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The paper establishes a sharp dichotomy for log Calabi-Yau manifolds: a smooth boundary divisor preserves compact-CY behavior, while two proportional boundary divisors break all four classical properties and make the universal cover of infi

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For log Calabi-Yau pairs, a smooth boundary gives holonomy SU(n), Bochner principle, and stability, while two boundary components break all three and make the universal cover non-compactifiable.

T0 review reviewed 2026-08-04 challenge →

load-bearing objection A very solid paper on log CY manifolds; main theorems are new and well-proved, but the key non-compactifiability claim relies on an external local computation that is misstated for n≠4 and should be fixed before publication. the 2 major comments →

arxiv 2509.07508 v1 pith:TY66KYTJ submitted 2025-09-09 math.AG math.CVmath.DG

Log Calabi--Yau manifolds: holomorphic tensors, stability and universal cover

Tristan C. Collins , Henri Guenancia This is my paper
classification math.AG math.CVmath.DG MSC 14J3232Q20
keywords log Calabi-Yau pairscomplete Ricci-flat Kähler metricsholonomy SU(n)Bochner principlelogarithmic tangent bundlepolystabilityuniversal cover infinite topological typevanishing cycles
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

This paper studies log Calabi-Yau manifolds, meaning a compact Kähler manifold $X$ with a simple normal crossing divisor $D$ such that $K_X+D$ is trivial, so $X\setminus D$ carries a holomorphic volume form. The authors compare two cases. When $D$ is a single smooth anticanonical divisor, they prove that the complete Ricci-flat Kähler metric has full $\mathrm{SU}(n)$ holonomy, every holomorphic tensor with logarithmic poles is parallel (Bochner principle), and the logarithmic tangent bundle is stable. When $D$ splits into two proportional components—for example two transverse hypersurfaces in $\mathbb{P}^n$ whose degrees add to $n+1$—a complete Ricci-flat metric still exists, but all four classical properties from compact Calabi-Yau theory fail: the quasi-Albanese map is not locally trivial, the logarithmic tangent bundle is not polystable, and the universal cover has infinite topological type, with $H_n$ of infinite rank, so it cannot be biholomorphic to a Zariski open subset of any compact complex manifold. The paper also describes the asymptotic Riemannian geometry of the universal cover: its tangent cone at infinity is $(\mathbb{R}_{>0})^2 \times \mathbb{R}$ with an explicit warped metric and volume growth of order $R^{6n/(n+2)}$.

Core claim

The central discovery is a structural dichotomy. In the one-divisor case ($X$ Fano, $D$ smooth anticanonical), the complete Ricci-flat metric of Tian-Yau type behaves like a compact Calabi-Yau metric: the holonomy is the full special unitary group, every holomorphic tensor with logarithmic poles is parallel, and the logarithmic tangent bundle $T_X(-\log D)$ is stable with respect to $-K_X$. In the two-divisor case, with $D=D_1+D_2$ and the two components numerically proportional, the complete Ricci-flat metric of Collins-Li type also has $\mathrm{SU}(n)$ holonomy, yet each of the four classical consequences fails. The strongest statement is Theorem 4.14: for $n$ at least 3, if $D_1$ and $D_2$ are smooth transverse hype

What carries the argument

The two load-bearing constructions are the complete Ricci-flat metrics. For one smooth divisor, the metric has subquadratic volume growth and model asymptotics near the divisor; its asymptotics force any parallel intermediate-degree $p$-form on any finite cover to vanish, which yields full $\mathrm{SU}(n)$ holonomy and the Bochner principle. For two divisors, the metric is asymptotically a homogeneous optimal-transport solution on $(\mathbb{R}_{>0})^2$ with volume growth $R^{4n/(n+2)}$. The universal-cover result uses the quasi-Albanese fibration $f = s_1/s_2: M \to \mathbb{C}^*$, its infinite cyclic cover obtained by pulling back the exponential map $\mathbb{C} \to \mathbb{C}^*$, and the vanishing cycle in a Lefschetz pencil. The key mechanism is to

Load-bearing premise

The claim that the universal cover has infinite rank middle homology relies on an unpublished local asymptotic computation near an ordinary double point, namely that the integral of the volume form over the vanishing cycle behaves like a nonzero constant times $t$ as $t \to 0$; if that computation is wrong, the homological independence argument does not go through.

