Analytic torsion for irreducible holomorphic symplectic fourfolds with involution, II: the singularity of the invariant (with an Appendix by Ken-Ichi Yoshikawa)

Dai Imaike

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Analytic torsion for irreducible holomorphic symplectic fourfolds with involution, II: the singularity of the invariant (with an Appendix by Ken-Ichi Yoshikawa)

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Pith's one-line read The invariant $\tau_{M,K}$ of $K3^{[2]}$-type fourfolds with antisymplectic involution acquires a rational-log singularity along the discriminant divisor, and in the Hilbert-square cases it is a constant multiple of a power of the…

desk verdict A serious, mostly sound extension of Yoshikawa's torsion program to K3[2] fourfolds; the main theorems are new, but the referee should pin down the equivariant L2 asymptotic and the Part I dependency. read the letter →

arxiv 2411.13911 v1 pith:BRQHFLI4 submitted 2024-11-21 math.AG

Dai Imaike This is my paper · ORCID

classification math.AG MSC 14J2814J3258J5232Q15

keywords analytictorsionequivariantK3[2]-typemanifoldantisymplecticinvolutionQuillenmetricL2-metricBorcherdsproduct2-elementaryK3surface

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Starting from the invariant $\tau_{M,K}$ that the author constructed for manifolds of $K3^{[2]}$-type with antisymplectic involution, this paper establishes a precise asymptotic for its behavior as a period approaches the discriminant divisor: along any curve not contained in the boundary or the discriminant, the logarithm of the invariant is a rational multiple of $\log |s|^2$ up to an $O(\log\log|s|^{-1})$ error. The proof combines singular asymptotics for equivariant Quillen metrics and equivariant $L^2$-metrics, and exhibits the coefficient as a combination of topological and monodromy-theoretic data. When the fourfold is the Hilbert scheme of two points on a 2-elementary K3 surface, the invariant is shown, in the relevant cases, to be a constant multiple of a power of the invariant $\tau_{M0}$ of the K3 surface. Consequently the fourfold invariant is written as the Petersson norm of a Borcherds product times a Siegel modular form in those cases. A sympathetic reader would care because the result turns an a priori transcendental torsion invariant into an algebraic, number-theoretic object, and because the same boundary coefficients feed into the BCOV invariant of the crepant resolution of the quotient.

What carries the argument

The load-bearing mechanism is the comparison of two extensions of the equivariant determinant line bundle over a one-parameter degeneration: the Kaehler extension $\lambda_{\mu_2}(\widetilde{\Omega}^1_{X/C})$ and the logarithmic extension coming from the limiting Hodge filtration. Each has a known metric asymptotic: the equivariant Quillen metric gives a topological coefficient $\gamma_\iota(X_0,\Omega^1_{X/C})$ (Appendix A, Propositions 1.6-1.9), while the equivariant $L^2$-metric gives a coefficient $\sum_q (-1)^q(\alpha^{1,q}_+ - \alpha^{1,q}_-)$ with $\alpha^{p,q}_{\pm} = -\frac{1}{2\pi i}\mathrm{Tr}(\log T_s\mid \mathrm{Gr}^p_{F_\infty,\pm}H^{p+q}(X_\infty)_{\pm})$ (Proposition 1.14). The equivariant analytic torsion is their difference, and Proposition 1.15 compares the two extensions up to a known twist, yielding Theorem 1.16. On the moduli side, the invariant $\tau_{M,K}$ is matched to $\tau_{M0}$ because both satisfy the same curvature equation and the same boundary asymptotics, so their ratio is a pluriharmonic function that extends to a compact Baily-Borel model and is therefore constant.

