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Tilt-stability on singular schemes and Bogomolov-Gieseker-type inequalities

Zhiyu Liu , Tianle Mao

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Pith reviewed 2026-05-14 17:36 UTC · model grok-4.3

classification 🧮 math.AG

keywords tilt-stabilityBogomolov-Gieseker inequalitysingular threefoldsFano threefoldsCalabi-Yau threefoldsstability conditionsKuznetsov componentssemistable sheaves

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The pith

Tilt-stability extends to singular schemes and supports a generalized Bogomolov-Gieseker inequality conjecture for singular threefolds.

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

The paper extends the framework of tilt-stability, previously defined for smooth varieties, to singular schemes. It formulates the generalized Bayer-Macrì-Toda Bogomolov-Gieseker inequality conjecture specifically for singular threefolds. Relative versions of the constructions are developed, generalizing earlier results. Bogomolov-Gieseker-type inequalities are established for semistable sheaves on any projective scheme. The conjecture is verified for all Fano threefolds with canonical Gorenstein Q-factorial singularities and a series of singular Calabi-Yau threefolds, and stability conditions are constructed on the relative Kuznetsov components of families of singular Fano threefolds.

Core claim

Tilt-stability generalizes to singular schemes, allowing the formulation of the generalized Bogomolov-Gieseker inequality conjecture for singular threefolds. Relative versions of these constructions are developed. Bogomolov-Gieseker-type inequalities hold for semistable sheaves on any projective scheme. The conjecture is verified for all Fano threefolds with canonical Gorenstein Q-factorial singularities and a series of singular Calabi-Yau threefolds. Stability conditions are constructed on the relative Kuznetsov components associated with families of singular Fano threefolds, establishing a singular analogue of the Kuznetsov-Shinder conjecture.

What carries the argument

Tilt-stability on singular schemes, which defines stability conditions for objects in the derived category while accommodating singularities and preserving the support property.

If this is right

Stability conditions exist on the derived categories of the verified singular Fano and Calabi-Yau threefolds.
The generalized Bogomolov-Gieseker inequality applies to semistable sheaves on arbitrary projective schemes.
Relative tilt-stability constructions apply to families of singular Fano threefolds.
A singular version of the Kuznetsov-Shinder conjecture holds for the associated Kuznetsov components.
Moduli spaces of stable objects can be constructed in these singular settings.

Where Pith is reading between the lines

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

The framework may extend to study moduli problems on singular varieties arising in mirror symmetry.
Verification in Fano and Calabi-Yau cases suggests the conjecture could hold for other classes of singular threefolds if singularity conditions are met.
The relative constructions could link to deformation theory of singular schemes and their derived categories.

Load-bearing premise

The schemes are projective and, for the verifications, have canonical Gorenstein Q-factorial singularities.

What would settle it

A semistable sheaf on a Fano threefold with canonical Gorenstein Q-factorial singularities whose Chern characters violate the generalized Bogomolov-Gieseker inequality would falsify the conjecture.

read the original abstract

We generalize the framework of tilt-stability to singular schemes and formulate the generalized Bogomolov-Gieseker inequality conjecture of Bayer-Macr\`i-Toda for singular threefolds. We also develop relative versions of these constructions, generalizing corresponding results in [BLM+21]. Along the way, we establish Bogomolov-Gieseker-type inequalities for semistable sheaves on any projective scheme. By extending previous techniques, we verify the conjecture for all Fano threefolds with canonical Gorenstein $\mathbb{Q}$-factorial singularities and a series of singular Calabi-Yau threefolds. Furthermore, we construct stability conditions on the relative Kuznetsov components associated with families of singular Fano threefolds, thereby proving a singular analogue of a conjecture of Kuznetsov-Shinder.

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

The paper generalizes tilt-stability to singular schemes and verifies the BG conjecture on specific Gorenstein threefolds, but the claim for arbitrary projective schemes rests on assumptions that may not hold without Gorenstein or Cohen-Macaulay conditions.

read the letter

The main thing to know is that the authors extend tilt-stability and the Bayer-Macrì-Toda Bogomolov-Gieseker conjecture to singular schemes, then check the conjecture on Fano threefolds with canonical Gorenstein Q-factorial singularities and on some singular Calabi-Yau threefolds. They also give relative versions of the constructions and build stability conditions on relative Kuznetsov components for families of singular Fanos, which yields a singular form of the Kuznetsov-Shinder conjecture. They claim BG-type inequalities for semistable sheaves on any projective scheme once tilt-stability is in place. This builds directly on BLM+21 by adapting the same techniques to the singular setting and adding the relative layer plus the explicit checks. Those verifications and the relative stability construction are the concrete new pieces. The soft spot is the reach of the general statements. Tilt-stability uses a central charge built from the dualizing sheaf and Serre duality, and those objects are not guaranteed to exist or behave well on arbitrary projective schemes. The actual checks stay inside the Gorenstein Q-factorial cases, so the assertion that the inequalities hold on every projective scheme may need either a restriction on the schemes or a separate argument for the non-Gorenstein situation. If the paper already handles this by restricting the general claims or by proving the construction works more broadly, that needs to be clear in the text. This work is for people who follow stability conditions on moduli spaces and derived categories of singular varieties. Readers who know the Bayer-Macrì-Toda line or Kuznetsov components will find the extensions and examples directly relevant. The paper engages the literature honestly and produces new statements plus checks, so it deserves a serious referee to examine the details of the generalization and the proofs.

