For a regular function f on a smooth complex quasi-projective variety, J.-D. Yu introduced in [Yu14] 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 E 1 of the spectral sequence attached to the irregular Hodge filtration, by using the method of [Sab10]. We also make explicit the relation with a complex introduced by M. Kontsevich and give details on his proof of the corresponding E 1 -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 latter statement with a possibly non reduced normal crossing pole divisor.

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