{"id":"50239210-6c9a-4658-9539-7f3ca70517f4","arxiv_id":"2508.06750","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Fano varieties with an snc anticanonical divisor whose dual intersection complex is pure-dimensional, the mirror symmetric Gamma conjecture holds for the structure sheaf and skyscraper sheaf.","lead":"This paper proves a version of the mirror symmetric Gamma conjecture for Fano varieties with a good anticanonical divisor, using a new relative mirror theorem and generalized theta functions. The result connects the periods of a Landau-Ginzburg model to Gromov-Witten invariants and Gamma classes.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 13 does not derive the Gamma class: cone-wise Γ factors become \\hatΓ_X only via an unproven 'We must have'.","rationale":"The reader's weakest assumption, Corollary 3.14, is a real gap: the degeneration proof is sketched and the mirror maps in D_i are not explicitly constructed. However, I think the single most load-bearing issue is the final Gamma-class identification in Section 13. The explicit computation for a maximal cone produces products of Γ(1+D_i^{λ_i}(σ)) over anticanonical divisor components, not the full Gamma class \\hatΓ_X over Chern roots of TX. The transition from these cone-wise factors to \\hatΓ_X is asserted through a differential-equation sentence ending in 'We must have'. This transition is the mathematical content of the mirror symmetric Gamma conjecture, not a routine step; it requires a nontrivial localization/asymptotic matching that the paper does not provide. The proposed P^1 test is a finite, concrete check: it either verifies that the cone-wise Γ factors combine to Γ(1+2H) after summing chambers, in which case the concern is repairable, or it exposes a genuine failure of the proof of Theorem 1.9. I therefore maintain the reader's CONDITIONAL verdict rather than moving to accept or reject.","tokens_in":60377,"tokens_out":21045,"duration_ms":241768,"concrete_test":"Take X=P^1 with D={0,∞}, so the assumptions of Theorem 1.9 hold and the relative mirror map is trivial. Compute the left side of (6) for φ=1, τ_{0,2}=0 using the paper's definitions: W=ϑ_{e_1}+ϑ_{e_2} (= x+x^{-1} by Theorem 10.2), with the cone-wise formula of Section 13. Explicitly sum the two maximal-cone contributions and take the non-equivariant limit λ_i→0. Compare the first several coefficients in the z-expansion with the right side computed from the known small J-function of P^1 and \\hatΓ_{P^1}=Γ(1+2H). If the coefficients disagree, the 'We must have' step is invalid; if they agree, the missing Gamma-class identification can be filled in and the concern is reduced to a presentation gap.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Even granting Corollary 3.14, the proof of Theorem 1.9 has a gap at the last step. In Section 13, for a fixed maximal cone σ, the computation yields an expression of the form ∫_X (z^{c1} z^{deg/2} J_{X,β}(τ_{0,2},-z)φ) ∪ ∏_{i∈I_σ} Γ(1+D_i^{λ_i}(σ)). The product is over the divisor components meeting at the zero-dimensional stratum D_σ, not over Chern roots of TX. The text then says the same computation works for each σ and, because the oscillatory integral and the equivariant J-function satisfy the same differential equation, 'We must have' the desired identity with \\hatΓ_X. This 'must have' is exactly the nontrivial Gamma-class identification: one must sum the cone-wise contributions, take the non-equivariant limit, and match the asymptotic expansion of the fundamental solution. No such calculation is supplied. The gap is visible already for X=P^1 with D={0,∞}: \\hatΓ_X=Γ(1+2H), while a fixed cone contributes a single Γ(1+D_i^{λ_i}); the factor 2 (or the correct Chern-root replacement) must come from the missing localization/summation step. Thus Identity (6) is not proven by the displayed computation even assuming the relative mirror theorem.