{"id":"eab55f25-e131-473b-849f-7e1640fdf245","arxiv_id":"2411.13911","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"The singularity of the equivariant analytic torsion invariant on K3[2]-type fourfolds with involution is algebraic, and on Hilbert squares the invariant equals a fixed power of Yoshikawa's K3 invariant up to a constant.","lead":"This paper analyzes the boundary behavior of an analytic torsion invariant for four-dimensional K3[2]-type manifolds equipped with an antisymplectic involution. It proves that the singularity is algebraic and that, in many cases, the invariant is proportional to Yoshikawa's invariant for K3 surfaces, hence to the norm of automorphic forms.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 1.14's O(log log) L2 bound is imported from [39]/[37] for μ2-eigenframes without rederivation; if (1.15) fails on one eigenspace, the rational coefficient in Theorem 2.15 is not defined.","rationale":"The reader's weakest assumption points to the imported equivariant L2 asymptotic from [17]; my concern overlaps but is more specific. I do not claim the paper's conclusion is false: the appendix carefully proves the equivariant Quillen-metric part, and the missing verification is likely a routine adaptation of [17, §2]. However, because Theorem 0.1 and the comparison Theorems 4.6/4.9 both rely on the rational coefficient coming from Proposition 1.14, and because the O(log log) estimate (1.15) is not re-derived for the ±-eigenframes, the manuscript carries a real correctness risk. Lemma 4.5 is also asserted with a one-line proof, but it is secondary: once Theorem 2.15 and [45, Thm. 6.5] are available, the comparison argument is structurally sound. Since the reader already returned CONDITIONAL, and my check would either confirm or close this gap, no change to the verdict is needed.","tokens_in":54419,"tokens_out":27624,"duration_ms":295788,"concrete_test":"Analytical check: for the μ2-equivariant degeneration obtained from the nodal-sextic family (M0=⟨2⟩), take the semistable reduction g:Y→Δ' and write the shifted eigenframe t^{-Σ_j b^{p,q}_{j,±}} ρ^* θ^{p,q}_± (p=1, each q) in the standard Deligne basis {e^{-τN} v_i} of R^{p+q}g_* C ⊗ O_{Δ'}, separately on the + and − eigenspaces. Schmid's bound (1.15) holds iff each such frame is obtained from that basis by an invertible holomorphic matrix at t=0; compute the Gram determinant and verify log ||t^{-Σ_b}ρ^*θ||^2 = O(log log |t|^{-1}) with the coefficient α^{p,q}_± from [17, Cor. 2.8]. If a power of |t| appears in the Gram determinant on either eigenspace, Proposition 1.14 fails; if not, the rational coefficient κ is justified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 0.1 (Theorem 2.15) asserts a rational logarithmic singularity for log τ_{M,K}. Its proof applies Theorem 2.12, whose coefficient κ combines c, δ and \\hat c. The only place where the μ2-equivariant degeneration theory is imported rather than proved is Proposition 1.14. In the semistable reduction, Lemma 1.12 constructs frames θ^{p,q}_± of det R^q g_* Ω^p_{Y/Δ'}(log)_± such that t^{-Σ_j b^{p,q}_{j,±}} ρ^* θ^{p,q}_± is a nowhere vanishing holomorphic section. Proposition 1.14 then asserts, via (1.15), that the L2 norm of this section is O(log log |t|^{-1}), citing [39,(6.6)] and [37, Prop. 2.2.1]. Those estimates are proved in the non-equivariant setting for sections adapted to the Deligne extension and the monodromy logarithm. The paper does not explicitly verify that the ±-eigenframes produced by applying [17, Thm. 2.6] to the μ2-invariant filtration satisfy the required admissibility hypothesis separately on the + and − eigenbundles. If an extra integral or non-integral power of |t| survived on one eigenspace and were not cancelled by the explicit shifts b^{p,q}_{j,±}, then the quantity Σ_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 be ill-defined. The argument is probably completable by invoking [17] at the level of eigenbundles, but as written this is the load-bearing imported premise.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":54798,"tokens_out":26928,"duration_ms":246895,"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":[{"comment":"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'_±.","section":"§1.3, Proposition 1.14 and Eq. (1.15)"},{"comment":"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.","section":"§3, Eq. (3.1)"},{"comment":"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.","section":"§2.1.2, Definition 2.5 and Theorems 2.6–2.8"}],"minor_comments":[{"comment":"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}).","section":"§2.3, Theorem 2.15"},{"comment":"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^∘).","section":"§2.3, Proposition 2.14"},{"comment":"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.","section":"§0 and §2.1.2"},{"comment":"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.","section":"§4.1, Lemma 4.3"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically substantial and within the journal's scope. I agree with the stress-test concern: the equivariant L2 estimate in Proposition 1.14 is the main load-bearing imported point, and the paper should make the eigenbundle verification explicit. The formula for g and k in (3.1) appears to be a straightforward typo, but it is used in a central way and must be fixed. The reliance on the companion preprint [23] is also worth addressing in revision. None of these issues seems fatal; the central argument is plausible and the appendix is a genuine contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nThe soft spots, in order of importance:\n\n1. 