Logarithmic Bogomolov–Tian–Todorov theorem

Treat log dimension zero separately. Locally the characteristic map of a saturated log-étale morphism is Kummer of equal rank. Saturation forces its index to be one: otherwise adjoining a missing lattice root in a saturated base pushout contradicts the defining saturation property. Thus its sharp characteristic map is an isomorphism and the morphism is strict étale. A strict étale morphism has a unique nilpotent lift locally by lifting an étale presentation and its invertible Jacobian; uniqueness on intersections glues these lifts and their central identifications. The log structure is the strict pullback of the given base log structure. Therefore every given deformation lifts across every Artin surjection in this dimension. This is the zero-dimensional clause of the chart calculation in F2 and does not use a positive-dimensional BV assertion.

Suppose $d\ge 1$. Apply F2 with the specified central log volume. This produces $G_A^{p,q}$ and its form calculus for every $A$, uniformly bounded in both degrees, flat over $A$ and compatible with arbitrary Artin base maps. It has curvature $\mathrm{\ell }$, predifferential $\bar{\mathrm{\partial }}$, BV operator $\mathrm{\Delta }$ and volume correction $y$, with exactly the identities in F4. The construction uses strict-étale chart descent inside affine opens and a finite Zariski affine TW cover, so it applies to the fs log structures in F1 without a global-chart assumption. Its central columns compute precisely the coherent log form cohomology.

The first conclusion of F3 supplies finite central Hodge groups and central E1 degeneration. Its second conclusion applies to the absolute extension, nilpotent contraction gauges and fixed base changes supplied by F2; hence the corrected form total complex surjects in cohomology onto its central fibre for every Artin $A$. The volume isomorphism $\kappa$ identifies that complex with $(G_A,\bar{\mathrm{\partial }}+\mathrm{\Delta }+(\mathrm{\ell }+y)\wedge )$ up to the fixed degree shift $d$. Thus the corrected BV complex has the same cohomology-surjectivity property. This establishes every hypothesis of F4 BEFORE choosing any MC solution or global deformation. It uses neither an existing-family degeneration theorem as a replacement nor a central E1 assertion alone.

Let $B\to A$ be any surjection and let $X_A$ be ANY given deformation in the Statement. Under the natural bijection of F2 choose a classical MC representative ${\varphi }_A\in {\mathfrak{m}}_A{G}_A^{-1,1}$ of its marked isomorphism class. F4, with the hypotheses verified above, supplies ${\varphi }_B$ reducing to precisely ${\varphi }_A$. Its geometric class under F2 is a log-smooth deformation $X_B$, and naturality of that bijection on the actual map $B\to A$ identifies its reduction with the original marked class of $X_A$. No preferred class, quotient of the base category, or projective subfamily has been substituted.

The preceding argument applies to every surjection and every given class, so the functor is smooth in the exact sense of F1. For an SNC pair the log-volume hypothesis is $K_X(D)\cong {\mathcal{O}}_X$ by F5; none of the construction replaces it by $K_X\cong {\mathcal{O}}_X$. Both conclusions are precisely the original Statement.