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arxiv: 2602.04866 · v2 · pith:JBLILUVBnew · submitted 2026-02-04 · 🧮 math.SG · math.AG

Homological mirror symmetry for orbifold log Calabi-Yau surfaces

Bogdan Simeonov This is my paper

Pith reviewed 2026-05-21 13:39 UTC · model grok-4.3

classification 🧮 math.SG math.AG
keywords homological mirror symmetryorbifold log Calabi-YauLefschetz fibrationMcKay correspondencedel Pezzo surfacesrational surfacesstacky anticanonical divisorlarge complex structure limit
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The pith

Constructing an abstract Lefschetz fibration proves homological mirror symmetry for orbifold log Calabi-Yau surfaces.

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

The paper establishes homological mirror symmetry for orbifold log Calabi-Yau surfaces at the large complex structure limit. It does so by associating an abstract Lefschetz fibration to each pair consisting of a projective rational surface with isolated cyclic quotient orbifold points and a stacky anticanonical divisor. If correct, this provides a systematic way to match the derived categories on both sides of the mirror for these singular geometries. Sympathetic readers would care because it enlarges the class of spaces where mirror symmetry can be verified through fibration techniques.

Core claim

We prove homological mirror symmetry for orbifold log Calabi-Yau surfaces at the large complex structure limit by constructing an abstract Lefschetz fibration associated to each pair (X, D) with X a projective rational surface with isolated cyclic quotient orbifold points and D a stacky anticanonical divisor. We describe a Lefschetz stabilization procedure which on the mirror corresponds to the special McKay correspondence of Ishii and Ueda. We also relate the abstract construction to an explicit Laurent polynomial mirror for a family of orbifold del Pezzo surfaces.

What carries the argument

The abstract Lefschetz fibration associated to the pair (X, D), which encodes the homological data at the large complex structure limit and supports the stabilization procedure.

Load-bearing premise

That an abstract Lefschetz fibration can be associated to every such pair in a manner that encodes the homological data at the large complex structure limit, with the stabilization corresponding precisely to the special McKay correspondence.

What would settle it

Take a specific family of orbifold del Pezzo surfaces, construct both the abstract Lefschetz fibration and the explicit Laurent polynomial mirror, and compare their associated homological invariants or categories; disagreement in the data would show the proof does not hold.

read the original abstract

We prove homological mirror symmetry for orbifold log Calabi-Yau surfaces at the large complex structure limit by constructing an abstract Lefschetz fibration associated to each pair $(\mathcal{X},\mathcal{D})$ with $\mathcal{X}$ a projective rational surface with isolated cyclic quotient orbifold points and $\mathcal{D}$ a stacky anticanonical divisor. We describe a Lefschetz stabilization procedure which, on the mirror, corresponds to the special McKay correspondence of Ishii and Ueda arXiv:1104.2381v2 [math.AG]. Moreover, we relate our abstract construction to an explicit Laurent polynomial mirror in an example consisting of a family of orbifold del Pezzo surfaces.

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

0 major / 2 minor

Summary. The paper proves homological mirror symmetry for orbifold log Calabi-Yau surfaces at the large complex structure limit by constructing an abstract Lefschetz fibration associated to each pair (X,D) with X a projective rational surface with isolated cyclic quotient orbifold points and D a stacky anticanonical divisor. It describes a Lefschetz stabilization procedure which corresponds to the special McKay correspondence of Ishii and Ueda, and relates the abstract construction to an explicit Laurent polynomial mirror in an example of a family of orbifold del Pezzo surfaces.

Significance. If the construction and correspondences hold, this would extend homological mirror symmetry to orbifold log Calabi-Yau surfaces via abstract Lefschetz fibrations, providing a framework that integrates the special McKay correspondence and offers concrete verification through the Laurent polynomial example for orbifold del Pezzo surfaces. This could facilitate further work on mirror symmetry for singular varieties.

minor comments (2)
  1. [Abstract and §1] The abstract and introduction could clarify how the abstract Lefschetz fibration is defined from the input data (X,D) to make the central construction more immediately accessible.
  2. [Example section] In the explicit example of orbifold del Pezzo surfaces, additional computational steps showing how the homological equivalence at the large complex structure limit is checked would strengthen the illustration.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful summary of the manuscript and for recommending minor revision. We are pleased that the referee recognizes the potential of the abstract Lefschetz fibration construction and its relation to the special McKay correspondence for extending homological mirror symmetry to orbifold log Calabi-Yau surfaces.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper constructs an abstract Lefschetz fibration directly from each input pair (X, D) consisting of a projective rational surface with isolated cyclic quotient orbifold points and a stacky anticanonical divisor. It then defines a Lefschetz stabilization procedure and verifies that this matches the special McKay correspondence of Ishii and Ueda (an independent external citation, arXiv:1104.2381v2). The homological mirror symmetry statement at the large complex structure limit is obtained by explicit verification on this construction, with an illustrative example using explicit Laurent polynomial mirrors for orbifold del Pezzo surfaces. No equation or central step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the argument remains independent of the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is inferred from the high-level claims. The central claim rests on the ability to associate an abstract Lefschetz fibration to each pair (X,D) and on the correspondence between stabilization and the special McKay correspondence.

axioms (1)
  • domain assumption An abstract Lefschetz fibration can be constructed for every projective rational surface X with isolated cyclic quotient orbifold points and stacky anticanonical divisor D such that it encodes homological mirror symmetry at the large complex structure limit.
    This is the load-bearing premise invoked to prove the main statement.

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discussion (0)

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