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a relative mirror symmetry framework for a smooth projective Fano variety X equipped with an snc anticanonical divisor D, using orbifold Gromov–Witten theory of root stacks. It states a relative mirror theorem for snc pairs without assuming nefness of the components D_i (Theorem 3.1, Corollary 3.14), introduces generalized mirror functions and theta functions (Sections 5–7), and uses these ingredients to prove the mirror symmetric Gamma conjecture for X for O_X and O_pt (Theorems 1.7 and 1.9). The main result, Identity (6), equates an oscillatory integral of the theta-function Landau–Ginzburg potential with the Gromov–Witten J-function of X capped with the Gamma class. The manuscript is written as a research announcement: several central arguments are presented as sketches, and the final Gamma-class identification is asserted rather than derived.","tokens_in":60692,"tokens_out":4821,"duration_ms":63901,"significance":"If fully substantiated, the paper would be a substantial contribution: it removes the nefness assumption in relative mirror theorems, proposes new higher-degree mirror functions, and gives a new route to the mirror symmetric Gamma conjecture for a large class of Fano varieties. The construction of the functions in Section 6 and the product rules in Sections 5–7 are interesting and potentially reusable. The paper is also honest about several limitations: Remark 1.5 states that mirror maps in D_i are not described explicitly, Remark 1.12 states that the mirror map for φ is not computed, and Section 11 explicitly says that the quantum differential operator is not described. These admissions are directly relevant to the proof of the main theorem, not merely to presentation. The manuscript currently does not contain a complete, verifiable proof of the central identity; it is more a detailed program with strong evidence.","major_comments":[{"comment":"The relative mirror theorem is the load-bearing bridge used in Sections 10, 12, and 13, but its proof is only sketched. The comparison of the two degeneration formulas (19) and (20) is not carried out in detail, and the passage 'the difference is governed by the mirror map of (Y,D_0)' (Section 3.3.3) is asserted rather than proved. Remark 1.5 concedes that the mirror maps in D_i are not described. Since Corollary 3.14 is used verbatim to translate the relative invariants into the J-function of X, the main theorems inherit this incompleteness. A complete proof of Corollary 3.14, or at least of its log Calabi–Yau specialization used here, is required.","section":"Corollary 3.14 and Section 3.3"},{"comment":"The derivation of Identity (6) has a gap at the last step. For a fixed maximal cone σ, the computation yields a factor ∏_{i∈I_σ} Γ(1+D_i^{λ_i}(σ)), where I_σ indexes the divisor components containing the zero-dimensional stratum D_σ. The text then says the same computation works for each σ and, because the oscillatory integral and the equivariant J-function satisfy the same differential equation, 'We must have' the desired identity with R;hat Γ_X. This is exactly the nontrivial Gamma-class identification: the product over divisor components in a given cone is not the Gamma class of T X, whose Chern roots are not the restrictions D_i|_Dσ. One must sum the cone-wise contributions, take the non-equivariant limit, and match the asymptotic expansion of the fundamental solution. No such calculation is supplied. The P^1 example with D={0,∞} shows the issue: a single cone contributes Γ(1+H), whi","section":"Section 13, especially the final paragraph"},{"comment":"There is a risk that the main identity is close to being definitional rather than a theorem. The functions ˇφτ,⃗s are defined in Definition 6.1 using the same relative invariants that later appear in the J-function of (X,D), and Proposition 7.1 is engineered so that the product of ˇφ with exp(-W/z) reproduces those invariants. After applying Corollary 3.14, the computation in Section 13 transforms this into the J-function of X times cone-wise Gamma factors. The only genuinely new content left is the passage from those factors to R;hat Γ_X, which is missing. Section 11 also states 'we do not give an explicit description of the quantum differential operator', so the claim that the oscillatory integral and the equivariant J-function satisfy the same differential equation is not verifiable from the manuscript. A nontrivial worked example, where both sides of (6) are computed independently, w","section":"Sections 6, 11, and 13"}],"minor_comments":[{"comment":"The statement of Proposition 7.1 contains corrupted typesetting with repeated '⇂⟨⟨⟪rl⟫l⟩⟩' symbols. As printed, the displayed identity is unreadable and must be corrected.","section":"Proposition 7.1"},{"comment":"The notation R;vec r is used both for rooting parameters and for contact orders, and Remark 4.1 only partially resolves the conflict. In formulas such as (12)–(13) and Definition 5.6, the reader must repeatedly infer which meaning is intended; a systematic change of notation would help.","section":"Remark 