Relative mirror symmetry, theta functions and the Gamma conjecture

Fenglong You

REVIEW 3 major objections 4 minor 2 references

Relative mirror symmetry, theta functions and the Gamma conjecture

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2508.06750 v1 pith:T7P3SJLX submitted 2025-08-08 math.AG

Fenglong You This is my paper · ORCID

classification math.AG MSC 14N3514J3353D45

keywords mirrorsymmetricGammaconjecturerelativesymmetrythetafunctionsFanovarietiesGromov-WittentheorylogCalabi-Yaupairsrootstacksquantumperiods

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

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 $X$.

What carries the argument

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

What would settle it

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 $D_i\cdot\beta<0$.

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Extended reading notes

Core claim

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 $$ \int_{\Gamma_R} \check{\varphi}_\tau(-z) $e^{{-W_\tau/z}}$\,\omega = \int_X \bigl($z^{{c_1}}$ $z^{{\deg/2}}$ J_X(\tau_{0,2},-z)\varphi\bigr)\cup \widehat\Gamma_X, $$ 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

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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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

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.

But 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.

There 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.

Still, 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.

For 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.

Referee Report

3 major / 4 minor

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.

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 (3)

[Corollary 3.14 and Section 3.3] 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 13, especially the final paragraph] 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
[Sections 6, 11, and 13] 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

minor comments (4)

[Proposition 7.1] 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.
[Remark 4.1 and Section 5] 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 11] 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.
[Theorem 10.1 proof] 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.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the derivation is carried by an independent relative mirror theorem; the Gamma-class identification is an unproven gap, not a definitional reduction.

full rationale

The paper's central claim (Theorem 1.9) is not shown to be circular in the sense of the rubric. The relative mirror theorem (Corollary 3.14) is stated and proved separately via degeneration, and it is not the same as the Gamma conjecture. The mirror functions and theta functions in Definitions 5.6 and 6.1 are generating series of relative/orbifold invariants, not objects defined so that the target Gamma-class identity holds by construction. The computation in Section 12 expresses relative invariants through the S-extended I-function, which is a hypergeometric modification of the J-function of X; the Gamma class is not inserted there. The real weakness is the final step in Section 13: after computing a fixed-cone contribution with factors Γ(1+D_i^{λ_i}(σ)), the text says the oscillatory integral is a solution of the same differential equation and 'We must have' the desired identity with \hatΓ_X. That is an omitted proof (the cone-wise Gamma factors are not summed/localized to Chern roots, and the linear combination coefficients of the J-basis are not determined). However, an omitted proof is a correctness gap, not a circular reduction: the target identity is not used to define the inputs, and the relative mirror theorem is not derived from the Gamma conjecture. The self-citations to [TY23a], [TY23b], [You24a] are load-bearing but are published prior results with independent content, not a self-citation chain that forces the Gamma conjecture. Accordingly, the circularity score is low.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper's central claim depends on the geometric hypothesis on D, the validity of orbifold/root-stack invariants and the relative-orbifold correspondence, and the correctness of the degeneration argument in the non-nef case. These are largely imported from prior work by the author and collaborators, and the non-nef degeneration proof is only sketched.

assumptions (4)

domain assumption Existence of an snc anticanonical divisor D with pure-dimensional dual intersection complex B such that every stratum of D contains a zero-dimensional stratum.
Main geometric hypothesis. Invoked in Section 1.2 and Section 4.
domain assumption The relative-orbifold correspondence and degeneration formula are valid in the non-nef setting.
Used in the proof of the relative mirror theorem (Section 3.3); imported from [ACW17], [TY20], [FWY20].
domain assumption Mid-age invariants are well-defined and independent of rooting parameters for large r (You24a Theorem 5.9).
Basis for the theta functions with mid-ages, Section 2.2 and Section 5.
standard math Orbifold structure constants define a commutative associative algebra on R_I (TY23a Theorem 37).
Used to define the mirror family, Section 4.

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Pith. "Pith review of Relative mirror symmetry, theta functions and the Gamma conjecture." pith.science (2026). https://pith.science/paper/T7P3SJLX

@misc{pith2026250806750,
  author       = {Pith},
  title        = {Pith review of: Relative mirror symmetry, theta functions and the Gamma conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T7P3SJLX}},
  note         = {Machine review of arXiv:2508.06750}
}

abstract

Let $X$ be a Fano variety, and $D\subset X$ be an snc anticanonical divisor. We study relative mirror symmetry for the log Calabi--Yau pair $(X,D)$. (1) We prove a relative mirror theorem for snc pairs without assuming the divisors are nef. (2) We study theta functions associated with the pair $(X,D)$. (3) We introduce functions on the mirror that are obtained from the higher-degree part of the big relative quantum cohomology. As an application, we use these new ingredients in relative mirror symmetry to prove a version of the mirror symmetric Gamma conjecture for $X$ for $\mathcal O_X$ and $\mathcal O_{\operatorname{pt}}$ in this setting, where the Landau--Ginzburg potential is defined as a sum of theta functions.

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Works this paper leans on

2 extracted references · 2 canonical work pages

[307]

↑8, 13, 21, 27 [FWY21] H. Fan, L. Wu, and F. You,Higher genus relative Gromov-Witten theory and double ramification cycles , J. Lond. Math. Soc. (2) 103 (2021), no. 4, 1547–1576. MR4273479 ↑27 [FWZ23] Bohan Fang, Junxiao Wang, and Yan Zhou, Mirror symmetric Gamma conjecture for del Pezzo surfaces , arXiv preprint arXiv:2309.02154 (2023). ↑3 [GGI16] Sergey...

work page Pith review arXiv 2021
[500]

↑13, 21 [AFW25] Konstantin Aleshkin, Bohan Fang, and Junxiao Wang, Mirror symmet- ric Gamma conjecture for toric GIT quotients via Fourier transform , arXiv preprint arXiv:2501.14222 (2025). ↑3 RELATIVE MS, THETA AND GAMMA 71 [AGIS20] Mohammed Abouzaid, Sheel Ganatra, Hiroshi Iritani, and Nick Sheri- dan, The Gamma and Strominger-Yau-Zaslow conjectures: a...

work page arXiv 2025

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Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [307]

↑8, 13, 21, 27 [FWY21] H. Fan, L. Wu, and F. You,Higher genus relative Gromov-Witten theory and double ramification cycles , J. Lond. Math. Soc. (2) 103 (2021), no. 4, 1547–1576. MR4273479 ↑27 [FWZ23] Bohan Fang, Junxiao Wang, and Yan Zhou, Mirror symmetric Gamma conjecture for del Pezzo surfaces , arXiv preprint arXiv:2309.02154 (2023). ↑3 [GGI16] Sergey...

work page Pith review arXiv 2021

[2] [500]

↑13, 21 [AFW25] Konstantin Aleshkin, Bohan Fang, and Junxiao Wang, Mirror symmet- ric Gamma conjecture for toric GIT quotients via Fourier transform , arXiv preprint arXiv:2501.14222 (2025). ↑3 RELATIVE MS, THETA AND GAMMA 71 [AGIS20] Mohammed Abouzaid, Sheel Ganatra, Hiroshi Iritani, and Nick Sheri- dan, The Gamma and Strominger-Yau-Zaslow conjectures: a...

work page arXiv 2025