Arithmetic Mixed Sheaves

We give a formalism of arithmetic mixed sheaves, including the case of arithmetic mixed Hodge structures which are recently studied by P. Griffiths, M. Green [22], [23] and M. Asakura [1] using other methods. This notion became necessary for us to describe the image of Griffiths’ Abel-Jacobi map for a generic hypersurface (inspired by previous work of M. Green [21] and C. Voisin [45], [46]), and also to prove the following variant of results of D. Mumford [31] and A. Roitman [34] suggested (and proved in the case dimX = 2) by S. Bloch in Exercise of Appendix to Lecture 1 of [7]: Let X be a smooth proper complex algebraic variety. If there is a morphism of complex varieties S → X inducing a surjective morphism CH0(S)Q → CH0(X)Q, then Γ(X,Ω j X) = 0 for j > dimS. Actually, it turns out that the usual Hodge theory [12] is enough for this. See Prop. 1.4 of Lect. 6 in [46]. But the attempt led us to the following formulation. For a subfield A of R, and a subfield k of C with finite transcendence degree, the category of k-finite mixed A-Hodge structures MHS(A)〈k〉 is defined to be the inductive limit of the category of admissible variations of mixed A-Hodge structures on S = SpecR over kR := k∩R, with R running over finitely generated smooth k-subalgebras of C. (Here an admissible variation on S means an admissible variation of mixed A-Hodge structures on SC := S ⊗kR C in the sense of [27], [44] such that the Hodge filtration, the connection and the polarizations are defined on S/kR.) More generally, for a complex algebraic variety X , there is a k-subalgebra R of C as above such that X is defined over R, and the category MHM(X,A)〈k〉 of k-finite mixed Hodge Modules onX is defined similarly. See (2.1). These can be extended to the mixed sheaves in the sense of [37] where l-adic sheaves can also be included, and the main theorems of this paper hold in the generalized situation. (In fact, they apply even to the subcategory consisting of the objects of geometric origin.) A similar category was defined in [37,(1.9)] for a subfield K of C finitely generated over k, by assuming that the fractional field of R isK in the above definition. (This was inspired by [7], p. 1.20.) Then it is enough to take further the inductive limit over K in order to get the above category of arithmetic mixed Hodge Modules (or sheaves, more generally).