  We give a formalism of arithmetic mixed sheaves including the case of
arithmetic mixed Hodge structures, and show the nonvanishing of certain higher
extension groups, and also the nontriviality of the second Abel-Jacobi map for
zero cycles on a smooth proper complex variety of any dimension under the
existence of a nontrivial global two-form. For two codimensional cycles, the
injectivity of the cycle map is reduced to that of the Abel-Jacobi map for
smooth projective varieties over number fields. (This shows that Asakura's
additional hypothesis is unnecessary). Here it is also possible to use the
systems of realizations in the definition of the cycle map and the Abel-Jacobi
map. Some arguments can be extended to higher Chow groups, and we get evidence
for a conjecture of C. Voisin on the countability of indecomposable higher
cycles.
