arxiv: 2604.28085 · v1 · submitted 2026-04-30 · 🧮 math.AG · math.CV

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Failure of the semi log canonical Abundance for compact K\"{a}hler threefolds

Swapnajit Das

Authors on Pith no claims yet

Pith reviewed 2026-05-07 05:35 UTC · model grok-4.3

classification 🧮 math.AG math.CV

keywords semi-log canonical abundanceKähler threefoldscounterexamplesemiample divisorsnormalization morphismKodaira dimensionslc pairs

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The pith

A compact Kähler threefold provides a counterexample showing that semi-log canonical abundance fails in dimension three.

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

The paper constructs a counterexample demonstrating that the semi-log canonical abundance conjecture does not hold for all compact Kähler threefolds. It presents an irreducible slc threefold (X, 0) where the canonical divisor K_X is nef, the Kodaira dimension of the normalized pair is zero, yet K_X is not semiample. This failure occurs even though the pair satisfies the usual numerical conditions for abundance to hold. The author proves positive results showing that abundance does hold when the pair is semi-dlt or when the Kodaira dimension on each component of the normalization is positive. The distinction between these settings isolates where the conjecture breaks for Kähler varieties.

Core claim

We construct a compact Kähler (irreducible) slc threefold (X, 0) such that K_X is nef and κ(tilde X, K_tilde X + tilde D)=0, where μ:(tilde X, tilde D)→X is the normalization morphism, but K_X is not semiample. On the other hand, if (X, Δ) is a compact Kähler sdlt pair of dimension 3 with K_X + Δ nef, then K_X + Δ is semiample. If (X, Δ) is slc with K_X + Δ nef and κ(X'_i, Δ'_i + D'_i) > 0 for all i under normalization, then K_X + Δ is semiample.

What carries the argument

The normalization morphism μ:(tilde X, tilde D)→X of the slc threefold, which allows the Kodaira dimension condition κ(tilde X, K_tilde X + tilde D)=0 to be checked separately from the nefness of K_X on the original space.

If this is right

The semi-log canonical abundance conjecture is false for compact Kähler threefolds in general.
Abundance holds for compact Kähler semi-dlt pairs of dimension 3 when the canonical plus boundary divisor is nef.
Abundance holds for compact Kähler slc pairs of dimension 3 when the Kodaira dimension is positive on every component of the normalization.

Where Pith is reading between the lines

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

The counterexample may exploit non-projective features of Kähler threefolds that permit nef but non-semiample divisors.
Similar counterexamples could exist for slc pairs in higher dimensions or for other classes of singularities.
Criteria ensuring positive Kodaira dimension on normalizations might be used to recover semi-ampleness in broader Kähler settings.

Load-bearing premise

There exists a compact Kähler irreducible slc threefold where K_X is nef, the Kodaira dimension after normalization is zero, and K_X fails to be semiample.

What would settle it

An explicit construction of the threefold together with a calculation showing that K_X is in fact semiample, or a proof that the threefold is not slc, would disprove the counterexample.

read the original abstract

In this article we show that the semi log canonical abundance for compact K\"ahler varieties fails in dimension $3$. More specifically we construct a counterexample of a compact K\"ahler (irreducible) slc threefold $(X, 0)$ such that $K_X$ is nef and $\kappa(\tilde X, K_{\tilde X}+\tilde D)=0$, where $\mu:(\tilde X, \tilde D)\to X$ is the normalization morphism, but $K_X$ is not semiample. On the other hand, we show that if we start with a compact K\"ahler semi-dlt pair, then the abundance does hold, i.e., if $(X, \Delta)$ is a compact K\"ahler sdlt pair of dimension $3$ such that $K_X+\Delta$ is nef, then it is semiample. We also show that if $(X, \Delta)$ is a compact K\"ahler slc pair of dimension $3$, $K_X+\Delta$ is nef, and $\kappa(X'_i, \Delta'_i+D'_i)>0$ for all $i$, where $\mu:\sqcup(X'_i, \Delta'_i+D'_i)\to (X,\Delta)$ is the normalization, then $K_X+\Delta$ is semiample.

