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Toric Mirror Symmetry for Homotopy Theorists

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Pith's one-line read Toric mirror symmetry holds over the sphere spectrum.

desk verdict The spectral lift and monoidal structures are real, but the proof of the image theorem uses a false convolution-invertibility claim for polytope sheaves, so the main result is unproven as written. read the letter →

arxiv 2501.06649 v1 pith:Y2TOYEY4 submitted 2025-01-11 math.AG math.ATmath.CT

Qingyuan Bai , Yuxuan Hu This is my paper · ORCID
classification math.AGmath.ATmath.CT MSC 14M2514F0855P42
keywords toricmirrorsymmetrycoherent-constructiblecorrespondencespectralalgebraicgeometryconstructiblesheavesofspectrasingularsupportFLTZskeletonde-equivariantizationBeilinsontheorem
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 establishes a spectral version of toric mirror symmetry: the classical dictionary between torus-equivariant sheaves on a toric variety and constructible sheaves on a real vector space continues to work when the ground ring is upgraded from the complex numbers to the sphere spectrum, the universal ring in stable homotopy theory. The main theorem constructs a fully faithful functor that preserves tensor products, from quasi-coherent sheaves on the quotient stack of a flat toric scheme over the sphere spectrum to sheaves of spectra on a real vector space, and it describes the image exactly as the sheaves that are constructible for the hyperplane stratification attached to the fan and have singular support inside the associated FLTZ skeleton. If correct, this lifts the known coherent-constructible correspondence to spectral algebraic geometry and supplies a genuine instance of mirror symmetry over the sphere spectrum, where the symmetric monoidal structure and functoriality are new even over a field.

What carries the argument

The machinery has two independent legs that are glued cone-by-cone. On the coherent side, for each cone $\sigma$ the category $\mathrm{QCoh}([X_\sigma/T])$ is shown to be equivalent to the presheaf category $\mathrm{Fun}(\Theta(\sigma)^{\mathrm{op}},\mathrm{Sp})$, where $\Theta(\sigma)$ is the poset of integral translates $m+\sigma^\vee$ of the dual cone; the proof follows the geometry-of-filtrations strategy and gives a symmetric monoidal equivalence. On the constructible side, a relative-homology functor $\Gamma_{M_{\mathbb{R}}}$ built from the six-functor formalism sends each translate $m+\sigma^\vee$ to the sheaf $\omega_{m+\sigma^\vee}$, and after left Kan extension produces a symmetric monoidal functor $\Psi_\sigma$ into the category of modules over the idempotent algebra $\omega_{\sigma^\vee}$ in $\mathrm{Shv}(M_{\mathbb{R}};\mathrm{Sp})$. Taking limits over the fan and invoking a sheaf-theoretic counterpart of Zariski descent assembles these into $\kappa$. The image characterization is carried by the probing sheaves $\omega(D_x)$ associated to divisors at points $x$, together with the non-characteristic deformation lemma, which together show that every constructible sheaf with singular support in $\Lambda_\Sigma$ lies in the image.

What would settle it

Run the corepresentability calculation of Theorem 5.3.3 on a concrete constructible sheaf of spectra: take $F$ to be the extension-by-zero of the constant spectrum along the open half-plane $\{x_1>0\}$ in $\mathbb{R}^2$ with the standard grid FLTZ stratification, and check whether $\mathrm{map}(\omega(D_x),F)[n]$ is isomorphic to the stalk $F_x$ for a point $x$ on the boundary hyperplane. A single $x$ where the two spectra are not equivalent would break the equality $\mathrm{Im}(\kappa)=\mathrm{Shv}_{\Lambda_\Sigma}$, since these probing sheaves are what force vanishing of the right orthogonal.

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Extended reading notes

Core claim

The central claim is Theorem A: for a smooth projective fan $\Sigma$, there is a fully faithful symmetric monoidal functor $\kappa$ from $\mathrm{QCoh}([X_\Sigma/T])$, the quasi-coherent sheaves on the flat toric scheme $X_\Sigma$ over the sphere spectrum modulo the flat torus $T$, to $\mathrm{Shv}(M_{\mathbb{R}};\mathrm{Sp})$, the category of sheaves of spectra on the real vector space $M_{\mathbb{R}}$. The image is characterized explicitly: it is the full subcategory of sheaves constructible for the FLTZ stratification $\mathcal{S}_\Sigma$ (the affine hyperplane arrangement determined by the fan's one-dimensional cones) and with singular support contained in the FLTZ skeleton $\Lambda_\Sigma$ (the associated conic Lagrangian subset of the cotangent bundle). This is a spectral lift of the classical coherent-constructible correspondence, and the paper derives from it a compatible action of the classifying stack $BT$ (Theorem B), a non-equivariant version on the real torus by de-equivariantization (Theorem C), and a proof of Beilinson's linear-algebraic description of quasi-coherent sheaves on projective spaces over the sphere spectrum (Theorem D).