What would settle it

On a one-parameter family of degree-2 K3 surfaces whose double-cover branch sextic acquires one node, form the induced family of Hilbert squares of the smooth fibers and compute the coefficient $a$ in Theorem 2.12 from the topological formula (2.15). If the resulting $a$ is not rational, or if the logarithm of the Petersson norm of the corresponding Borcherds-Siegel product has a different coefficient, the theorem fails.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 0.1 (Theorem 2.15): for an irreducible projective curve $C$ in the Baily-Borel compactification of the moduli space of marked $K3^{[2]}$-type fourfolds with involution, with $C$ avoiding the boundary and the discriminant divisor, and for a smooth point $p$ of $C$ on the discriminant, there is a rational number $a$ such that $\log\tau_{M,K}|_C(s) = a\log|s|^2 + O(\log\log|s|^{-1})$ as $s\to 0$. The argument goes through the equivariant determinant-of-cohomology line bundle: Theorem 1.16 expresses the singular coefficient of the equivariant analytic torsion of the relative cotangent bundle as $c(X_0,\Omega^1_{X/C}) = \gamma_\iota(X_0,\widetilde{\Omega}^1_{X/C}) + (\mu_+-\mu_-) - \sum_q (-1)^q(\alpha^{1,q}_+ - \alpha^{1,q}_-)$, where the $\alpha$ terms come from the monodromy action on the graded pieces of the limiting Hodge filtration. The comparison theorems then state that in the Hilbert-square cases the invariant satisfies $\tau_{\widetilde{M}_0,K}(Y^{[2]},\sigma^{[2]}) = C_{M_0}\tau_{M0}(Y,\sigma)^{-2(\mathrm{rk}(M_0)-9)}$ under the hypotheses of Theorems 4.6 and 4.9; for $M_0 = \langle 2\rangle$ the same formula holds for both the Hilbert square and its Mukai flop with different constants. Equivalently, in these cases the invariant is the Petersson norm of a Borcherds product and a Siegel modular form.

Load-bearing premise

The equivariant $L^2$-metric asymptotic must hold as stated, with the involution preserving the limiting Hodge filtration; if the $\mu_2$-action twisted these inputs, the rational coefficient $a$ would not be defined.

Editorial extensions

If this is right

The asymptotic makes $\tau_{M,K}$ a meromorphic object on the compactified moduli space: each curve through the discriminant has a well-defined rational order $a$ of vanishing or growth.
For the 23 types of primitive hyperbolic 2-elementary lattices with $\mathrm{rk}(M_0)\leq 17$ and a single $O(M_0^\perp)$-orbit, the identity $\tau_{M,K}(Y^{[2]},\sigma^{[2]}) = C_{M_0}\tau_{M0}(Y,\sigma)^{-2(\mathrm{rk}(M_0)-9)}$ holds for every fiber.
In those cases $\tau_{M,K}$ is the Petersson norm of a Borcherds product times a Siegel modular form, so it can be evaluated by automorphic methods rather than by spectral geometry.
For $M_0=\langle 2\rangle$ the same power formula holds on both deformation components, so the Mukai flop does not change the exponent, only the constant.
The boundary coefficients computed here are exactly the data needed to pass to the BCOV invariant of the crepant resolution of the quotient $X/\iota$.
The matching of curvature and boundary asymptotics is likely to force the same power law in cases beyond the enumerated list, as long as the fixed-locus Hodge bundle of the fourfold is a direct sum of copies of the K3 fixed-locus Hodge bundle; the list of 23 types should be sufficient rather than necessary.
If $a$ is interpreted as an intersection number on the Baily-Borel compactification, the formula gives a way to compute $\tau_{M,K}$ at cusps directly from monodromy and Chern numbers of the degeneration, which could be checked on explicit families.
One can test the comparison theorem numerically by computing both sides on a one-parameter family of plane sextic double covers; a mismatch in the constant ratio would signal either a failure of the equivariant $L^2$ asymptotic or a missing term in the fixed-locus contribution.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

First thing to know: this paper is a real step forward. It proves that the analytic torsion invariant tau_{M,K} from Part I has a rational logarithmic singularity at the discriminant, and it identifies tau on Hilbert squares with a power of Yoshikawa's K3 invariant, up to a constant. Those results are new, and the comparison with Yoshikawa's automorphic forms is the payoff.

What it does well: the singularity theorem (Thm 2.15) is built on the equivariant Quillen singularity computed in Yoshikawa's appendix, which is careful and largely self-contained. The extension to the non-natural chamber via the Mukai flop is a nice application, and the curvature comparison in Section 3 is clearly argued. The paper is honest about what it imports.

