Smoothing toroidal crossing spaces

TL;DR

This paper establishes a method for smoothing toroidal crossing spaces by linking log structures with infinitesimal deformations, leading to new insights in algebraic geometry and potential applications in constructing Calabi-Yau and Fano manifolds.

Contribution

It introduces a novel approach connecting log structures with deformations, proves a Hodge-de Rham degeneration for certain log spaces, and settles a conjecture by Danilov.

Findings

Proved existence of smoothings for toroidal crossing spaces.

Established a Hodge-de Rham degeneration theorem for specific log spaces.

Provided a framework for constructing new Calabi-Yau and Fano manifolds.

Abstract

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.