Homological Mirror Symmetry

• Homological Mirror Symmetry is a categorical framework that establishes an equivalence between the Fukaya category of a symplectic manifold and the derived category of coherent sheaves on its mirror.
• It employs dg- and A∞-category techniques alongside functorial descent, patching local equivalences on toric charts for rigorous categorical constructions.
• This framework has enabled precise computations and new insights in toric, Calabi–Yau, and hypertoric settings, advancing our understanding of mirror symmetry and categorical gluing.

Homological Mirror Symmetry (HMS) is a categorical framework unifying symplectic geometry and algebraic geometry, originally proposed by Kontsevich. It postulates an equivalence between two intrinsically defined derived categories: the Fukaya category of a symplectic manifold ("the A-model") and the derived category of coherent sheaves on a mirror complex variety ("the B-model"). HMS organizes, generalizes, and often subsumes previous geometric mirror symmetry phenomena within the language of dg/A∞-categories, functoriality, and categorical descent.

1. Categorical Formulation and Large Complex-Structure Limits

The classical mirror construction employs a pair (X, ω) of a (typically Calabi–Yau or Fano) variety X and its symplectic form. The mirror, Ẋ, is realized as a family or union of toric varieties glued along their toric boundaries––formalized as a "fanifold" Φ, a stratified space with fans Σ_F on its strata. Algebraically, one builds the B-model as the colimit of toric varieties glued along these strata:

• For each F ∈ Φ, assign T(Σ_F), a toric variety.
• The exit-path category Exit(Φ) encodes the gluing data.
• The scheme X = colim_{Exit(Φ)^{op}} T(Σ_F) is a large-complex-structure limit, a union of toric pieces glued along boundaries.
• The sheaf-theoretic descent (using IndCoh formalism) identifies the derived category of coherent sheaves D^b Coh(X) as the colimit of the categories of coherent sheaves on local toric charts (Gammage et al., 2021).

2. Symplectic Side: Weinstein Manifolds, Skeleta, and Fukaya Categories

On the A-model, the mirror is constructed as a Weinstein manifold W = W(Φ), assembled inductively by gluing together cotangent (Weinstein) charts along specified Legendrian boundaries. The procedure is as follows:

• Start with disjoint unions of cotangent bundles over tori.
• For each stratum, attach Weinstein handles along Legendrians corresponding to the local boundary data, yielding a global manifold with skeleton L(Φ) projecting to Φ.
• Locally, FLTZ skeleta (due to Fang–Liu–Treumann–Zaslow) in T^*(Hom(M, S^1)) have explicit presentations and serve as the "skeleton at infinity" determining the wrapped Fukaya category.

The wrapped Fukaya category Fuk(W) localizes to sections of the Kashiwara–Schapira microsheaf category supported on the skeleton, with functorial sheaf-theoretic formulation: Fuk(M) ≃ Γ(L(Φ), μSh_{L(Φ)})^op (Gammage et al., 2021). The pushforward π*μSh{L(Φ)} yields a constructible sheaf of categories on Φ, with local models matching Fukaya categories of toric charts.

3. Equivalence: Homological Mirror Symmetry and Local-to-Global Principle

The foundational result is a dg-enhanced A∞-quasi-equivalence:

$$D^b\,\operatorname{Coh}(X) \;\simeq\; \operatorname{Fuk}(M)$$

This equivalence is constructed by patching local equivalences on toric charts (Fukaya = Coh) [FLTZ2, Ko, GS17], assembling sheaves of categories on the fanifold, and then applying descent to global sections (Gammage et al., 2021).

Notably, the equivalence is fully functorial: pullbacks (restriction functors) for coherent sheaves correspond to Viterbo restriction functors for Fukaya categories. Exact triangles, restriction diagrams, and descent squares commute under HMS. The formalism realizes Seidel's "local-to-global principle": the Fukaya category of the total space is recovered as a homotopy limit of subdomain Fukaya categories, precisely mirroring Zariski descent for coherent sheaves.

