Hodge number polynomials for nearby and vanishing cohomology

We recall the construction of the Hodge character and we show, using a result due to F. Bittner, that these can be constructed using classical pure Hodge theory only, sideskipping Deligne’s construction of functorial mixed Hodge structures on the cohomology of algberaic varieties. We then give a proof, based on the weak factorization theorem, that the nearby fibre is well-defined in the Grothendieck group of varieties. Finally, various formulas for numerical invariants of Hodge-theoretic origin are presented which are associated to one-parameter degenerations of projective manifolds.