Rigidity of Log Morphisms

Abstract

In the paper [5], we proved Kato’s conjecture, that is, the finiteness of dominant rational maps in the category of log schemes as a generalization of Kobayashi-Ochiai theorem [4]. It guarantees the finiteness of K-rational points of a certain kind of log smooth schemes for a big function field K, which gives rise to an evidence for Lang’s conjecture. In the proof of the above theorem, the most essential part is the rigidity theorem of log morphisms. In this paper, we would like to generalize it to a semistable scheme over an arbitrary noetherian scheme. Let f : X → S be a scheme of finite type over a locally noetherian scheme S. We assume that f : X → S is a semistable scheme over S, namely, f is flat and, for any morphism Spec(Ω) → S with Ω an algebraic closed field, the completion of the local ring of X ×S Spec(Ω) at every closed point is isomorphic to a ring of the type