  We prove the existence of a smoothing for a toroidal crossing space under
mild assumptions. By linking log structures with infinitesimal deformations,
the result receives a very compact form for normal crossing spaces. The main
approach is to study log structures that are incoherent on a subspace of
codimension two and prove a Hodge-de Rham degeneration theorem for such log
spaces which also settles a conjecture by Danilov. We show that the homotopy
equivalence between Maurer-Cartan solutions and deformations combined with
Batalin-Vilkovisky theory can be used to obtain smoothings. The construction of
new Calabi-Yau and Fano manifolds as well as Frobenius manifold structures on
moduli spaces are potential applications.
