On the Hodge filtration of Hodge modules

We show an estimate of the generating level of the Hodge filtration on pure Hodge modules in terms of the dimension of the support and the level of the generic variation of Hodge structure. We also prove a rather explicit formula for the Hodge filtration in the constant case if the support has only weighted-homogeneous isolated singularities. This implies a counterexample to a conjecture of Brylinski on the Hodge filtration of pure Hodge modules.