We introduce the concept of a viable generically Gorenstein toroidal crossing (ggtc) space $Y$. This generalizes the concept of Gorenstein toroidal crossing scheme, which in turn generalizes that of a simple normal crossing scheme. On such a space $Y$, we define a sheaf $\mathcal{LS}_Y$, intrinsic to $Y$, by means of an explicit construction. Our main theorem establishes a bijection between the set $\operatorname {LS}_{k^\dagger } (Y)$ of isomorphism classes of log structures on $Y$ over the log point $\operatorname {Spec} k^\dagger$ that are compatible with the ggtc structure and the set $\Gamma (Y,\mathcal{LS}_Y^\times )$ of nowhere-vanishing global sections of $\mathcal{LS}_Y$. The definition of $\mathcal{LS}_Y$ by explicit construction permits the effective construction of log structures on $Y$; it also enables logarithmic birational geometry, in particular the construction, in some cases, of resolutions of singular log structures. Our work generalizes [Gross and Siebert, J. Differential Geom. 72 (2006), 169–338, Theorem 3.22], adapting the original proof with techniques from the theory of $2$-groups and local line bundle systems.

We define the notion of a morphism of generalized semi-stable type, which is a generalization of the notion of a semistable degeneration over a curve. We partially generalize Steenbrink‘s results on the limit of Hodge structures to the case of such a morphism. As an application we prove the E1-degeneration of the relative Hodge–De Rham spectral sequence for this case.