General HMS for a specified compact mirror datum

Assume AC. Choose a compact smooth symplectic manifold $(X,\omega )$ and a specified smooth $\omega$-compatible almost-complex structure $J:TX\to TX$: $J^2=-\mathrm{id}$ and $g(v,w)=\omega (v,Jw)$ is a symmetric positive definite metric. Regard $TX$ as a complex vector bundle using $J$, and define $c_1(X,J)=c_1((TX,J))\in H^2(X;\mathbb{Z})$ componentwise by the topological Chern-class convention on CW-type bases; smooth manifolds here are second countable and have CW homotopy type. Require $c_1(X,J)=0$. No choice-independence assertion is needed because $J$ is supplied data.

For a fixed mirror datum $(X,\omega ,\mathrm{\Lambda })\leftrightarrow (Y,{\mathcal{O}}_Y)$ with these data, $Y$ smooth projective with $K_Y\cong {\mathcal{O}}_Y$, compatible gradings and coefficient completion, and a weakly unobstructed Fukaya A-infinity category specified, ask whether there exists a split-closed quasi-equivalence

$D^{\pi }\mathcal{F}(X,\omega ;\mathrm{\Lambda })\simeq D^b\mathrm{Coh}(Y)$

compatible with the mirror-parameter map in the following specified sense. In addition to the geometric mirror datum, supply small parameter categories $P_A,P_B$ and a functor $\mathrm{\Phi }:P_A\to P_B$ representing its map $\phi :U_A\to U_B$. Concretely, specify parameter points $t_a:1\to U_A$ and $s_b:1\to U_B$ in the declared parameter category, with $s_{\mathrm{\Phi }(a)}=\phi t_a$; the morphisms in $P_A,P_B$ are the chosen parameter transitions and their geometric identifications are part of the datum. Require a common coefficient ring after the declared base changes, and diagrams $\mathcal{A}:P_A\to \mathcal{D}$ and $\mathcal{B}:P_B\to \mathcal{D}$. Here $\mathcal{D}$ is the category obtained from small dg categories and dg functors by formally inverting quasi-equivalences; a dg functor is a quasi-equivalence when it induces quasi-isomorphisms on all Hom complexes and is essentially surjective on $H^0$. The value $\mathcal{A}(a)$ is a specified dg enhancement of the split-closed derived Fukaya category at $t_a$, and $\mathcal{B}(b)$ is a specified dg enhancement of the coherent derived category at $s_b$. An enhancement here means a dg category with a specified equivalence from its $H^0$ to the indicated triangulated category. Supply the transition morphisms and functor laws for these diagrams; they are additional hypotheses, not data inferred from $\phi$ alone. For such additional diagram data, the parameter problem is to determine whether a comparison exists, to identify obstructions when it does not, and to construct it when existence holds. No assertion that every supplied pair of diagrams admits a comparison is made. The requested comparison, when it exists, is a natural isomorphism $F:\mathcal{A}\Rightarrow \mathcal{B}\circ \mathrm{\Phi }$ in $\mathcal{D}$: each $F_a$ is an isomorphism there, and for every $u:a\to a^{\mathrm{\prime }}$ one requires $\mathcal{B}(\mathrm{\Phi }u)F_a=F_{a^{\mathrm{\prime }}}\mathcal{A}(u)$. At the distinguished parameter this must induce the displayed equivalence. Even pointwise quasi-equivalence of the fibres is only a necessary condition for this diagram problem. This fixes the compatibility condition at the level of the localized dg category; no extra analytic flat connection or chain-level higher naturality is implicit. A single specified pair is the case of terminal parameter categories. This asks for a comparison for the specified data, and does not assert that every $X$ has a mirror $Y$.

Specified mathematical mirror datum

Compact Calabi–Yau manifold conventions

Blocked construction interface for multi-object Fukaya categories and polygon morphisms

Small dg categories, perfect modules, and Morita deformation

Symplectic and almost-complex curve data

Chern classes from the projective-bundle relation

Smooth manifolds have CW homotopy type

The Axiom of Choice

The single-pair equivalence question is the general HMS question motivated by the cited Kontsevich formulation. The extra diagram problem is a locally specified extension of that formulation; existence for arbitrary transition diagrams is not a conjecture attributed to Kontsevich. The proved identity-versus-zero transition obstruction is recorded separately in Identity and zero transitions obstruct a diagram comparison: identical fibres do not suffice for an isomorphism of the supplied diagrams. Such elementary obstructions must be separated from the unresolved equivalence problem for a genuine geometric mirror pair and from the additional geometric conditions needed for family compatibility. The source-specific elliptic, quartic, quintic and toric-subcategory theorems on the companion A page have their own hypotheses and do not solve this question. Its log-surface material provides geometric and categorical interfaces, without a log-surface equivalence theorem in that page inventory. The exact multi-object Fukaya category and source-specific analytic construction remain prerequisites rather than consequences of the question. Reviewed 2026-10-26; no proof or refutation is supplied here.