  We describe a conjectural construction (in the spirit of Hilbert's 12th
problem) of units in abelian extensions of certain base fields which are
neither totally real nor CM. These base fields are quadratic extensions with
exactly one complex place of a totally real number field F, and are referred to
as Almost Totally real (ATR) extensions.
  Our construction involves certain null-homologous topological cycles on the
Hilbert modular variety attached to F. The special units are the images of
these cycles under a map defined by integration of weight two Eisenstein series
on GL_2(F). This map is formally analogous to the higher Abel-Jacobi maps that
arise in the theory of algebraic cycles. We show that our conjecture is
compatible with Stark's conjecture for ATR extensions; it is, however, a
genuine strengthening of Stark's conjecture in this context since it gives an
analytic formula for the arguments of the Stark units and not just for their
absolute values. The last section provides numerical evidences for our
conjecture.
