In this talk, a compact Kaehler manifold of even-dimension 2n with h^{2,0}=h^{2n,0}=0
is called an Enriques manifold if it is a free quotient of either a Calabi-Yau manifold or
a (simple) hyperkaehler manifold. We construct an invariant of Enriques manifold using
analytic torsion. It turns out that the function on the moduli space thus obtained is a potential
function of the Weil-Peterson metric. From this, we conclude that the moduli space of
Enriques manifolds contains no compact subvarieties of positive dimension, which
partially extends the related result of Borcherds to higher dimension. If time allows,
we will also give an explicit formula for the invariant in terms of modular forms for some
interesting examples of Enriques manifolds such as those associated to hyper elliptic
curves and those associated to Enriques surfaces.