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.