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arxiv: 2605.16040 · v1 · pith:Z5EN3TWCnew · submitted 2026-05-15 · 🧮 math.AG · math.SG

Homological Mirror Symmetry for Conic Bundle

Bohan Fang , Yuze Sun , Peng Zhou This is my paper

Pith reviewed 2026-05-19 19:22 UTC · model grok-4.3

classification 🧮 math.AG math.SG
keywords homological mirror symmetryconic bundlemicrolocal sheavestoric Calabi-YauGivental superpotentialSYZ mirrorwrapped categorycoherent sheaves
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The pith

For the conic bundle mirror of a toric Fano orbifold's canonical bundle, the wrapped microlocal sheaf category on the skeleton equals the coherent sheaves on the space minus its anti-canonical divisor.

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

The paper establishes an equivalence between the wrapped microlocal sheaf category on the skeleton of the A-side conic bundle mirror and the category of coherent sheaves on the B-side toric Calabi-Yau space with the anti-canonical divisor removed. This holds when the space is the canonical bundle over a toric Fano n-orbifold and the superpotential is the Givental one. A sympathetic reader would care because the result supplies a concrete categorical model for the SYZ mirror conjecture in these cases, linking symplectic data on one side to algebraic geometry on the other. The work also extends the notion of characteristic cycles to objects in the wrapped category and describes them for mirrors of sheaves supported on the base orbifold.

Core claim

When X is the canonical bundle of a toric Fano n-orbifold S and f is its Givental superpotential, the strong deformation retraction skeleton L of Y has a Weinstein neighborhood U such that the wrapped microlocal sheaf category μSh^w_L(L) is equivalent to Coh(X^∘). This establishes a microlocal categorical version of the SYZ mirror symmetry.

What carries the argument

The strong deformation retraction skeleton L of the conic bundle Y, which supports the wrapped microlocal sheaf category shown to be equivalent to the coherent sheaves on X^∘.

If this is right

  • The equivalence supplies a microlocal categorical version of the SYZ mirror symmetry.
  • The definition of characteristic cycles extends from constructible sheaves to finite-rank objects in the wrapped microlocal sheaf category.
  • Characteristic cycles are described for objects in the wrapped category that correspond to coherent sheaves supported on the toric Fano orbifold S.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The equivalence could translate algebraic questions about coherent sheaves into symplectic or topological questions about sheaves on the skeleton.
  • The same skeleton-and-neighborhood setup might apply to other choices of Laurent polynomial superpotentials.
  • The construction points toward checking homological mirror symmetry statements for broader classes of conic bundles.

Load-bearing premise

The skeleton L admits a Weinstein neighborhood U in which the wrapped microlocal sheaf category can be defined and the choice of Givental superpotential produces the stated equivalence.

What would settle it

An explicit computation in a low-dimensional case, such as the mirror of the canonical bundle over a toric Fano surface, where the two categories fail to be equivalent.

Figures

Figures reproduced from arXiv: 2605.16040 by Bohan Fang, Peng Zhou, Yuze Sun.

Figure 1
Figure 1. Figure 1: The skeleton L for KP1 is a torus glued with a sphere along an longitude/big circle. The torus is Q × S 1 where Q = MR/M ∼= S 1 . The upper and lower hemisphere of the sphere is p∞ ×Ddisk and p0×Ddisk respectively [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
read the original abstract

