On the mixed Hodge structure on the cohomology of the Milnor fibre

In [18], the second author introduced a mixed Hodge structure on the cohomology of the Milnor fibre of an isolated hypersurface singularity. (For the definition of a mixed Hodge structure, el. [17, Sect. 3.4].) The weight filtration is essentially the "monodromy weight filtration" [18, Sect. 4] and thus simplifies the Jordan normal form of the unipotent part of the monodromy. The significance of the Hodge filtration, however, is not so dear, and its description seems difficult to use. This paper gives another description of the Hodge filtration which we hope is easier to understand and apply. Its definition does not use resolution of singularities. Instead it relies on the theory of holonomic ~-modules in one variable with regular singularities. This paper arose from conversations at the 1980 Arbeitstagung in Bonn, where Varchenko's conjectured Hodge filtration [22] had been discussed in Brieskorn's seminar. We have since learned that Varchenko obtained similar results to those in this paper in the summer and autumn of 1980 [23-25]. In Sect. 1 the Milnor fibration is embedded in a family of smooth projective hypersurfaces. The Hodge theory of a smooth projective hypersurface is explained in the fashion of Brylinski [2] in Sect. 2. In Sect. 3 we recall the description of the Gauss-Manin system from [8]. Our new formula for the Hodge filtration is explained in Sects. 4, 5. In Sect. 6 we prove that it gives the same result as [18]. We prove a "Thom-Sebastiani" formula for the Hodge filtration in Sect. 7 which leads to a proof of conjecture (5.4) of [18]. In Sect. 8 we show how to prove Varchenko's result about the Jordan normal forms of multiplication by f in the Jacobian ring of f and the logarithm of the unipotent monodormy of f with our method. In the last chapter the mixed Hodge structure is calculated for two examples:

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