4.1 and Section 5"},{"comment":"The differential-equation argument is invoked as a substitute for an explicit identification of the fundamental solution. Since the operator is not written down, the statement 'the components of the J-function span a basis of solution to the differential system' is not checkable from the text.","section":"Section 11"},{"comment":"The proof of Theorem 10.1 says that after applying the relative mirror theorem the corresponding coefficient is 'precisely the RHS of (52)', but the extraction of the coefficient from the extended I-function is not displayed. This is shorter than the analogous computation in Section 12 and would benefit from the same level of detail.","section":"Theorem 10.1 proof"}],"recommendation":"major_revision","confidential_remarks":"The paper is ambitious and likely important if the gaps can be filled. The most serious issue is not the relative mirror theorem sketch itself, but the final Gamma-class identification in Section 13: the sentence 'We must have' is not a proof, and the missing summation/localization step is essential. I would advise the editor that acceptance should require either a complete derivation of the Gamma-class matching or a clear restriction of Theorem 1.9 to a setting where the cone-wise factors are already known to assemble into R;hat Γ_X."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper gives a real extension of the relative mirror theorem: removing the nefness assumption on the divisor components via degeneration to the normal cone is the right idea, and the new functions from the higher-degree part of relative quantum cohomology are a genuine addition. The treatment of the mirror map in D, though not explicit, is honest about what is being assumed. The proof of the mirror symmetric Gamma conjecture for O_pt (Theorem 1.7) is more convincing: the residue computation and the use of the extended I-function line up. The paper is worth a serious referee.\n\nBut the main theorem, Theorem 1.9 for O_X, has a real gap at the last step. For each maximal cone sigma, the computation lands on an integral against the product of Gamma factors Gamma(1 + D_i^{lambda_i}(sigma)) over the divisors meeting at that cone. The text then says that because the oscillatory integral and the equivariant J-function satisfy the same differential equation, \"we must have\" the desired identity with the full Gamma class \\hatΓ_X. That is exactly the nontrivial step: summing cone contributions, taking the non-equivariant limit, and matching the asymptotic expansion of the fundamental solution. No such calculation appears. The P^1 example makes the gap visible: a single cone contributes a factor like Gamma(1 + H), while \\hatΓ_X = Gamma(1+2H); the factor 2 has to come from the missing summation over the two cones. So Identity (6) is not established by the displayed computation.\n\nThere is also some unfinished business around Corollary 3.14: the mirror map in D_i is referred to but never explicitly constructed. The proof via degeneration is sketched rather than written out. That alone would be survivable if the final step were solid, but the two together mean the central application rests on two under-specified planks. The corrupted display in Section 7 (the garbled formula in Proposition 7.1) makes it hard to verify the product rule, though that looks like a TeX issue rather than a mathematical one.\n\nStill, the architecture is plausible and the relative mirror theorem is valuable even if Theorem 1.9 needs more work. The paper uses prior results heavily, but that is not a flaw when the citations are apt; the author is transparent about what is imported. I would not desk-reject this. I would send it to referees and ask them to focus on the cone-summation/Gamma-class identification and on making the mirror map in D explicit. If that step can be supplied, this would be a strong paper. Until then, it is a conditional proof.