Smoothing toroidal crossing spaces

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