Some remarks on limit mixed Hodge structures and spectrum

In this note we clarify some subtle points on the limit mixed Hodge structures and on the spectrum. These are more or less well-known to the specialists, but do not seem to be stated explicitly in the literature. However, as they do not seem to be obvious to the beginners, we consider them to be worth writing down explicitly. The general constructions are exemplified by considering the isolated (weighted) homogeneous singularities in detail. 1. Limit mixed Hodge structure 1.1. In [19], [23], the limit mixed Hodge structures were constructed in the unipotent monodromy case. For the non-unipotent case, we can combine it with [20] as follows. Here we describe the limit of the mixed Hodge structure on the cohomology with compact supports using the Cech-type construction, since this seems to be the easiest way to explain the relation with the theory of motivic Milnor fibers [4]. For the usual cohomology (i.e. without compact supports), we can use the commutativity of the dualizing functor D and the passage to the limit mixed Hodge structure, i.e. D ◦ψt = ψt ◦D (up to a Tate twist), see [23], [15] as well as [6] for generalities on constructible or perverse sheaves. Of course, we can also use the two weight filtrations on the logarithmic complex associated with the divisor with V -normal crossings [20] as in [23]. It also follows from the theory of mixed Hodge modules [14], [15].