S. For and . Ansp, (C) sm /S) which is by definition given by -for U/S a smooth morphism

, (C)/S) is by definition a natural extension of

S. Let and . Var-;-c)-and-h-1-:-u-1-?-s, Denote h 12 : U 12 := U 1 × S U 2 ? S and p 112 : U 1 × S U 2 ? U 1

, ) are usu local equivalence by proposition 10, ? since h : U ? S is a smooth morphism, the inclusion ? U/S : h * O S ? ? ? U/S is a quasi-isomorphism, ? since h ? : U T ? T is a smooth morphism

]. ?-[1,-·-·-·-l, S. I-=-?-i?i, and S. , S I ?? S the open embedding. Let i i : S i ??S i closed embeddings

M. Then and N. Da-c,

?. T-(f-f-dr-s,an and N. )-:-f-f-dr-s,an-(m-)-?-l-os-f-f-dr-s,

. ?-?-f-f-dr-s,an-(m-?-n,

N. S-i-=-?-i?i-s-i-and-j-i-:-s-i-??-s-the-open-embedding-;-m, F. Da(s)-and, and G. , Let i i : S i ??S i closed embeddings, withS i ? SmVar(C), ) sm /S)

F. Gm-s, (. Ld-s-m-)-?-l-os-f-gm-s, and (. Ld-s-n,

T. , A. , ?. Ldsm, and . Ldsn-),

T. , A. , and F. F-dr-s,

. /-/-f-f-dr-s,an-(m-)-?-l-os-f-f-dr-s,

?. T-(f-f-dr-s,an and N. Gm-s,an-(d-s-l(m-?-n,

T. , A. , and F. F-dr-s,

. /-/-f-f-dr-s,an-(m-?-n,

S. Os, AnSp(C) sm /S an )

. Then and . Da-c-;-f-f-dr, T an )) ? D D(1,0)f il,? (T an /(Y an × S an I )), see definition 122 and definition 143 commutes g * mod Hdg, the following diagram in ? T (D(M HM

M. Bti, we define the Betti realization functor as

S. Re, S cw ) * Cw * S M = e(S cw ) * sing I * Cw et)(F )

, For the Corti-Hanamura weight structure on DA ? (S), we have by functoriality of (i) the functor

*. An, *. Cw, and . Trivially, Note that, by considering the explicit D 1 S local model for presheaves on AnSp(C) sm /S an

D. , S an ) ; by considering the explicit I 1 S local model for presheaves on CW sm /S cw

D. ,

T. T-?-s-a-morphism and S. Var, Definition 153, We have, for M ? DA(S), (F, W ) ? C f il (Var(C) sm /S) such that (M, W ) = D(A 1 , et)(F, W ), and an equivalence (A 1 , et) local e : f * (F, W ) ? (F ? , W ) with (F ? , W ) ? C f il (Var(C) sm /S) such that (f * M, W ) = D

B. T-(f,

S. and W. ,

. ?-?-dr,

?. S-*-e-usu-(?-?,

*. Hom-;-an, *. Ld-s-l(f, and W. ,

. ?-?-dr,

S. and W. ,

W. S-*-l?-s-*-?-s-*-r-(x-*-,d-*-)/s-(?-*-s-l(f,

, AnS *

D. S-*-l?-s-*-?-s-*-r-(x-*,

. ?-?-dr,

, C f il (Var(C) sm /S) is such that (M, W ) = D(A 1 , et)(F, W )

, S )) is the isomorphism of theorem 23

?. S-*-e-usu-(?-?,

?. and W. , Gr 12

D. S-*-l?-s-*-?-s-*-r-(x-*, W ))) ? q is an equivalence (A 1 , et) local by lemma 5

F. ?-t-(an and . Dr-s, M ) : (F DR S (M )) an ? ? ? F DR S,an (M ) is an isomorphism by theorem 41

J. Ayoub, Note sur les opérations de Grothendieck et la realisation de Betti, Journal of the Institute of Mathematics of Jussieu, vol.9, issue.02, pp.225-263, 2010.