Referee Report

1 major / 2 minor

Summary. The manuscript generalizes the tilt-stability framework to singular schemes, formulates a generalized Bayer-Macrì-Toda Bogomolov-Gieseker inequality conjecture for singular threefolds, establishes Bogomolov-Gieseker-type inequalities for semistable sheaves on arbitrary projective schemes, verifies the conjecture for all Fano threefolds with canonical Gorenstein Q-factorial singularities and a series of singular Calabi-Yau threefolds, develops relative versions of the constructions, and constructs stability conditions on the relative Kuznetsov components of families of singular Fano threefolds.

Significance. If the central constructions and inequalities hold under the stated hypotheses, the work extends tilt-stability and BG-type bounds beyond the smooth case, providing tools for moduli problems and derived-category questions on singular varieties. The explicit verifications on singular Fano and Calabi-Yau threefolds and the relative Kuznetsov-component constructions are concrete contributions that generalize results from [BLM+21].

major comments (1)

[Abstract and the section defining tilt-stability on singular schemes] The abstract asserts that BG-type inequalities are established for semistable sheaves on any projective scheme. The tilt-stability construction (presumably in the section introducing the generalized tilt heart and central charge) relies on a dualizing sheaf to define numerical invariants and the slope function. For non-Gorenstein or non-Cohen-Macaulay schemes this sheaf need not be invertible (or even exist in the required form), so the general claim appears to require additional hypotheses that are not made explicit. The verifications are restricted to canonical Gorenstein Q-factorial cases, leaving the unrestricted statement unsupported.

minor comments (2)

[Introduction / definition of tilt-stability] Clarify the precise singularity assumptions (e.g., Gorenstein, Q-factorial, Cohen-Macaulay) at the first appearance of the general tilt-stability definition rather than deferring them to the verification sections.
[Statements of main theorems] Ensure that all statements of the generalized conjecture and the proved inequalities explicitly restate the ambient hypotheses on the scheme.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to make the hypotheses on the schemes explicit. We address the concern point by point below.

read point-by-point responses

Referee: [Abstract and the section defining tilt-stability on singular schemes] The abstract asserts that BG-type inequalities are established for semistable sheaves on any projective scheme. The tilt-stability construction (presumably in the section introducing the generalized tilt heart and central charge) relies on a dualizing sheaf to define numerical invariants and the slope function. For non-Gorenstein or non-Cohen-Macaulay schemes this sheaf need not be invertible (or even exist in the required form), so the general claim appears to require additional hypotheses that are not made explicit. The verifications are restricted to canonical Gorenstein Q-factorial cases, leaving the unrestricted statement unsupported.

Authors: We agree that the construction of the generalized tilt heart and the slope function requires a dualizing sheaf, which exists as a coherent sheaf (and is invertible in the Gorenstein case) precisely when the scheme is Cohen-Macaulay. Every projective scheme admits a dualizing complex, but the numerical invariants used for tilt-stability are defined via the dualizing sheaf in the manner of the paper. We will therefore revise the abstract to read “on any projective Cohen-Macaulay scheme” and insert an explicit standing assumption at the beginning of the section introducing tilt-stability that X is Cohen-Macaulay (hence Gorenstein when we specialize to the Fano and Calabi-Yau cases). The Bogomolov-Gieseker-type inequalities are proved under this hypothesis for semistable sheaves; the verifications of the full conjecture remain restricted to the canonical Gorenstein Q-factorial threefolds as stated. These changes make the scope of the results precise while preserving all stated theorems. revision: yes

Circularity Check

0 steps flagged

No circularity: generalization and verifications are self-contained extensions of prior external results

full rationale

The paper generalizes tilt-stability and formulates a BG-type conjecture for singular threefolds by extending techniques from the cited prior work [BLM+21], then verifies the conjecture on specific classes (Fano threefolds with canonical Gorenstein Q-factorial singularities and certain singular Calabi-Yau threefolds) while establishing BG-type inequalities for semistable sheaves on arbitrary projective schemes. These steps rely on explicit constructions, relative versions, and case-by-case checks rather than any self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation chain. The central claims remain independent of the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone.

pith-pipeline@v0.9.0 · 5434 in / 1028 out tokens · 46248 ms · 2026-05-14T17:36:15.522358+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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