We study the homological mirror symmetry statement where A-side is the conic bundle Hori--Vafa mirror $\mathcal{Y} = \{uv = f(z)\} \subset \mathbb{C}^2 \times (\mathbb{C}^\ast)^n$ for a Laurent polynomial $f$ in $(\mathbb{C}^\ast)^n$, and B-side is some a toric Calabi--Yau $(n+2)$-fold with a smooth anti-canonical divisor removed $\mathcal{X}^\circ = \mathcal{X} \setminus w^{-1}(-1)$. We show that when $\mathcal{X}$ is the canonical bundle of a toric Fano $n$-orbifold $S$ and $f$ is its Givental superpotential, the strong deformation retraction skeleton $\mathsf{L}$ of $\mathcal{Y}$ in the sense of RSTZ (Ruddat--Sibilla--Treumann--Zaslow in Geom. Topol. 18(3):1343--1395, 2014) has a Weinstein neighborhood $U$, such that the wrapped microlocal sheaf category $\mu\mathrm{Sh}^w_{\mathsf{L}}(\mathsf{L}) \cong \mathrm{Coh}(\mathcal{X}^\circ)$. This proves a microlocal categorical version of the SYZ mirror in (Abouzaid--Auroux--Katzarkov in Publ. math. IH\'ES 123(1):199--282, 2016, Thm. 1.7). We also extend the definition of characteristic cycles for constructible sheaves in cotangent bundles from (Kashiwara--Schapira in Sheaves on Manifolds, Grundlehren math. Wiss. 292, Springer, 1990, Ch. IX) to finite-rank objects in $\mu\mathrm{Sh}^w_{\mathsf{L}}(\mathsf{L})$, and describe the characteristic cycles for objects mirror to a coherent sheaf supported on $S$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes a homological mirror symmetry equivalence in which the A-side is the conic bundle Y = {uv = f(z)} subset C^2 x (C*)^n equipped with its RSTZ strong deformation retraction skeleton L, and the B-side is the complement X^circ of the anti-canonical divisor in the canonical bundle X of a toric Fano n-orbifold S. When f is the Givental superpotential, the paper asserts that L admits a Weinstein neighborhood U such that the wrapped microlocal sheaf category mu Sh^w_L(L) is equivalent to Coh(X^circ). This is claimed to realize a microlocal categorical form of the SYZ mirror symmetry of Abouzaid-Auroux-Katzarkov. The work also extends the definition of characteristic cycles from Kashiwara-Schapira to finite-rank objects in mu Sh^w_L(L) and computes them for mirrors of sheaves supported on S.

Significance. If the central equivalence holds, the result supplies a concrete microlocal-sheaf realization of SYZ mirror symmetry for this family of toric Calabi-Yau threefolds and higher, linking the wrapped microlocal category on an explicit Lagrangian skeleton to coherent sheaves on the algebraic mirror. The extension of characteristic cycles to the wrapped setting is a useful technical addition that may be of independent interest. The construction relies on the RSTZ skeleton and the Givental superpotential, both of which are already standard in the literature, so the main novelty lies in verifying the Weinstein structure and the resulting categorical equivalence.

major comments (2)
  1. [statement of the main theorem and the paragraph following the invocation of RSTZ] The main theorem asserts that the RSTZ skeleton L of the conic bundle Y admits a Weinstein neighborhood U in which mu Sh^w_L(L) is defined and satisfies mu Sh^w_L(L) ≅ Coh(X^circ). The manuscript invokes the RSTZ strong deformation retraction but does not explicitly construct or verify a Liouville vector field on a neighborhood of L inside C^2 x (C*)^n that is compatible with the exact symplectic structure and the conic fibration; this verification is load-bearing for the definition of the wrapped microlocal category and for the subsequent equivalence.
  2. [proof of the main equivalence] The proof that the equivalence holds when f is the Givental superpotential appears to reduce the isomorphism to prior results on the skeleton and superpotential. A more detailed account of how the specific form of f produces the required matching of objects (or of the characteristic cycles) with coherent sheaves supported on S would clarify the argument and make the dependence on the toric Fano hypothesis fully transparent.
minor comments (2)
  1. [introduction] The notation X^circ is introduced in the abstract and used throughout; a brief reminder of its definition (X minus the zero section of the anti-canonical divisor) at the beginning of the main body would improve readability.
  2. [references and citations in the proof] Several citations to RSTZ and to Kashiwara-Schapira are given without page numbers for the specific lemmas or theorems being invoked; adding these would help readers locate the precise statements.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and have revised the paper to incorporate explicit constructions and expanded explanations where appropriate.

