E 1 -degeneration of the irregular Hodge filtration

© De Gruyter 2017. 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 C 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.