What would settle it

Compute directly, for the pencil of quadrics in $\mathbb{P}^3$ cut out by two smooth quadrics, the period integral $h(t) = $ integral over the vanishing cycle of the pulled-back holomorphic volume form as $t \to 0$. If the leading coefficient $\alpha$ in $h(t) \sim \alpha t$ were zero, the exponential-period integrals would no longer separate the cycles $L_k$, and the infinite-rank conclusion of Theorem 4.14 would fail.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • For a smooth anticanonical divisor in a Fano manifold, every holomorphic section of a logarithmic tensor bundle is parallel for the complete Ricci-flat metric, and the space of such tensors is exactly the SU(n)-invariant subspace at a point.
  • The logarithmic tangent bundle T_X(-log D) is stable with respect to -K_X when D is smooth, while for two proportional components it is only semistable and its Jordan-Hölder filtration can be described explicitly.
  • The universal cover of P^n minus two transverse hypersurfaces of degrees summing to n+1 has infinite rank in middle homology, and therefore cannot be realized as a Zariski open set in any compact complex manifold.
  • The asymptotic geometry of the universal cover is explicit: its tangent cone at infinity is (R_{>0})^2 times R with a warped product metric, and its volume growth is R^{6n/(n+2)}.
  • The quasi-Albanese map in the two-component case is not locally trivial; for very general hypersurfaces of high degree, two general fibers need not even be birational to each other.

Where Pith is reading between the lines

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

  • The dichotomy suggests that holonomy SU(n) alone is too coarse to define a non-compact analogue of an irreducible Calabi-Yau manifold: the two-divisor examples have full SU(n) holonomy yet violate the Bochner principle, stability, and compactifiability.
  • The construction of homologically independent cycles by parallel transporting a vanishing cycle along an infinite cyclic cover may apply to other fibrations over C* with a degenerate fiber, as the paper itself notes that only one fiber with an ordinary double point is needed.
  • The volume-growth distinction (subquadratic for one divisor, superquadratic for two) could serve as a testable heuristic: subquadratic growth may force the Bochner principle, while superquadratic growth permits non-parallel closed holomorphic forms.
  • The exponential-period separation method used to prove independence of the cycles L_k resembles a period-integral argument that might be adapted to other complete Calabi-Yau metrics built from logarithmic pairs.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper studies log Calabi-Yau pairs (X,D), i.e. log smooth pairs with K_X+D=0, in two configurations: D smooth (Tian-Yau metric) and D=D1+D2 with D1,D2 proportional (Collins-Li metric). In the smooth-divisor case it proves SU(n) holonomy for the complete Tian-Yau metric, a Bochner principle for logarithmic tensor bundles, and stability of T_X(-log D). In the two-divisor case it identifies the quasi-Albanese map with a Lefschetz pencil, shows non-local triviality and non-polystability of T_X(-log D), determines the asymptotic tangent cone and volume growth of the universal cover, and constructs examples (complements of two transverse hypersurfaces in P^n of degrees summing to n+1) whose universal cover has infinite topological type, in particular infinite-rank middle homology, hence cannot be a Zariski open set in a compact complex manifold.

Significance. If the main theorems hold, the paper draws a striking contrast between the two Log Calabi-Yau settings: the smooth-anticanonical case behaves like the compact Calabi-Yau case (Bochner, stability, finite fundamental group), while the two-component case violates all these properties and yields the first algebraic examples of Calabi-Yau universal covers of infinite topological type with complete Ricci-flat metrics. The proofs are largely self-contained and use established machinery: holonomy classification, Bochner techniques, Collins-Li asymptotics, and Lefschetz vanishing cycles. The paper also gives a useful description of the Riemannian geometry at infinity. However, one of the two families of examples in Theorem 4.14 rests on an unproven and dimensionally inconsistent local period estimate, and the statement of Theorem 4.14 is broader than what the proof actually establishes.

major comments (2)
  1. [§4.3, Claim 4.17] The nonvanishing integral ∫_L τΩ is load-bearing for Claim 4.18 and hence for Theorem 4.14. The proof invokes the asymptotic h(t)∼αt, α≠0, as t→0, with a reference to [Col25, Section 4]. For the standard local model f=Σ_{i=1}^n z_i^2 on C^n, with Ω=dz_1∧...∧dz_n and L_t={|x|^2=t, y=0}, the residue period ∫_{L_t} Ω/df scales as t^{(n-2)/2}; it is proportional to t only for n=4 (e.g. n=3 gives √t). Thus the estimate as stated is not scale-consistent for general n≥3. The contradiction argument in Claim 4.17 would still work with any nonzero leading term αt^p, p>0, so the theorem is likely repairable, but the central nonvanishing step is currently not established in the paper.
  2. [§4.3, Theorem 4.14] The theorem is stated for arbitrary smooth transverse D1,D2 in the cases (d1,d2)=(1,n) and d1=d2=(n+1)/2 with n odd. However, the proof of the equal-degree case begins 'Since D1,D2 are general, each fiber ... has a single ordinary double point singularity.' The statement does not include a generality assumption, and no degeneration argument is supplied. As written, the proof only covers general hypersurfaces; either the theorem statement must be amended to include 'general', or the proof must be extended to all smooth transverse pairs. This is a genuine gap between statement and proof.
minor comments (3)
  1. [Lemma 2.4] The hypothesis says 'assume additionally that k=1', but the statement concerns the two-divisor case; it should read k=2.
  2. [§3.2, proof of Theorem 3.4] There are stray symbols in the displayed integration after equation (3.3) ('2 Z X χR|∇f|^2'); the intended displayed equation should be cleaned up.
  3. [§4.3, Claim 4.18] The displayed identity for ∫_{L_k} e^{s bf} bΩ contains an apparent typo: the right-hand side has ε^{s bf} where it should be e^{s bf}. More importantly, the identity depends on the Galois action shifting bf by k, which is used to conclude that the holomorphic function Σ a_k e^{sk} vanishes; this should be spelled out for the reader.