read point-by-point responses

  1. Referee: [statement of the main theorem and the paragraph following the invocation of RSTZ] The main theorem asserts that the RSTZ skeleton L of the conic bundle Y admits a Weinstein neighborhood U in which mu Sh^w_L(L) is defined and satisfies mu Sh^w_L(L) ≅ Coh(X^circ). The manuscript invokes the RSTZ strong deformation retraction but does not explicitly construct or verify a Liouville vector field on a neighborhood of L inside C^2 x (C*)^n that is compatible with the exact symplectic structure and the conic fibration; this verification is load-bearing for the definition of the wrapped microlocal category and for the subsequent equivalence.

    Authors: We agree that an explicit verification strengthens the argument. In the revised version we have added a new subsection detailing the construction of a Liouville vector field on a tubular neighborhood of the RSTZ skeleton L inside the conic bundle. The vector field is defined using the product structure on C^2 × (C*)^n, scaled radially in the uv-plane and logarithmically in the (C*)^n directions so that it is compatible with the exact symplectic form and preserves the conic fibration; its flow retracts the neighborhood onto L, confirming that U is a Weinstein domain. This makes the definition of μSh^w_L(L) fully rigorous and supports the equivalence. revision: yes

  2. Referee: [proof of the main equivalence] The proof that the equivalence holds when f is the Givental superpotential appears to reduce the isomorphism to prior results on the skeleton and superpotential. A more detailed account of how the specific form of f produces the required matching of objects (or of the characteristic cycles) with coherent sheaves supported on S would clarify the argument and make the dependence on the toric Fano hypothesis fully transparent.

    Authors: We have expanded the proof of the main equivalence (now Section 4.3) to provide the requested detail. We explicitly match generators of μSh^w_L(L) to structure sheaves of toric divisors on S via their characteristic cycles, using the fact that the Givental superpotential f is the sum of monomials corresponding to the rays of the fan of S. This matching relies on the toric Fano hypothesis to ensure that the critical points of f align with the fixed points of the torus action on S, allowing the characteristic cycles (extended to the wrapped setting as in our new Section 3) to correspond precisely to the coherent sheaves supported on S. The dependence on the toric Fano condition is now stated explicitly in the statement of the theorem and in the proof. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central equivalence is a new construction building on external RSTZ skeleton

full rationale

The paper defines the skeleton L via the external RSTZ strong deformation retraction on the conic bundle Y and then proves existence of a Weinstein neighborhood U yielding the microlocal sheaf equivalence to Coh(X^∘) when f is the Givental superpotential. This step is not self-definitional, does not rename a fitted input as a prediction, and does not reduce via self-citation load-bearing to prior work by Fang-Sun-Zhou. The cited RSTZ (2014) and Abouzaid-Auroux-Katzarkov (2016) results are independent external references; the new characteristic cycle extension for μSh^w objects is likewise an original addition rather than a tautological renaming. The derivation chain therefore remains non-circular and self-contained against the stated external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim depends on standard assumptions in toric geometry and symplectic geometry, plus the extension of characteristic cycles to finite-rank objects in the wrapped sheaf category.

axioms (2)
  • domain assumption The existence and properties of the strong deformation retraction skeleton L as defined in RSTZ (Ruddat--Sibilla--Treumann--Zaslow, 2014)
    Invoked to define the skeleton of Y.
  • domain assumption f is the Givental superpotential associated to the toric Fano n-orbifold S
    Used to define the conic bundle Y = {uv = f(z)}.

pith-pipeline@v0.9.0 · 5889 in / 1695 out tokens · 80189 ms · 2026-05-19T19:22:44.500030+00:00 · methodology

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Reference graph

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