Smoothing toroidal crossing spaces

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 2 and prove a Hodge-de Rham degeneration theorem for such log spaces that 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 provides potential applications.

Automated summary

In this paper, the existence of a smoothing for a toroidal crossing space under mild assumptions is proven. By linking log structures with infinitesimal deformations, the result is presented in a very compact form for normal crossing spaces. The main approach involves studying log structures that are incoherent on a subspace of codimension 2 and proving a Hodge-de Rham degeneration theorem, settling a conjecture by Danilov. This work demonstrates that the homotopy equivalence between Maurer-Cartan solutions and deformations, combined with Batalin-Vilkovisky theory, can be utilized to obtain smoothings, with potential applications in constructing new Calabi-Yau and Fano manifolds, as well as Frobenius manifold structures on moduli spaces.