Compactifications of log morphisms (Q1771516)

Let \(T\) be a log scheme. The author calls a \(T\)-log scheme with boundary the following data: a morphism of log schemes \(X \rightarrow T\) together with an open log schematically dense embedding of log schemes \(i\colon X\rightarrow \overline{X}\) so that \(\overline{X}\) coincides with the schematic image of \(i.\) The relative logarithmic de Rham complex on \(X\) is extended naturally on \(\overline{X},\) the compactification of \(X.\) The aim of the paper is to obtain a smoothness criterion for \(T\)-log schemes with boundary formulated in terms of morphisms of monoids. The author underlines that his criterion is very similar to Kato's criterion for log smoothness [\textit{K. Kato}, in: Algebraic analysis, geometry, and number theory, Proc. JAMI Inaug. Conf., Baltimore/MD (USA) 1988, 191--224 (1989; Zbl 0776.14004)]. He then considers in some details semistable \(k\)-log schemes with boundary (\(k\) is a field) in the context of the de Rham and crystalline cohomology theories, discusses relations between crystalline and convergent cohomologies [\textit{A. Ogus}, Compos. Math. 97, No. 1--2, 187--225 (1995; Zbl 0849.14008)], with the theory of integrable log connections with regular singularities, etc.