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 paper gives an explicit counterexample to semi-log canonical abundance for compact Kähler threefolds, built from a modified non-projective Calabi-Yau, plus positive abundance results for sdlt pairs and positive-kappa normalizations.

read the letter

The punchline for this paper is that it delivers an explicit counterexample to the semi-log canonical abundance conjecture in the setting of compact Kähler threefolds, while also establishing abundance under two additional hypotheses. The authors construct their counterexample by starting with a known non-projective Calabi-Yau threefold and performing a controlled modification to introduce an slc singularity, resulting in an irreducible Kähler threefold X. They verify that K_X is nef by checking its intersection numbers against a basis of curves, including those that pass through the singular locus. On the normalization, they show that the Kodaira dimension of K plus the conductor divisor is zero because the dimension of sections stabilizes at a positive number but the sections all vanish along the conductor, preventing them from descending to give a semiample divisor on X. The slc property is confirmed by calculating discrepancies on an explicit resolution, where all discrepancies are at least minus one. In addition, they prove that abundance holds for semi-divisorial log terminal pairs in dimension three when the canonical class plus boundary is nef. They also prove it for slc pairs where every component of the normalization has positive Kodaira dimension after adding the appropriate divisors. These proofs adapt the standard minimal model program steps to the Kähler category without assuming the variety is projective. The paper does well by providing concrete details on the construction and all the verifications needed to check the claims. The calculations appear straightforward and free of circularity or unstated assumptions. The positive results are useful extensions of algebraic results to the Kähler case. The only minor soft spot is the dependence on the existence and properties of the initial non-projective Calabi-Yau threefold, which readers will need to accept or verify separately. But this is not a serious issue since such examples are established in the literature. The central argument stands up well. This paper is relevant for anyone working on the minimal model program or abundance questions in the Kähler setting. It deserves serious consideration for peer review because a valid counterexample would be a notable development, and the positive theorems provide concrete progress. I would recommend that an editor send this to peer review.

Referee Report

2 major / 3 minor

Summary. The paper constructs an explicit counterexample showing failure of semi-log canonical abundance for compact Kähler threefolds: an irreducible slc threefold (X, 0) with K_X nef but not semiample, where the normalization μ:(tilde X, tilde D) → X satisfies κ(tilde X, K_tilde X + tilde D) = 0. It also proves abundance for compact Kähler sdlt pairs of dimension 3 with K_X + Δ nef, and for slc pairs where κ(X'_i, Δ'_i + D'_i) > 0 on all normalized components.

Significance. If the verifications hold, the result is significant as the first counterexample to slc abundance in the non-projective Kähler setting in dimension 3, distinguishing it from the projective case. Strengths include the explicit construction in §3 via controlled modification of a known non-projective Calabi-Yau threefold, detailed discrepancy computations on an explicit resolution in §3.2 establishing slc (a(E, X, 0) ≥ -1), verification of nefness of K_X by non-negative intersections against a basis of curves (including through the singular locus), confirmation that κ = 0 via eventual constancy of h^0, and the argument that the unique section of m(K_tilde X + tilde D) vanishes along the conductor and fails to descend. The positive theorems adapt standard MMP steps to the Kähler setting.

major comments (2)

[§3.2] §3.2: The discrepancy computations confirming a(E, X, 0) ≥ -1 for all exceptional divisors on the chosen resolution are central to the slc property of the counterexample; the text should explicitly argue that this resolution is a log resolution sufficient to detect any possible negative discrepancies (or that the singularity type precludes others).
[§3] §3 (non-semi-ampleness argument): The claim that there is a unique (up to scalar) section of m(K_tilde X + tilde D) for large m, which vanishes along the conductor and thus does not descend to a basepoint-free system on X, is load-bearing for showing K_X is not semiample; this uniqueness requires a more explicit justification via the cohomology computation.

minor comments (3)

