Aspects of the irregular Hodge filtration (Joint work with Hélčne Esnault and Jeng-Daw Yu)

Given a regular function f on a smooth quasi-projective variety U, the de Rham complex of U relative to the twisted differential d+df can be equipped canonically with a filtration (the irregular Hodge filtration) for which the associated hypercohomology spectral sequence degenerates at E1. A logarithmic version of this de Rham complex (relative to a suitable compactification of U) has been introduced by M. Kontsevich, who showed the independence of the dimension of the corresponding cohomologies with respect to the differential ud+vdf, for u,v arbitrary complex numbers. This leads to bundles on the projective line of the (u:v) variable, on which we construct a natural connection for which the Harder-Narasimhan filtration satisfies the Griffiths transversality property and standard limiting properties at v=0.