Florian Ivorra

Perverse, Hodge and motivic realizations of étale motives

Let k = C be the field of complex numbers. In this article, we construct Hodge realization functors defined on the triangulated categories of etale motives with rational coefficients. Our construction extends, to every smooth quasi-projective k-scheme, the construction done by M. Nori over a field and relies on the original version of the basic lemma proved by A. Bĕilinson. As in the case considered by M. Nori, the realization functor factors through the bounded derived category of a perverse version of the Abelian category of Nori motives.

Microlocalization of ind-sheaves

Let X be a C∞-manifold and T*X its cotangent bundle. We construct a microlocalization functor μ X: Db(I(\( \mathbb{K}_X \) )) → Db(I(\( \mathbb{K}_{T*X} \) )), where Db(I(\( \mathbb{K}_X \) )) denotes the bounded derived category of ind-sheaves of vector spaces on X over a field \( \mathbb{K} \) . This functor satisfies Rℌom(μ X(F), μ X(G)), ⋍ μhom(F,G) for any F,F ∈ Db(\( \mathbb{K}_X \) ), thus generalizing the classical theory of microlocalization. Then we discuss the functoriality of μ X. The main result is the existence of a microlocal convolution morphism

Réalisation ℓ-adique des motifs triangulés géométriques II

\mu _{X \times Y} \left( {\mathcal{K}_1 } \right)_ \circ ^a \mu _{Y \times Z} \left( {\mathcal{K}_2 } \right) \to \mu _{X \times Z} \left( {\mathcal{K}_1 \circ \mathcal{K}_2 } \right)

Réalisation ℓ-adique des motifs mixtes

Abstract For a field of characteristic zero Levine has proved in [M. Levine, Mixed motives, Math. Surv. Monogr. 57, American Mathematical Society, 1998.], Part I, Ch. VI, 2.5.5, that the triangulated tensor categories of motives defined in [V. Voevodsky, Triangulated categories of motives over a field, Cycles, transfers, and motivic homology theories, Ann. Math. Stud. 143, Princeton University Press, Princeton, NJ, 2000.] and [M. Levine, Mixed motives, Math. Surv. Monogr. 57, American Mathematical Society, 1998.] are equivalent. Using some results of [F. Ivorra, Réalisation ℓ-adique des motifs triangulés géométriques I, Preprint K-theory/0762, January 2006.], in this paper we show that the strategy of Levines proof can also be applied on every perfect field to the categories of triangulated motives with rational coefficients or to the pseudo-abelian hulls of the integral tensor subcategories generated by motives of smooth projective schemes.