The Hodge realization functor on the derived category of relative motives

2020/07/05 by Bouali, Johann
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2007.02446

Abstract

We give, for a complex algebraic variety S, a Hodge realization functor \mathcal F_S^Hdg from the derived category of constructible motives DA_c(S) to the derived category D(MHM(S)) of algebraic mixed Hodge modules over S. Moreover, for f:T→ S a morphism of complex quasi-projective algebraic varieties, \mathcal F_-^Hdg commutes with the four operation f^*,f_*,f_!,f^! on DA_c(-) and D(MHM(-)), making the Hodge realization functor a morphism of 2-functor wich for a given S sends DA_c(S) to D(MHM(S), moreover \mathcal F_S^Hdg 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 realtive version of the comparaison theorem of Grothendieck between the algebaric De Rham cohomology and the analytic De Rham cohomology.

Related

• The De Rham, complex Hodge and p-adic Hodge realization functors on the derived category of relative motives over a field of characteristic zero
• Réalisations de Hodge des motifs de Voevodsky
• The four operations on perverse motives
• On the realization functor of the derived category of mixed motives
• Motives and algebraic de rham cohomology