Algebraic connections on logarithmic schemes

In the theory of $\text{O}$-modules with an integrable logarithmic connection in the context of log schemes (over a field of characteristic zero), one of the first problems is that, contrary to the classical case, an object of these categories which is $\text{O}$-coherent is not necessarily locally free. We present some sufficient conditions for the local freeness, based essentially on the notion of residues of a log connection. Then we handle the problem of stability of local freeness under derived direct image for morphisms of log schemes; we prove a generalization of the Deligne–Illusie results on degeneration of the “Hodge to de Rham” spectral sequence, local freeness, compatibility with base change.