\n\nFor a reading group, maybe: the ideas are good but the missing details will frustrate a close read. I would not cite it as a proof of the Gamma conjecture yet, but I might cite the relative mirror theorem after checking the degeneration argument more carefully.","headline":"Plausible and genuinely new relative mirror theorem, but the final Gamma-class step in Theorem 1.9 is asserted rather than proved, so the headline conjecture should be treated as conditional.","tokens_in":61132,"tokens_out":2048,"would_cite":false,"duration_ms":27931,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14N35","14J33","53D45"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves the mirror symmetric Gamma conjecture for Fano varieties with simple normal crossing anticanonical divisors, using a relative mirror theorem and theta functions.","keywords":["mirror symmetric Gamma conjecture","relative mirror symmetry","theta functions","Fano varieties","Gromov-Witten theory","log Calabi-Yau pairs","root stacks","quantum periods"],"falsifier":"Take a Fano threefold $X$ with an snc anticanonical divisor having a non-nef component $D_i$, and compute the first nonzero $x^{\\vec d}x_{\\vec b}$-coefficient of the two sides of Corollary 3.14; if the coefficient predicted by the $I$-function after any valid mirror map in $D_i$ differs from the corresponding relative invariant of $(X,D)$, the theorem fails, and a concrete starting point is the one-point invariant $\\langle [pt]\\psi^k\\rangle^X_{0,1,\\beta}$ for a class $\\beta$ with $D_i\\cdot\\beta<0$.","tokens_in":60289,"feed_emoji":"🧮","tokens_out":5895,"duration_ms":56668,"temperature":0.7,"pith_summary":"This paper sets out to prove a version of the mirror symmetric Gamma conjecture for Fano varieties, using the relative mirror symmetry of a log Calabi--Yau pair $(X,D)$ where $D$ is a simple normal crossing anticanonical divisor. The central claim is that the oscillatory integral of the Landau--Ginzburg potential $W=\\sum_i \\vartheta_{[D_i]}$, defined through $\\theta$ functions built from orbifold Gromov--Witten invariants, equals the $J$-function of $X$ capped with the Gamma class and a mirror map. This matters because the proof avoids the nefness assumptions that earlier relative mirror theorems required, and because it packages the Gamma conjecture as a statement about relative invariants of pairs. If correct, the identity gives a computational bridge from the mirror of the complement $X\\setminus D$ back to the quantum cohomology of $X$.","feed_headline":"Theta functions prove mirror Gamma conjecture for Fano pairs","feed_subtitle":"Oscillatory integrals of a theta-function superpotential equal the Gamma-corrected J-function when an anticanonical divisor is snc.","key_machinery":"The argument is carried by the relative mirror theorem for snc pairs (Corollary 3.14), proved by degenerating both the root stack and its ambient bundle to the normal cone of $D$. The theorem asserts that an $S$-extended $I$-function, after a possibly nontrivial mirror map in each divisor $D_i$, lies in the Givental Lagrangian cone of $(X,D)$; this connects integrals over the mirror to the $J$-function of $X$. The complementary machinery is the orbifold $\\theta$-function calculus: $\\theta$ functions $\\vartheta_{\\tau,\\vec s}$ are defined as sums of mid-age orbifold invariants, and the functions $\\check\\varphi_\\tau$ in Definition 6.1 extend them to higher-degree cohomology classes and satisfy the pr","core_discovery":"On the author's own terms, the paper establishes that for a Fano variety $X$ with an snc anticanonical divisor $D$ whose dual intersection complex is pure dimensional of real dimension $\\dim X$, the mirror symmetric Gamma conjecture holds for the structure sheaf $\\mathcal O_X$ and for $\\mathcal O_{\\mathrm{pt}}$. The identity is $$\n\\int_{\\Gamma_R} \\check{\\varphi}_\\tau(-z) $e^{{-W_\\tau/z}}$\\,\\omega =\n\\int_X \\bigl($z^{{c_1}}$ $z^{{\\deg/2}}$ J_X(\\tau_{0,2},-z)\\varphi\\bigr)\\cup \\widehat\\Gamma_X,\n$$ with $W_\\tau$ a sum of $\\theta$ functions and $\\check{\\varphi}_\\tau$ defined as a mirror function attached to the higher-degree part of relative quantum cohomology. A companion result identifies the regularized quant","pith_inferences":["If the undetermined mirror maps in $D_i$ can be shown to be the standard Birkhoff-factorization maps of the local model $N_{D/X}$, the relative mirror theorem becomes an effective computational tool rather than an existence statement; this is not proved in the paper.","The approach suggests a route toward the modified Gamma conjecture for Fano varieties with a log Calabi--Yau compactification: any failure of the absolute Gamma conjecture would have to be invisible on the relative side, since the paper notes the known counterexample to the absolute conjecture is not a counterexample to this mirror-symmetric version.","The identity (Theorem 1.9) may admit a testable strengthening: when $\\varphi$ ranges over a basis of $H^*(X)$, the full system of oscillatory integrals should span the solution space of the