We determine the irregular Hodge filtration, as introduced by Sabbah, for the purely irregular hypergeometric${\mathcal{D}}$-modules. We obtain, in particular, a formula for the irregular Hodge numbers of these systems. We use the reduction of hypergeometric systems from GKZ-systems as well as comparison results to Gauss–Manin systems of Laurent polynomials via Fourier–Laplace and Radon transformations.

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We provide two Mayer-Vietoris-like spectral sequences related to the localization over the complement of a closed subvariety of an algebraic variety by using techniques from D-modules and homological algebra. We also give, as an application of the previous, a method to calculate the cohomology of the complement of any arrangement of hyperplanes over an algebraically closed field of characteristic zero.where Y I is the intersection of the components (not necessarily irreducible) Y i for i ∈ I. This way of dividing Y and taking the spectral sequence is completely analogous to hoẁ Alvarez Montaner, García López and Zarzuela Armengou acted with local cohomology of modules (with support in certain ideals) in [AGZ], work which was generalized by Lyubeznik in [Ly]. As the title says, there is another spectral sequence provided in theorem 4.5, very related to the one written above, but in a relative version. To achieve that, we work with D Xmodules, by using the direct image functor in the derived category of coherent D-modules associated with a morphism f : X → Z, denoted by f + . The spectral sequence takes a complex of D X -modules M ∈ D b c (D X ) and deals with complexes of D Z -modules like this:

We consider the Hodge filtration on the sheaf of meromorphic functions along free divisors for which the logarithmic comparison theorem holds. We describe the Hodge filtration steps as submodules of the order filtration on a cyclic presentation in terms of a special factor of the Bernstein–Sato polynomial of the divisor and we conjecture a bound for the generating level of the Hodge filtration. Finally, we develop an algorithm to compute Hodge ideals of such divisors and we apply it to some examples.