E₁-degeneration of the irregular Hodge filtration

For a regular function f on a smooth complex quasi-projective variety, J.-D. Yu introduced in [35] a filtration (the irregular Hodge filtration) on the de Rham complex with twisted differential d + df, extending a definition of Deligne in the case of curves. In this article, we show the degeneration at E₁ of the spectral sequence attached to the irregular Hodge filtration, by using the method of [26]. We also make explicit the relation with a complex introduced by M. Kontsevich and give details on his proof of the corresponding E₁-degeneration, by reduction to characteristic p, when the pole divisor of the function is reduced with normal crossings. In Appendix E, M. Saito gives a different proof of the E₁-degeneration.