The soft spots, in order of importance:

1. Proposition 1.14 is load-bearing and under-proved. The O(log log) bound on the L2 norm of the ±-eigenframes is imported from Schmid and Peters through [17], and the paper does not verify that the eigenframes are admissible separately on the + and − eigenbundles. The μ2-action commutes with monodromy, so this is almost certainly repairable, but as written it is a genuine gap. If the estimate failed on one eigenspace, the rational coefficient in Theorem 2.15 would be undefined. A referee should ask the author to spell this out.

2. The definition of tau_{M,K} and its curvature equations are quoted from the unpublished Part I [23]. That is normal for a series, but it means the reader cannot fully check the foundation from this paper alone. Part I needs to be available and ideally refereed.

3. Lemmas 4.5 and 4.8, which are needed for the comparison theorems, are dismissed as 'straightforward.' They probably are, but since they are the link between Theorem 2.15 and the constant ratio, a few lines of justification would be welcome.

4. The comparison constants are undetermined. The theorems give equality up to a constant depending only on M0, not the value of the constant. This is typical for the method and not a flaw, but it limits what you can extract without extra work.

My take: the central argument holds up if the imported equivariant asymptotic is accepted, and Yoshikawa's appendix is a solid piece of formal work. The paper deserves a serious referee. I would send it to review and ask for a strengthened Proposition 1.14 and a detailed version of Lemma 4.5 before publication.

Referee Report

3 major / 4 minor

Summary. The paper studies the invariant τ_{M,K} attached to K3[2]-type fourfolds with antisymplectic involution, which was constructed in the companion paper [23] using equivariant analytic torsion. The main new result, Theorem 2.15, asserts that along a curve in the Baily-Borel compactification of the moduli space, and near a smooth point of the discriminant divisor, the logarithm of τ_{M,K} has a rational logarithmic singularity: log τ_{M,K}|_C(s) = a log |s|^2 + O(log log |s|^{-1}) with a ∈ Q. The proof combines a singularity formula for μ2-equivariant Quillen metrics, supplied by Yoshikawa's appendix, with an equivariant L2-metric asymptotic (Proposition 1.14) following the work of Eriksson-Freixas i Montplet-Mourougane. In Sections 3 and 4, the author shows that for 2-elementary K3 surfaces (Y,σ) of type M0, the fourfold invariant satisfies the same curvature equation as a power of Yoshikawa's invariant; using the boundary behavior and the residue theorem, he then proves exact proportionality in Theorem 4.6 and in the M0=⟨2⟩ case in Theorem 4.9, including the Mukai flop. Thus, in those cases, τ_{M,K} is a constant multiple of a power of Yoshikawa's invariant and hence of the Petersson norm of a Borcherds product and a Siegel modular form.

Significance. If the central results are correct, this is a significant step in the analytic torsion approach to irreducible holomorphic symplectic fourfolds. It provides the first boundary-behavior theorem for the equivariant analytic torsion invariant on K3[2]-type moduli spaces, and it identifies the fourfold invariant with a known automorphic invariant in a nontrivial range. The comparison is genuinely non-circular: τ_{M,K} and τ_{M0} are defined from independent torsion data on different manifolds, and the constancy of their ratio is deduced from curvature equations and boundary asymptotics rather than from the definitions. The appendix by Yoshikawa is a valuable, detailed contribution that proves the singularity formula for μ2-equivariant Quillen metrics needed in the body. The main weaknesses are that one key asymptotic input, Proposition 1.14, is imported from the non-equivariant literature without a complete verification of its hypotheses on the individual ± eigenspaces, and that the definition and curvature properties of τ_{M,K} are quoted from the unpublished companion preprint [23]. Both issues appear fixable, but as written they make the main theorems conditional.

major comments (3)

