# The Hodge realization functor on the derived category of relative motives

Abstract : We give, for a complex algebraic variety S, a Hodge realization functor F Hdg S from the (un-bounded) derived category of constructible motives DAc(S) over S to the (undounded) derived category D(M HM (S)) of algebraic mixed Hodge modules over S. Moreover, for f : T → S a morphism of complex quasi-projective algebraic varieties, F Hdg − commutes with the four operations f * , f * , f ! , f ! on DAc(−) and D(M HM (−)), making in particular the Hodge realization functor a morphism of 2-functor on the category of complex quasi-projective algebraic varieties which for a given S sends DAc(S) to D(M HM (S)), moreover F Hdg S commutes with tensor product. We also give an algebraic and analytic Gauss-Manin realization functor from which we obtain a base change theorem for algebraic De Rham cohomology and for all smooth morphisms a relative version of the comparaison theorem of Grothendieck between the algebraic De Rahm cohomology and the analytic De Rahm cohomology.
Document type :
Preprints, Working Papers, ...
Domain :

https://hal.archives-ouvertes.fr/hal-02888285
Contributor : Johann Bouali <>
Submitted on : Wednesday, February 3, 2021 - 5:25:43 PM
Last modification on : Tuesday, June 1, 2021 - 11:02:14 AM

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DMmh448.pdf
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### Identifiers

• HAL Id : hal-02888285, version 5

### Citation

Johann Bouali. The Hodge realization functor on the derived category of relative motives. 2021. ⟨hal-02888285v5⟩

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