Toroidal crossings and logarithmic structures

We generalize Friedman's notion of d-semistability, which is a necessary condition for spaces with normal crossings to admit smoothings with regular total space. Our generalization deals with spaces that locally look like the boundary divisor in Gorenstein toroidal embeddings. In this situation, we replace d-semistability by the existence of global log structures for a given gerbe of local log structures. This leads to cohomological descriptions for the obstructions, existence, and automorphisms of log structures. We also apply toroidal crossings to mirror symmetry, by giving a duality construction involving toroidal crossing varieties whose irreducible components are toric varieties. This duality reproduces a version of Batyrev's construction of mirror pairs for hypersurfaces in toric varieties, but it applies to a larger class, including degenerate abelian varieties.