  In this paper, we define a certain Hodge-theoretic structure for an arbitrary
variety X over the complex number field by using the theory of mixed Hodge
module due to Morihiko Saito. We call it an arithmetic Hodge structure of X. It
is shown that extension groups of arithmetic Hodge structure do not vanish even
for degree $\geq2$. Moreover, we define higher Abel-Jacobi maps from Bloch's
higher Chow groups of X to these extension groups. These maps essentially
involve the classical Abel-Jacobi maps by Weil and Griffiths, and Mumford's
infinitesimal invariants of 0-cycles on surfaces.
