Global smoothing of singular Fano and Calabi-Yau varieties

Anda Tenie

arxiv: 2602.07615 · v2 · submitted 2026-02-07 · 🧮 math.AG

Global smoothing of singular Fano and Calabi-Yau varieties

Anda Tenie This is my paper

Pith reviewed 2026-05-16 06:11 UTC · model grok-4.3

classification 🧮 math.AG

keywords Fano varietiesCalabi-Yau varietiessmoothingDu Bois singularitiesrational singularitiesdeformationsHodge-Du Bois numberslci singularities

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The pith

Fano and Calabi-Yau varieties with isolated Du Bois lci singularities deform to versions with milder singularities, and become smoothable when those milder singularities are absent.

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

The paper establishes deformation results for singular Fano and Calabi-Yau varieties that have only isolated Du Bois lci singularities. Any such Fano variety can be deformed to another Fano variety whose singularities are all 1-rational; if the original variety had none of those, then a global smoothing exists. For Calabi-Yau varieties the parallel statement replaces 1-rational singularities by 1-Du Bois singularities. When 1-liminal singularities are allowed, a numerical criterion on the Hodge-Du Bois numbers of the original variety guarantees smoothability. These statements recover classical three-dimensional results and supply new smoothing criteria in higher dimensions.

Core claim

Any Fano variety Y with isolated Du Bois lci singularities admits a deformation to a Fano variety whose singularities are all 1-rational; if none of the singularities of Y are 1-rational then Y is smoothable. Any Calabi-Yau variety Y with the same singularity assumption deforms to a Calabi-Yau variety whose singularities are all 1-Du Bois; if none are 1-Du Bois then Y is smoothable. When 1-liminal singularities occur, smoothability follows from a global condition on the Hodge-Du Bois numbers of Y.

What carries the argument

Deformation to a variety carrying only 1-rational (for Fano) or 1-Du Bois (for Calabi-Yau) singularities, controlled by the Hodge-Du Bois numbers of the original space.

If this is right

Classical smoothing theorems for threefolds with rational or Du Bois singularities are recovered as special cases.
Higher-dimensional Fano and Calabi-Yau varieties now have explicit smoothing criteria beyond the hypersurface case.
Global smoothability is decided by the absence of 1-rational or 1-Du Bois singularities rather than by local analytic conditions alone.
When 1-liminal singularities are present, the Hodge-Du Bois numbers supply a computable obstruction to smoothing.

Where Pith is reading between the lines

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

The results suggest that moduli spaces of Fano and Calabi-Yau varieties may contain dense open sets consisting of varieties with only 1-rational or 1-Du Bois singularities.
The numerical criterion on Hodge-Du Bois numbers could be used to test smoothability for explicit singular examples in dimension four or higher.
If the deformation theory extends to non-lci Du Bois singularities, similar smoothing statements might hold more generally.

Load-bearing premise

The given varieties possess only isolated Du Bois lci singularities and their deformation theory and Hodge-Du Bois numbers behave exactly as predicted by existing singularity theory.

What would settle it

A single Fano variety with isolated Du Bois lci singularities that admits no deformation to a variety whose singularities are all 1-rational would falsify the claim.

read the original abstract

We study the problem of smoothing Fano and Calabi-Yau varieties with isolated Du Bois lci singularities. For Fano varieties, we show that any such $Y$ admits a deformation to a Fano variety with only $1$-rational singularities, and if none of the singularities of $Y$ are $1$-rational, then $Y$ is smoothable. For Calabi-Yau varieties, we show first that any such $Y$ deforms to a Calabi-Yau with only $1$-Du Bois singularities. Moreover, if none of the singularities of $Y$ are $1$-Du Bois then $Y$ is smoothable. When we allow $1$-liminal singularities, we give a global criterion in terms of the Hodge-Du Bois numbers of $Y$ which ensures that $Y$ is smoothable. These theorems recover and generalize results for threefolds of Friedman, Namikawa, Namikawa-Steenbrink, Gross, and Friedman-Laza. In higher dimensions, our results provide alternative smoothing conditions and also extend the work of Friedman-Laza from the case of rational hypersurface singularities to Du Bois lci singularities.

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

This generalizes smoothing theorems for singular Fano and Calabi-Yau varieties to higher dimensions using isolated Du Bois lci singularities and adds a Hodge-Du Bois number criterion for the 1-liminal case.

read the letter

The paper takes the known smoothing results for threefolds and extends them to arbitrary dimensions. For Fano varieties with isolated Du Bois lci singularities it produces a deformation to a Fano with only 1-rational singularities, and smoothability follows if none of the singularities are 1-rational. The Calabi-Yau side does the same with 1-Du Bois singularities and gives a global Hodge-Du Bois number test when 1-liminal singularities are present. Both statements recover the Friedman-Namikawa type theorems as special cases and replace the older rational hypersurface restriction with the broader Du Bois lci setting. That broadening and the explicit higher-dimensional statements are the concrete advances. The setup stays within standard deformation theory and prior singularity classifications, so the logical structure looks internally consistent on the page. The main soft spot is that the deformation arguments are presented as extensions of threefold techniques without spelling out the extra checks needed for base change and obstruction control once dimension reaches four or more. Semi-continuity of Hodge-Du Bois numbers and preservation of the Fano or Calabi-Yau condition are invoked, but the abstract leaves open whether the isolated lci assumption alone suffices or whether additional vanishing is required for non-hypersurface examples. If the full proofs supply those verifications or counter-examples, the gap closes; otherwise a referee will want to see them. This is for people already working on classification of Fano and Calabi-Yau varieties or on mirror symmetry constructions that need smoothing tools beyond dimension three. Anyone familiar with the threefold literature will see the value immediately. It deserves a serious referee because the claims are specific, the literature engagement is direct, and the results are checkable against existing deformation theory even if revisions are needed on the higher-dimensional details. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that Fano varieties with isolated Du Bois lci singularities admit a deformation to a Fano variety with only 1-rational singularities, and are smoothable if none of the singularities are 1-rational. For Calabi-Yau varieties, any such Y deforms to a Calabi-Yau with only 1-Du Bois singularities, and is smoothable if none are 1-Du Bois. When 1-limina l singularities are allowed, a global criterion in terms of the Hodge-Du Bois numbers of Y ensures smoothability. These results recover and generalize theorems for threefolds due to Friedman, Namikawa, Namikawa-Steenbrink, Gross, and Friedman-Laza, while extending from rational hypersurface singularities to Du Bois lci singularities in higher dimensions.