Circularity Check

0 steps flagged

No circular reduction: central results derive from external existence theorems and standard topology; the only same-author citation is for a local period computation, not a target-equivalent input.

full rationale

The paper's derivation chain is self-contained modulo standard external theorems. Theorem A uses the Tian–Yau metric and its asymptotics to prove holonomy SU(n), then derives Bochner and stability; Theorem B uses the identification of the quasi-Albanese map with the pencil f=s1/s2, the existence of singular fibers, and Proposition 2.7; Theorem C uses the Collins–Li asymptotic model and standard holonomy arguments; Theorem 4.14 uses topological properties of Lefschetz pencils, Ehresmann triviality, and homological independence arguments. No fitted parameter is renamed as a prediction, and no quantity is defined in terms of the result it is supposed to establish. The one load-bearing same-author citation is in Claim 4.17, where the local period asymptotics h(t) ~ α t near an ordinary double point is delegated to [Col25, Section 4]. This is a standard local residue computation, not a statement equivalent to the theorem being proved; even if the printed exponent were scale-inconsistent for general n, the argument only needs h(t) nonzero near t=0, so the issue is a correctness risk, not circularity. The other self-citations ([CL24b], [CTY24]) supply external, parameter-free existence/asymptotic results, not the paper's own conclusions. Consequently, the circularity score is 0.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The paper rests on deep existence and asymptotic results for complete Ricci-flat metrics (Tian-Yau, Collins-Li) and on standard topological and Hodge-theoretic tools. No empirical constants are fitted, and no new entities (particles, forces, dimensions) are postulated. The main non-standard dependency is the unpublished local vanishing-cycle computation of [Col25].

axioms (6)
  • standard math Existence of the complete Tian-Yau Ricci-flat Kähler metric omega_TY on X\D for X Fano and D smooth anticanonical, with the asymptotic model (3.1).
    Invoked in Section 3 as a black box from [TY90]; the asymptotic expansion is used in Claims 3.3 and in the proof of the Bochner principle.
  • standard math Existence of the complete Collins-Li Ricci-flat Kähler metric omega_CL on X\(D1 union D2) with the stated asymptotics (generic region (4.4), non-generic region (4.5), polynomial decay).
    Invoked in Section 4 from [CL24b]; used to prove Theorems 4.7 and 4.12 and the volume growth estimates.
  • standard math Local vanishing-cycle asymptotic: for the Lefschetz pencil, the period integral satisfies h(t) ~ alpha t as t approaches 0, with alpha nonzero.
    Used in Claim 4.17 to prove non-vanishing of the integral over the constructed cycle; delegated to [Col25, Section 4]. If false, the construction of independent homology classes fails.
  • standard math Berger-Simons holonomy classification for Riemannian manifolds.
    Used in Proposition 3.2 and Theorem 4.7 to constrain the holonomy group to SU(n) or Sp(n/2).
  • standard math Nori's theorem on fundamental groups of complements of ample divisors.
    Used in Lemma 2.1 to compute pi_1(X\D) and in Corollary 2.2.
  • standard math Stable irrationality of very general hypersurfaces of degree d >= log_2(n-1)+2 in P^n (Schreieder [Sch19], Shinder [Shi22]).
    Used in Proposition 4.4(3) to show that two very general fibers of the quasi-Albanese map are not birational.

reviewed 2026-08-04 · how reviews work

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Cite this review

Pith. "Pith review of Log Calabi--Yau manifolds: holomorphic tensors, stability and universal cover." pith.science (2026). https://pith.science/paper/TY66KYTJ

@misc{pith2026250907508,
  author       = {Pith},
  title        = {Pith review of: Log Calabi--Yau manifolds: holomorphic tensors, stability and universal cover},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TY66KYTJ}},
  note         = {Machine review of arXiv:2509.07508}
}
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abstract

We study various geometric properties of log Calabi-Yau manifolds, i.e. log smooth pairs $(X,D)$ such that $K_X+D=0$. More specifically, we focus on the two cases where $X$ is a Fano manifold and $D$ is either smooth or has two proportional components. Despite the existence of a complete Ricci flat K\"ahler metric on $X\setminus D$ in both cases, we will show that the geometric properties of the pair $(X,D)$ are vastly different, e.g. validity of Bochner principle, local triviality of the quasi-Albanese map, polystability of $T_X(-\log D)$ and compactifiability of the universal cover of $X\setminus D$. When $D$ has two components we show that the universal cover of $X\setminus D$ is a Calabi-Yau manifold of infinite topological type, and we describe the geometry at infinity from a Riemannian point of view.

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    work page 2024

This paper was first reviewed by deepseek-v4-flash on August 4, 2026.