Load-bearing premise

The load-bearing premise is that classical singular-support and non-characteristic-deformation results, written for bounded derived categories of sheaves of complex vector spaces, extend unchanged to the large category of sheaves of spectra; the paper notes in Remark 5.1.1 that the reference it uses is 'not directly applicable' and that the needed results would be verified with the same proof in future work.

Editorial extensions

If this is right

  • Over any connective commutative ring spectrum $R$, the same construction gives a fully faithful symmetric monoidal coherent-constructible correspondence, and over $\mathbb{C}$ it recovers a large-category version of the classical theorem.
  • The de-equivariantized functor identifies quasi-coherent sheaves on the flat toric scheme $X_\Sigma$ with constructible sheaves of spectra on the real torus $M_{\mathbb{R}}/M$, making the singular-support description global.
  • Applying exodromy to the non-equivariant equivalence recovers Beilinson's description of $\mathrm{QCoh}(\mathbb{P}^n_S)$, in particular $\mathrm{QCoh}(\mathbb{P}^1_S) \simeq \mathrm{Fun}(\bullet \Rightarrow \bullet; \mathrm{Sp})$.
  • The image category $\mathrm{Shv}_{\Lambda_\Sigma}(M_{\mathbb{R}};\mathrm{Sp})$ is compactly generated, with explicit generators $\omega_{m+nP}$ attached to a moment polytope $P$, giving a concrete handle on mapping spectra in the mirror category.
  • The relative toric construction formalizes base change along symmetric monoidal functors out of $\mathrm{Fun}(M,\mathrm{Sp})$, recovering toric fibrations over schemes equipped with line bundles.

Reading between the lines

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

  • The paper leaves implicit that, if the correspondence extends from smooth projective fans to more general fans, the singular-support description would supply an $\mathbb{S}$-valued analogue of microlocal sheaf categories; one test is whether the FLTZ skeleton continues to produce compact generators for non-projective fans.
  • The probing-sheaf corepresentability result suggests a general recipe for finding compact generators of categories of sheaves with prescribed singular support: look for objects that corepresent microlocal stalk functors, then test whether the same recipe works on the torus quotient $M_{\mathbb{R}}/M$ rather than on $M_{\mathbb{R}}$.
  • One natural next step, which the paper notes is expected but does not carry out, is to interpret the relative toric construction as a category of twisted sheaves on the torus valued in a local system of categories; if that interpretation holds, it would unify the equivariant and non-equivariant versions of the theorem.
  • A direct proof of Beilinson's quiver presentation for $\mathbb{P}^n_S$ might be obtainable purely from exodromy together with a well-behaved theory of singular support for spectra, without first passing through the full equivariant correspondence.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper constructs a spectral lift of the toric coherent-constructible correspondence of Fang--Liu--Treumann--Zaslow. For a smooth projective fan Σ, it builds a fully faithful symmetric monoidal functor κ : QCoh([X_Σ/T]) → Shv(M_R; Sp) and claims that its image is exactly the category Shv_{Λ_Σ}(M_R; Sp) of sheaves constructible for the FLTZ stratification and with singular support contained in the FLTZ skeleton. The proof proceeds by a combinatorial model Θ(σ), an equivalence with quasi-coherent sheaves on affine toric quotient stacks, a sheaf-theoretic comparison via the relative homology functor Γ_{M_R}, gluing by idempotent algebras, and a singular-support argument adapted from work of Zhou. The final sections derive a de-equivariantized statement for QCoh(X_Σ), identify it with constructible sheaves on the torus M_R/M, and deduce Beilinson's description for projective spaces over the sphere spectrum.