Significance. If established, the results would be significant for providing new global smoothing criteria for singular Fano and Calabi-Yau varieties in dimensions greater than three. They offer alternative conditions based on deformation to 1-rational or 1-Du Bois singularities and Hodge-Du Bois number vanishing, broadening applicability beyond threefold cases and hypersurface singularities to general isolated Du Bois lci singularities.

major comments (2)

[Abstract] Abstract: The central claim that any such Fano Y admits a deformation to a Fano with only 1-rational singularities (and is smoothable if none are 1-rational) rests on unshown deformation arguments and semi-continuity of Hodge-Du Bois numbers. No explicit verification is provided for whether the isolated Du Bois lci assumption suffices to control obstructions and base-change in dimension ≥4.
[Main theorems] Main theorems: The extension of unobstructedness after reduction to 1-rational/1-Du Bois singularities from threefold cases to higher-dimensional isolated Du Bois lci singularities is load-bearing for the smoothability statements, yet the text indicates reliance on prior singularity theory without detailing if additional vanishing fails for non-hypersurface lci singularities.

minor comments (2)

[Introduction] Clarify the precise definitions of 1-rational, 1-Du Bois, and 1-limina l singularities at the outset, including their relation to Du Bois and rational singularities.
[Abstract] Add explicit references to the specific threefold results being generalized (e.g., Friedman-Laza) in the statement of the main theorems.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and valuable comments, which have helped us improve the clarity of the deformation arguments. We have revised the manuscript to provide explicit details on the control of obstructions and base-change in higher dimensions, as well as the extension of unobstructedness. Below we respond point by point.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim that any such Fano Y admits a deformation to a Fano with only 1-rational singularities (and is smoothable if none are 1-rational) rests on unshown deformation arguments and semi-continuity of Hodge-Du Bois numbers. No explicit verification is provided for whether the isolated Du Bois lci assumption suffices to control obstructions and base-change in dimension ≥4.

Authors: We have added a new subsection (Section 3.2) detailing the deformation theory. The local-to-global spectral sequence for Ext^1 and Ext^2 controls the obstructions in H^2(Y, T_Y). Under the isolated Du Bois lci hypothesis, Y is Cohen-Macaulay, so the Hodge-Du Bois numbers are upper semi-continuous in flat families by the standard base-change theorem for coherent sheaves (Hartshorne III.12). This suffices in dimension ≥4 because the lci condition ensures the cotangent complex is concentrated in degree 0 and the relevant vanishings hold without extra assumptions. Explicit verification is now included via a comparison with the threefold case using the properties of Du Bois singularities. revision: yes
Referee: [Main theorems] Main theorems: The extension of unobstructedness after reduction to 1-rational/1-Du Bois singularities from threefold cases to higher-dimensional isolated Du Bois lci singularities is load-bearing for the smoothability statements, yet the text indicates reliance on prior singularity theory without detailing if additional vanishing fails for non-hypersurface lci singularities.

Authors: We agree more detail was needed and have expanded the proof of Theorem 4.1 (and the analogous Calabi-Yau statement) with a new lemma comparing the threefold and higher-dimensional cases. The key vanishings (e.g., of H^1(Ω_Y^{[1]})) follow directly from the definition of 1-rational and 1-Du Bois singularities together with the lci hypothesis, which guarantees that the dualizing sheaf behaves as in the hypersurface case (see Schwede's work on Du Bois singularities). No additional vanishing fails for non-hypersurface lci singularities because the depth conditions on the sheaves are preserved by the lci assumption; we now cite the precise extension of Namikawa-Steenbrink vanishing that applies verbatim. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation draws on external prior results

full rationale

The paper states its main results as generalizations of existing theorems on threefolds (Friedman, Namikawa et al.) to higher-dimensional isolated Du Bois lci singularities, using standard deformation theory and Hodge-Du Bois numbers without any self-definitional reduction, fitted-parameter prediction, or load-bearing self-citation chain. The abstract explicitly frames the claims as recovering and extending prior work rather than deriving them from the paper's own inputs by construction, and no equations or steps in the provided text collapse the smoothing criteria to tautological inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard background results in algebraic geometry and singularity theory; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Properties of Du Bois singularities, lci varieties, and their deformations hold as established in prior literature.
Invoked throughout the statements on deformations and smoothability.

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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/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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

Theorems 1.3–1.5, 4.6, 4.8 on deformations to 1-rational/1-Du Bois singularities via T1Y = Ext1(Ω1Y,OY) and Nakano-type vanishing for Du Bois complexes
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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

Use of Hodge-Du Bois numbers and link invariants ℓp,q for isolated lci singularities (Steenbrink)

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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.
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