[§1.3, Proposition 1.14 and Eq. (1.15)] The estimate log‖σ'_±(t)‖²_{L2}=O(log log |t|^{-1}) is quoted from [39, (6.6)] and [37, Prop. 2.2.1], which are non-equivariant statements for sections adapted to the Deligne extension and the monodromy logarithm of a semistable degeneration. The proof applies these estimates to the ±-eigen-sections constructed in Lemma 1.12, but it does not verify that these eigen-sections satisfy the admissibility hypothesis of the cited estimates separately on the (+1)- and (−1)-eigenbundles. If an extra power of |t| survived on one eigenspace and were not cancelled by the explicit shifts b^{p,q}_{j,±}, then the coefficient ∑_q(-1)^q(α^{1,q}_+ − α^{1,q}_-) in (1.17) would not be the leading coefficient, and the rational number a in Theorem 2.15 would not be defined. The author should state and prove the μ2-equivariant analogue of [17, Thm. 2.6 and Cor. 2.8] for the eigenbundles, or otherwise show that the non-equivariant estimate applies verbatim to σ'_±.
[§3, Eq. (3.1)] The displayed formulas g = 11 − r + l/2 and k = r − l/2 are inconsistent with their use. For the lattice M0=⟨2⟩ of Example 2.3 and Theorem 4.9, one has M0^∨/M0 ≅ Z/2, hence r=1 and l=1, and (3.1) gives g=21/2 and k=1/2, although Lemma 3.1 and (3.2)–(3.3) require g to be the genus of the fixed curve (g=10 for the double cover of P²) and k to be the number of (−2)-curves (k=0). More generally, (3.1) gives 2(k−g+2)=4r−2l−18, while the proof of Corollary 3.6 needs 2(k−g+2)=t+1=2r−18. Since Corollary 3.6 and Theorem 3.7 feed directly into the comparison theorems of Section 4, the correct definitions (presumably g=(22−r−l)/2 and k=(r−l)/2) should be restored and the surrounding computations rechecked.
[§2.1.2, Definition 2.5 and Theorems 2.6–2.8] The definition of τ_{M,K}, its invariance, and the curvature equations (Theorems 2.6–2.8) are quoted from the companion preprint [23]. These statements are the starting point for Theorems 2.15, 4.6, and 4.9, so the main results are conditional on the availability and correctness of [23]. The author should either include the needed statements as part of the present paper, provide proofs or precise references to a published version of [23], or at least list exactly which results of [23] are used and state that [23] is publicly available in its current form.

minor comments (4)

[§2.3, Theorem 2.15] The statement as printed reads “τ_{M,K}(s) = a log|s|^2 + O(log log |s|^{-1})”; it should read log τ_{M,K}(s) = a log|s|^2 + O(log log |s|^{-1}).
[§2.3, Proposition 2.14] In Step 2 of the proof, the notation “B^∘ = B ∩ h^{-1}(Γ^∘)” is redundant and potentially confusing, since h is already defined as a map from B; it should simply be h^{-1}(Γ^∘).
[§0 and §2.1.2] The invariant in the introduction is written without the factors Vol(X,ω) and A(X,ι,h), while Definition 2.5 includes them. Please add a sentence explaining that the introduction normalizes h_X to be Ricci-flat with volume 1, under which A=1 and Vol(X,ω)=1.
[§4.1, Lemma 4.3] The sentence “Since ∂∂̄u = 0, ∂u is a holomorphic 1-form” should say that ∂u is a closed holomorphic 1-form; this makes the residue argument in the following paragraph easier to follow.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction found; the comparison theorem rests on independent boundary and curvature analysis, with only normal reliance on the author's Part I.

full rationale

The derivation chain is not circular. Theorem 2.15 obtains the algebraic singularity of tau_{M,K} from Theorem 1.16, whose Quillen-metric part is proved in the appendix by Yoshikawa (Proposition 1.6, Theorem A.4) and whose L2-metric part is imported from Schmid [39, (6.6)] and Peters [37, Prop. 2.2.1] through the semistable reduction; those are external, parameter-free estimates, and the equivariant eigenframe adaptation is explicitly carried out in Lemmas 1.10-1.13 and Proposition 1.14 rather than being defined into the conclusion. The comparison with Yoshikawa's invariant is likewise not circular: tau_{M,K} and tau_{M0} are independently defined invariants, and Theorem 4.6 proves their ratio is constant by matching curvature equations and then using the boundary residue theorem to kill the possible divisor current. The constant C_{M0} is shown to exist, not fitted or renamed from the data. The only caveat is that several key properties of tau_{M,K} (smoothness, curvature equation, independence of metric) are cited from the author's Part I [23], and Proposition 1.14 inherits an O(log log) estimate from non-equivariant sources without a fully written verification on the +/- eigenbundles; these are premises of the degeneration theory, not circular reductions. Hence the paper is self-contained relative to its external inputs and the comparison claim has genuine independent content.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central claim depends on the existence of tau_{M,K} from Part I and on the equivariant degenerations machinery from [17] and the appendix by Yoshikawa. No new physical or geometric entity is postulated; the only undetermined quantities are the comparison constants and the rational singularity coefficient, neither of which is computed.