Significance. If the main theorem is established, the paper would provide a genuinely new spectral and symmetric-monoidal enhancement of the classical FLTZ correspondence, and the functoriality and de-equivariantization results would be new even over a field. The construction of the symmetric monoidal structure via Day convolution and the relative homology functor is carefully written and is a useful contribution in its own right. The paper is also transparent about provenance: Remark 1.2.1 attributes the main strategy to earlier work, and the appendices supply technical categorical lemmas. However, the central proof has a load-bearing error: Proposition 5.3.15 uses a convolution inverse of the moment-polytope sheaf ω_P that does not exist under the paper's own definitions. Because Corollary 5.3.4 is exactly the image characterization asserted in Theorem A, and because Theorem C and the Beilinson application depend on it, the main results are not established by the present proof. The stress-test concern about convolution invertibility therefore lands.

major comments (2)
  1. [§5.3, Propositions 5.3.12 and 5.3.15] The proof of Corollary 5.3.4 uses the assertion that the moment-polytope sheaf ω_P is convolution-invertible in Shv(M_R; Sp). This assertion is false. By Lemma 4.3.1, ω_P ≃ S_{P°}[n] for a closed polyhedral set P with interior P°, and by Proposition 4.1.3, S_U ∗ S_V ≃ S_{U+V}[-n] for open polyhedral sets. Hence ω_P ∗ ω_P ≃ S_{2P°}[n], which is not the convolution unit (the skyscraper sheaf at 0). Moreover, for any sheaf L, the support of ω_P ∗ L is contained in the closure of P + supp(L), which has non-empty interior because P does; it cannot equal the unit, whose support is {0}. Therefore the functor (−) ∗ ω_P^{-1} used in Proposition 5.3.15 does not exist, and the equivalences G(x + εP°) ≃ map(ω(D_x), G)[n] in Proposition 5.3.12 are not justified. This is the step that upgrades the inclusion Im(κ) ⊆ Shv_{Λ_Σ} to equality, so Corollary 5.3.4, Theorem A, and the later consequences Theorem C and Example 6.2.1 are not established by the present proof.
  2. [§5.1, Remark 5.1.1 and §4.4, Remark 4.4.8] The definition of Shv_{Λ_Σ}(M_R; Sp) and the proof of Theorem 5.3.3 rely on extending classical singular-support results and the exodromy theorem from sheaves of spaces or from the bounded derived category to the large ∞-category of sheaves of spectra. Remark 5.1.1 states that Kashiwara--Schapira is 'not directly applicable' and that the needed facts 'could be verified with the same proof' but are deferred to future work; Remark 4.4.8 states that the exodromy theorem of Clausen--Jansen extends to spectra 'since the proof works verbatim'. These extensions are load-bearing for the explicit description of Im(κ): without them, the Fourier-Sato computation in Lemma 5.3.10 and the stalk-corepresentation theorem do not have a proven spectral counterpart. The manuscript should either supply these proofs or point to a precise reference where the spectral version is established.
minor comments (4)
  1. [Abstract] There is a typo: 'homolgoical' should be 'homological'.
  2. [Notation 1.4.4] The second sentence says 'We write Funlax⊗(C, D) for the category of symmetric monoidal functors'; it should say 'lax symmetric monoidal functors'.
  3. [§5.3, proof of Proposition 5.3.12] The displayed formula '(G∗ωP(P◦X))[−n]' has mismatched parentheses and an inconsistent use of X versus x; it should presumably read '((G ∗ ω_P)(P_x°))[−n]'.
  4. [§5.3, Proposition 5.3.15] In the sentence 'for each F ∈ Shv(M_R; Sp), F ∗ ω_P^{-1} ∈ C = Im(κ)', the quantifier should be over F ∈ Shv_{Λ_Σ}(M_R; Sp), matching the stated functor from Shv_{Λ_Σ} to C; as written it is inconsistent with the preceding line.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem A is a genuine spectral lift built from externally credited inputs, not a re-derivation of its own conclusions.