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.\n\n2. 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.\n\n3. 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.\n\n4. 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.\n\nMy 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.","headline":"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.","tokens_in":55318,"tokens_out":4121,"would_cite":true,"duration_ms":39275,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14J28","14J32","58J52","32Q15"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["analytic torsion","equivariant analytic torsion","K3[2]-type manifold","antisymplectic involution","Quillen metric","L2-metric","Borcherds product","2-elementary K3 surface"],"falsifier":"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.","tokens_in":54186,"feed_emoji":"📐","tokens_out":10155,"duration_ms":95591,"temperature":0.7,"pith_summary":"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.","feed_headline":"Symplectic fourfold torsion invariant is algebraic at boundary","feed_subtitle":"On Hilbert squares it equals a power of τ_M0, yielding Borcherds–Siegel norms and Calabi–Yau invariants.","key_machinery":"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.","core_discovery":"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.","pith_inferences":[],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the monodromy and Steenbrink-filtration asymptotics for $L^2$-metrics and the rational constants $\\alpha^{p,q}_{\\pm}$ used in Proposition 1.14.","marker":"[17]"},{"why":"Defines the invariant $\\tau_{M0}$ of 2-elementary K3 surfaces, its automorphic expression, and its boundary singularity.","marker":"[45]"},{"why":"Predecessor paper that constructs $\\tau_{M,K}$ and proves its smoothness, invariance, and curvature equation.","marker":"[23]"},{"why":"Provides the equivariant immersion and anomaly formulas for equivariant Quillen metrics used in Propositions 1.6-1.8 and Appendix A.","marker":"[6]"},{"why":"Gives the singularity formula for Quillen metrics along normal-crossing degenerations, used throughout for the $\\log|s|^2$ asymptotics.","marker":"[46]"},{"why":"Classifies deformation types of $K3^{[2]}$-type manifolds with antisymplectic involution and supplies the period map and chamber structure.","marker":"[25]"},{"why":"Furnishes the residue-theorem comparison argument used in the proof of the $\\langle 2\\rangle$ case.","marker":"[29]"},{"why":"Establishes the structure of $\\tau_{M0}$ as a Petersson norm of Borcherds products and Siegel modular forms.","marker":"[49]"}],"fun_headline_variants":["Torsion invariant for symplectic fourfolds is algebraic at singularities","Borcherds product and Siegel form express fourfold torsion invariant","Algebraic singularity of torsion invariant on symplectic fourfolds","On Hilbert squares, torsion invariant is Borcherds-Siegel norm"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Torsion invariant for symplectic fourfolds is algebraic at singularities","Borcherds product and Siegel form express fourfold torsion invariant","Algebraic singularity of torsion invariant on symplectic fourfolds","On Hilbert squares, torsion invariant is Borcherds-Siegel norm"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000832,"raw_usage":{"total_tokens":3688,"prompt_tokens":1058,"completion_tokens":2630,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":674,"completion_tokens_details":{"reasoning_tokens":2553}},"tokens_in":674,"tokens_out":2630,"duration_ms":18934,"temperature":1.0,"reasoning_tokens":2553,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:44:55.688017+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the monodromy and Steenbrink-filtration asymptotics for $L^2$-metrics and the rational constants $\\alpha^{p,q}_{\\pm}$ used in Proposition 1.14."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the invariant $\\tau_{M0}$ of 2-elementary K3 surfaces, its automorphic expression, and its boundary singularity."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the singularity formula for Quillen metrics along normal-crossing degenerations, used throughout for the $\\log|s|^2$ asymptotics."},{"cited_title":"Ma and K.-I","cited_arxiv_id":null,"evidence_quote":"Furnishes the residue-theorem comparison argument used in the proof of the $\\langle 2\\rangle$ case."}],"review_version":1}