Aspect	B-Model Interpretation	A-Model Interpretation
Building block	Toric variety T(Σ_F)	Weinstein chart, FLTZ skeleton
Gluing data	Exit-path colimit, closed immersions	Weinstein handle attachments
Local category	Coh(T(Σ_F))	Fukaya category for L(Σ_F)
Descent	Zariski/étale descent for Coh	Sheaf of categories via π_*μSh
Restriction functor	Pullback to open toric strata	Viterbo restriction on subdomain
Global object	D^b Coh(X)	Fuk(M)

4. Examples, Applications, and Computations

This framework unifies many prominent instances of HMS:

• In the case of toric Calabi–Yau hypersurfaces and their Batyrev mirrors, the full HMS is established by explicit matching of basis subcategories, deformation-theoretic extension to the whole category (via versal Hochschild deformation theory), and automatic split-generation arguments (Ganatra et al., 2024).
• For symmetric powers of punctured spheres (higher symmetric products of punctured S^2), HMS matches the wrapped Fukaya category of the symmetric product to the derived category of a singular boundary divisor Z ⊂ Y in a toric Landau–Ginzburg model (Lekili et al., 2021).
• In the setting of generalized Greene–Plesser mirrors, HMS is established for Calabi–Yau hypersurfaces and certain complete intersections (Batyrev–Borisov’s dual Gorenstein cones), reflecting the versatility of fanifold/Weinstein descriptions (Sheridan et al., 2017).
• For hypertoric varieties (both additive and multiplicative), the approach via Lagrangian skeleta and tilting/cosheaf resolutions produces equivalences between Fukaya categories and categories of coherent sheaves modules over noncommutative resolutions (Gammage et al., 2019, McBreen et al., 2018).

5. Functoriality, Descent, and Higher Structures

A central feature is compatibility of HMS with functorial operations:

• For closed constructible Z ⊂ Φ, the restriction Coh(U(Z)) corresponds to the Fukaya category of the complementary Weinstein domain W(Z); under HMS, restriction sequences and fiber-product squares are mapped accordingly.
• Zariski descent for categories of coherent sheaves (Cartier squares for open covers) are reflected by Cartesian diagrams for Fukaya categories, establishing that the mirror equivalence "passes through" the same functorial framework (Gammage et al., 2021).
• The gluing formalism uses modern sheaf-theoretic and microlocal tools: microsheaf techniques on skeleta, explicit computation of Fukaya categories via matching of cocores and their morphisms, and the use of sectorial descent to reduce global equivalences to compatibly glued local ones.

6. Mathematical Significance and Future Directions

The outlined approach to HMS at large volume realizes the philosophy that complex-algebraic and symplectic-topological invariants are equivalent at the categorical level, and are computable via gluing and descent—mirroring topological and algebraic methods. The match between Zariski and sectorial descent for categories suggests deep compatibilities between the algebraic and symplectic paradigms.

This framework applies in full generality to unions of toric varieties and their mirror Weinstein manifolds, providing a foundation for local-to-global principles, boundary and restriction phenomena, and categorical gluing beyond the Calabi–Yau or Fano domains. Applications include new cases of HMS for higher-dimensional mirrors, and the explicit computation of categories in toric and hypersurface settings.

The methods and equivalences extend naturally to the study of semiorthogonal decompositions, perverse schober structures, and the interplay between microlocal sheaf theory and noncommutative geometry.

• "Homological mirror symmetry at large volume" (Gammage et al., 2021)
• "Homological mirror symmetry for Batyrev mirror pairs" (Ganatra et al., 2024)
• "Homological mirror symmetry for the symmetric squares of punctured spheres" (Lekili et al., 2021)
• "Homological mirror symmetry for generalized Greene-Plesser mirrors" (Sheridan et al., 2017)
• "Homological mirror symmetry for hypertoric varieties II" (Gammage et al., 2019)