Log Structures on Generalized Semi-Stable Varieties

In this paper we study the log structures on generalized semistable varieties, generalize the result by F. Kato and M. Olsson, and prove the canonicity of log structure when it can be expected. In out text we first give the definitions of local chart and weakly normal crossing morphism. Then we study the invariants of complete noetherian local ring coming from weakly normal crossing morphisms. These invariants enable us to further define the refined local charts and prove that all log structures induced by refined local charts are locally isomorphic. Let f : X → S be a surjective, proper and weakly normal crossing morphism of locally noetherian schemes which satisfies the conditions (†) and (‡) in 3.3 and certain local conditions stated at the beginning of 5. Then the obstructions for the existence of semistable log structures on X is an invertible sheaf L (f) on a finite X-scheme E = E(f). The main result of local case with respect to base schemes is: Theorem. (1) There exists a semistable log structure on X if and only if L (f) ∼= OE . (2) The semistable log structure on X is unique up to (not necessarily canonical) isomorphisms if it exists. The main result of global case with respect to base schemes is: Theorem. Let X and S be locally noetherian schemes, f : X → S a surjective proper weakly normal crossing morphism without powers. If f satisfies the condition (†) in 3.3 and every fiber of f is geometrically connected, then (1) There exists a semistable log structure for f if and only if for every point y ∈ S, Lȳ is trivial on Eȳ. (2) Let (M1,N1, σ1, τ1, φ1) and (M2,N2, σ2, τ2, φ2) be two semistable log structures for f . Then there exist isomorphisms of log structures φ : M1 ∼ −→ M2 and ψ : N1 ∼ −→ N2 such that φ ◦ φ1 = φ2 ◦ f∗ψ, σ2 ◦ φ = σ1 and τ2 ◦ ψ = τ1. Moreover such pair (φ,ψ) is unique. We further prove that the existence of semistable log structures remains under fibred products, base extension, inverse limits, flat descent. Finally we study the semistable curves. The main result is: Theorem. Any semistable curve over a locally noetherian scheme is a weakly normal crossing morphism without powers and has a canonical semistable log structure.