quantum differential equation, giving an integral-normalized basis analogous to the Gamma-integral structure."],"forward_implications":["If the main theorem is correct, the mirror symmetric Gamma conjecture holds for $\\mathcal O_X$ and $\\mathcal O_{\\mathrm{pt}}$ for any Fano $X$ admitting such an snc anticanonical divisor.","The regularized quantum period of $X$ equals the classical period of the theta-function superpotential $W$, and when the relative mirror map is trivial $W$ is a Laurent polynomial in $n$ variables.","The relative mirror theorem applies without the nefness assumption on the irreducible components $D_i$, so the $J$-function of $X$ can be expressed through relative invariants with negative contact orders even when some $D_i$ have negative intersections with curve classes.","For Fano complete intersections inside a variety whose pair satisfies the hypotheses, the conjecture for the subvariety follows via a Laplace-transform argument.","When $D$ lacks zero-dimensional strata, the same computation extends to a maximally unipotent monodromy degeneration, giving Theorem 1.14."],"supporting_citations":[{"why":"Supplies the nef-case mirror theorem for multi-root stacks and the iterative snc construction that Section 3 generalizes by degeneration.","marker":"[TY23b]"},{"why":"Defines orbifold theta functions and mid-age invariants; the paper's Definition 5.1 is built on these and extended with a parameter $\\tau$.","marker":"[You24a]"},{"why":"Gives the intrinsic mirror construction for log Calabi--Yau pairs, the framework whose orbifold-invariant variant the paper uses for the mirror $\\check X$.","marker":"[GS19]"},{"why":"Proves the mirror theorem for root stacks under nefness; the non-nef generalization is obtained by degenerating the same root-stack/bundle setup.","marker":"[FTY19]"},{"why":"States the mirror symmetric Gamma conjecture that Theorem 1.9 proves in this relative setting.","marker":"[GI19]"},{"why":"Provides the integral-structure and differential-equation framework used to identify the oscillatory integrals with the Gamma-corrected $J$-function basis.","marker":"[Iri09]"},{"why":"Supplies the mirror theorem for toric stack bundles, which is the local model for the $I$-function computations in the degeneration.","marker":"[JTY17]"},{"why":"Gives the orbifold Gromov--Witten theory of snc pairs, including the degree-zero relative quantum product and the TRR used for the theta-function product rules.","marker":"[TY23a]"}],"fun_headline_variants":["Theta functions unlock mirror Gamma conjecture for Fano pairs","Gamma conjecture proven for Fano via theta-function mirror","Mirror Gamma conjecture for Fano varieties from theta functions","Higher-degree quantum cohomology powers Gamma conjecture proof"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The load-bearing premise is the relative mirror theorem, Corollary 3.14: the $S$-extended $I$-function, after applying a mirror map in each divisor $D_i$, lies on the Givental cone of $(X,D)$; the proof is only sketched and the mirror maps are not explicitly constructed, so if this bridge breaks, the Gamma-identity computation in Sections 12--13 collapses.","fun_headline_variants_meta":{"raw":{"variants":["Theta functions unlock mirror Gamma conjecture for Fano pairs","Gamma conjecture proven for Fano via theta-function mirror","Mirror Gamma conjecture for Fano varieties from theta functions","Higher-degree quantum cohomology powers Gamma conjecture proof"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000151,"raw_usage":{"total_tokens":1024,"prompt_tokens":715,"completion_tokens":309,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":459,"completion_tokens_details":{"reasoning_tokens":245}},"tokens_in":459,"tokens_out":309,"duration_ms":3701,"temperature":1.0,"reasoning_tokens":245,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T22:33:47.499975+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a Fano threefold $X$ with an snc anticanonical divisor having a non-nef component $D_i$, and compute the first nonzero $x^{\\vec d}x_{\\vec b}$-coefficient of the two sides of Corollary 3.14; if the coefficient predicted by the $I$-function after any valid mirror map in $D_i$ differs from the corresponding relative invariant of $(X,D)$, the theorem fails, and a concrete starting point is the one-point invariant $\\langle [pt]\\psi^k\\rangle^X_{0,1,\\beta}$ for a class $\\beta$ with $D_i\\cdot\\beta<0$.","supporting_citations":[],"review_version":1}