full rationale

I walked the paper's derivation chain. The coherent-combinatorial equivalence (Prop. 3.3.1) is proved by adapting [29] and a Beck-Chevalley argument; the sheaf-side functor ΨΣ (Prop. 4.3.4) is explicitly constructed from ΓMR and idempotent algebras; full faithfulness (Cor. 4.4.19) uses exodromy; gluing along idempotents (Prop. 4.5.4) is proved by stalk computations; and the image characterization (Cor. 5.3.4) follows the method of [42], which the paper credits explicitly. None of these steps is an instance of a fitted parameter later renamed a prediction, and no load-bearing uniqueness theorem is imported from the authors' own prior work. The spectral lift and symmetric monoidal structure are new even over a field, so the central claim has independent content beyond its inputs. The self-references [2] and [19] are illustrative or applications, not load-bearing. I note two correctness concerns that are not circularity: the proof of Cor. 5.3.4 uses ω_P as convolution-invertible in Prop. 5.3.15, which is false by the paper's own formulas (Prop. 4.1.3 and Lemma 4.3.1 give ω_P ∗ ω_P ≃ S_{2P°}[n], not the unit), and Remarks 5.1.1 and 4.4.8 explicitly defer the spectral extension of classical singular-support and exodromy results. These are proof gaps or missing supports, not circular reductions. Under the hard rules, they do not raise the circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the standard six-functor formalism and several extension-by-coherence assertions from spectra of the classical singular-support and exodromy theorems. These are flagged in the paper as requiring verification in future work, making them the main intellectual debt the reader is asked to accept.

assumptions (5)
  • domain assumption There exists a six-functor formalism for sheaves of spectra on locally compact Hausdorff spaces, including a lax symmetric monoidal structure on 'direct image with compact support'.
    The constructible side of the paper, including the convolution product (Construction 4.1.2) and the relative homology functor ΓMR (Definition 4.2.11), relies on the six-functor formalism from [41] and Lurie's Higher Algebra.
  • domain assumption The exodromy equivalence for constructible sheaves of spaces extends without modification to sheaves of spectra.
    Theorem 4.4.7 (from [7]) is stated for Spc-valued sheaves; Remark 4.4.8 says 'the proof works verbatim for Sp coefficient', which is asserted rather than proved.
  • domain assumption The classical singular support theory and non-characteristic deformation lemma for bounded derived categories of sheaves of complex vector spaces remain valid for the large category of sheaves of spectra with the Fourier-Sato definition of singular support.
    Remark 5.1.1 warns that [22] is not directly applicable and that the needed facts 'could be verified with the same proof' and 'we will revisit these facts in future work'. This assumption underlies the proof of the image characterization in Section 5.3.
  • domain assumption The fan Σ is smooth and projective, so a moment polytope exists and the combinatorial identities σ∨ + τ∨ = (σ ∩ τ)∨ hold.
    Theorems A, C, and D are stated under this hypothesis; the gluing of idempotents (Proposition 4.5.4) and the generating polytopes (Section 5.2) depend on it.
  • standard math The stacks [Xσ/T] and BT are perfect, and the base change and relative tensor product description of QCoh for perfect stacks applies in the spectral setting.
    Used in Theorem 6.1.2; the proof cites SAG Corollary 9.4.2.3 and checks the perfect stack conditions.

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Pith. "Pith review of Toric Mirror Symmetry for Homotopy Theorists." pith.science (2026). https://pith.science/paper/Y2TOYEY4

@misc{pith2026250106649,
  author       = {Pith},
  title        = {Pith review of: Toric Mirror Symmetry for Homotopy Theorists},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y2TOYEY4}},
  note         = {Machine review of arXiv:2501.06649}
}
abstract

We construct functors sending torus-equivariant quasi-coherent sheaves on toric schemes over the sphere spectrum to constructible sheaves of spectra on real vector spaces. This provides a spectral lift of the toric homolgoical mirror symmetry theorem of Fang-Liu-Treumann-Zaslow (arXiv:1007.0053). Along the way, we obtain symmetric monoidal structures and functoriality results concerning those functors, which are new even over a field $k$. We also explain how the `non-equivariant' version of the theorem would follow from this functoriality via the de-equivariantization technique. As a concrete application, we obtain an alternative proof of Beilinson's linear algebraic description of quasi-coherent sheaves on projective spaces with spectral coefficients.

Figures

Figures reproduced from arXiv: 2501.06649 by the authors.

Figure 1
Figure 1. An illustration of a sheaf in ShvΛP2 (R2/Z2 ), drawn in a fundamental domain of R2/Z2 . The short directional strokes—drawn along the edges and diagonal, fanning out at the cor￾ners—schematically represent ΛP2 in each cotangent fiber. Three distinguished stalks and ways that they are allowed to exit are drawn. 68 [PITH_FULL_IMAGE:figures/full_fig_p068_1.png] view at source ↗

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  1. An obstruction to lifting schemes to spectral schemes

    math.AG 2026-07 accept novelty 6.5 of 10

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.