Relative monodromy weight filtrations

In this paper we establish the existence of relative weight filtrations of cones of monodromy logarithms and identities between relative weight filtra tions in a context more general than M. Kashiwara's infinitesimal mixed Hed ge modules. This generalizes a theorem of M. Kashiwara and adds conceptual clarity to the problem. Our approach to working with relative weight filtra tions W(N, W') is to split the weight filtrations W' with canonical splitti ngs introduced by P. Deligne. These splittings agree with the splittings th at occur in the sI(2)-theory of variations of pure Hedge structure of W. Sc hmid and of E. Cattani, A. Kaplan, and W. Schmid, but are more general, as they don't require filtrations W' to be monodromy weight filtrations.