J. Ayoub, L'algebre de Hopf et le groupe de Galois motiviques d'un corps de caractéristique nulle I, Journal die reine (Crelles Journal), vol.2014, issue.693, pp.1-149

J. Ayoub, L'algebre de Hopf et le groupe de Galois motiviques d'un corps de caractéristique nulle II, Journal die reine (Crelles Journal), vol.2014, issue.693, pp.151-126

J. Ayoub, Les six opérations de Grothendieck et le formalisme des cyclesévanescents dans le monde motivique I et II, Société de Mathématiques de France, Astérisque, vol.314, 2006.

A. Beilinson, On the derived category of perverse sheaves, in K-theory, arithmetic and geometry, Lecture Notes in Mathematics, vol.1289, pp.27-41, 1987.

M. V. Bondarko, Weight structure vs. t-structure ; weight filtrations, spectral sequences, and complexes (for motives and in general), Journal of K-theory, vol.6, issue.3, pp.387-504, 2010.

J. Bouali, On the realization functor of the derived category of mixed motives

N. Budur, On the V-filtration of D-modules in algebra and number theory, Progr, Math, vol.235, pp.59-70, 2005.

J. Cirici and F. Guillen, Homotopy theory of mixed Hodge Complexes, 2013.

D. C. Cisinski and F. Deglise, Triangulated categories of mixed motived

A. Corti and M. Hanamura, Motivic decomposition and intersection Chow groups I, Duke math, J, vol.103, pp.459-522, 2000.

F. Jin, Borel-Moore Motivic homology and weight structure on mixed motives, Math.Z, vol.283, issue.3, pp.1149-1183, 2016.
URL : https://hal.archives-ouvertes.fr/ensl-01411015

R. Harshorne, On the De Rham cohomology of algebraic varieties, Inst. Hautes Etudes Sci. Publ. Maths, vol.45, pp.5-99, 1975.

A. Hatcher, Algebraic Topology, 2002.

H. Hironaka, Resolution of singularities of an algebraic variety over a field of caracteristic zero, Ann. of Math, vol.79, pp.205-326, 1964.

R. Hotta, K. Takeuchi, T. Tanisaki, and D. , Perverse Sheaves, and Representation Theory, 2008.

F. Ivorra, Perverse, Hodge and motivic realization of etale motives, Compos. Math, vol.152, issue.6, pp.1237-1285, 2016.

M. Kashiwara, The Riemann-Hilbert problem for holonomic systems, Publications of the Research Institute for Mathematical Sciences, vol.20, issue.2, pp.319-365, 1984.

M. Kashiwara and P. Schapira, Sheaves on manifolds, 1990.

F. Lecompte and N. Wach, Réalisation de Hodge des motifs de Voevodsky, manuscripta mathematica, vol.141, pp.663-697, 2012.

G. Laumont, Transformations canoniques et specialisation pour les V-modules filtrés

. Neeman, Homotopy limits in triangulated categories, Compos. Math, vol.86, pp.209-234, 1993.

M. Levine, Mixed motives, Handbook of K-theory, vol.1, pp.429-521, 2005.

C. Peters and J. Steenbrink, Mixed Hodge Structures, vol.52, 2008.

F. Pham, Singularités des systemes differentiels de Gauss-Manin, vol.2, 1979.

C. Mazza, V. Voevodsky, and C. Weibel, Lecture notes on motivic cohomology, vol.2, 2006.

M. Saito, Mixed Hodge Modules, Proc. Japan Acad. Ser, A.Math.Sci, vol.62, issue.9, pp.360-363, 1986.

J. P. Serre, Géometrie algébrique et géometrie analytique, Annales de l'institut Fourier, vol.6, pp.1-42

J. L. Verdier, Dualité dans la cohomologie des espaces localement compacts, Seminaire Bourbaki, vol.9, pp.337-349, 1995.

J. L. Verdier, Des categories dérivées de catgories abéliennes, Asterique, vol.239, 1996.

. Laga-umr-cnrs,

F. Villetaneuse,