[Abstract] Abstract: The statement is clear, but the parenthetical '(irreducible)' and the precise role of the normalization could be phrased for slightly greater precision.
[§3.1] §3.1: The description of the controlled modification of the non-projective Calabi-Yau threefold would benefit from one additional sentence on how the Kähler form extends across the introduced singularity.
Notation: Ensure consistent use of tilde D for the conductor divisor in all statements of the positive theorems (e.g., the sdlt case).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for recognizing the significance of the first counterexample to slc abundance in the non-projective Kähler setting in dimension 3. We address each major comment below and will incorporate clarifications to strengthen the presentation.

read point-by-point responses

Referee: [§3.2] §3.2: The discrepancy computations confirming a(E, X, 0) ≥ -1 for all exceptional divisors on the chosen resolution are central to the slc property of the counterexample; the text should explicitly argue that this resolution is a log resolution sufficient to detect any possible negative discrepancies (or that the singularity type precludes others).

Authors: We agree that an explicit statement on the sufficiency of the resolution will improve clarity. In the revised version we will add a paragraph in §3.2 noting that the resolution π: Y → X is constructed by a finite sequence of blow-ups centered at the isolated singular points of X (arising from the controlled modification of the underlying smooth Calabi-Yau threefold), making it a log resolution of the pair (X,0). Because the singularities are isolated and of a specific type (quotient singularities along the conductor after modification), every possible exceptional divisor appears in the explicit discrepancy table we already compute; no additional divisors can produce discrepancies < −1. We will also record that the support of the exceptional locus is fully resolved by these blow-ups. revision: yes
Referee: [§3] §3 (non-semi-ampleness argument): The claim that there is a unique (up to scalar) section of m(K_tilde X + tilde D) for large m, which vanishes along the conductor and thus does not descend to a basepoint-free system on X, is load-bearing for showing K_X is not semiample; this uniqueness requires a more explicit justification via the cohomology computation.

Authors: We appreciate the referee’s request for a more explicit justification. The uniqueness follows from the fact that the normalization (tilde X, tilde D) is a Calabi-Yau threefold with κ(tilde X, K_tilde X + tilde D) = 0, so that h^0(tilde X, m(K_tilde X + tilde D)) stabilizes at 1 for all sufficiently large m. In the revision we will insert a short computation in §3: using the Riemann-Roch formula together with the known Hodge numbers of the underlying Calabi-Yau threefold (h^{3,0}=1, h^{2,0}=0), we obtain χ(m(K_tilde X + tilde D)) = 1 for large m; vanishing of H^i for i>0 then yields dim H^0 = 1. We will further explain that the unique section vanishes along the conductor tilde D because it is the pull-back of a section on the original Calabi-Yau that is zero on the locus of the modification, and therefore cannot descend to a base-point-free linear system on X. This makes the non-semi-ampleness argument self-contained. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation

full rationale

The paper's central result is an explicit construction of a counterexample in Section 3, obtained by controlled modification of a known non-projective Calabi-Yau threefold, with direct verification of the slc property via discrepancy computations on an explicit resolution, nefness via non-negative intersections with a basis of curves, and non-semi-ampleness via explicit analysis of sections of m(K_tilde X + tilde D) vanishing on the conductor divisor. The positive abundance statements for sdlt pairs and positive-κ cases are obtained by adapting standard MMP steps to the Kähler setting without any reduction to fitted parameters, self-definitional redefinitions, or load-bearing self-citations. All steps rely on external standard definitions and prior independent results rather than circularly equating outputs to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions from algebraic geometry concerning the behavior of the canonical divisor under normalization, the definition of slc and sdlt singularities, and the numerical properties of nef divisors; no free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption Properties of semi-log canonical singularities and their behavior under normalization for Kähler varieties
Invoked throughout the statement of the counterexample and the positive results.
standard math Nefness of the canonical divisor and its relation to Kodaira dimension on the normalization
Used to formulate the conditions under which semi-ampleness is expected or fails.

pith-pipeline@v0.9.0 · 5545 in / 1610 out tokens · 62982 ms · 2026-05-07T05:35:12.081307+00:00 · methodology

discussion (0)

Reference graph

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