Algebraic Connections on Logarithmic Schemes

In the theory of $O$-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 $O$-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.