free parameters (2)

Comparison constant C_{M0} = not computed
Theorems 4.6 proves the existence of a positive constant C_{M0} depending only on M0 such that tau(Y^[2],sigma^[2]) = C_{M0} tau_{M0}(Y,sigma)^{-2(rk(M0)-9)}, but the value is not evaluated.
Comparison constants C1, C2 = not computed
Theorem 4.9 for M0 = <2> proves existence of positive constants C1 and C2 for the natural and non-natural chambers, but their values are not computed.

assumptions (6)

ad hoc to paper tau_{M,K} as defined in [23] is independent of the chosen invariant metric and satisfies the curvature equation in Theorem 2.6.
The invariant and its main properties are taken from the author's unpublished companion preprint [23] (arXiv:2406.18023); no proof appears here.
domain assumption Steenbrink's mixed Hodge structure and the L2-metric asymptotic for semistable degenerations hold in the equivariant setting with the mu2-action preserving the Hodge filtration.
Proposition 1.14 and equation (1.14) rely on [17, Corollary 2.8], [39], and [37]; the equivariant extension is justified only through Lemma 1.11 and Lemma 1.13.
standard math The Bismut immersion formula and anomaly formula for equivariant Quillen metrics are valid in the stated geometric situation.
These are used in Appendix A and in Propositions 1.6, 1.7, and 1.8, and are quoted from [6], [9], and [12].
domain assumption The fixed locus classification of 2-elementary K3 surfaces and their Hilbert squares given in equations (3.2) and (3.3) is correct.
Lemma 3.1 and Propositions 3.3 and 3.5 depend on this classification, which is cited from Nikulin [34] and Macdonald [31].
domain assumption Joumaah's deformation classification of antisymplectic involutions on K3[2]-type manifolds via admissible sublattices and Kaehler chambers is correct.
The period map, the modular variety M_{M,K}, and the comparison of chambers in Section 2 rely on [25].
domain assumption Yoshikawa's invariant tau_{M0} for 2-elementary K3 surfaces is expressed as the Petersson norm of a Borcherds product times a Siegel modular form.
This external result from [45], [48], and [49] is used to conclude that the fourfold invariant inherits the same automorphic expression.

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Pith. "Pith review of Analytic torsion for irreducible holomorphic symplectic fourfolds with involution, II: the singularity of the invariant (with an Appendix by Ken-Ichi Yoshikawa)." pith.science (2026). https://pith.science/paper/BRQHFLI4

@misc{pith2026241113911,
  author       = {Pith},
  title        = {Pith review of: Analytic torsion for irreducible holomorphic symplectic fourfolds with involution, II: the singularity of the invariant (with an Appendix by Ken-Ichi Yoshikawa)},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BRQHFLI4}},
  note         = {Machine review of arXiv:2411.13911}
}

abstract

We study the boundary behavior of the invariant of $K3^{[2]}$-type manifolds with antisymplectic involution, which we obtained using equivariant analytic torsion. We show the algebraicity of the singularity of the invariant by using the asymptotic of equivariant Quillen metrics and equivariant $L^2$-metrics. We prove that, in some cases, the invariant coincides with Yoshikawa's invariant for 2-elementary K3 surfaces. Hence, in these cases, our invariant is expressed as the Petersson norm of a Borcherds product and a Siegel modular form.

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Analytic torsion for irreducible holomorphic symplectic fourfolds with involution, III: relation with the BCOV invariant
math.AG 2024-11 accept novelty 7.0 of 10

The BCOV invariant of Camere-Garbagnati-Mongardi Calabi-Yau fourfolds is proportional to the author's equivariant torsion invariant and is expressed by Borcherds products in the Enriques and rational-curve-fixed-locus cases.

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