# Enriques manifolds and analytic torsion

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.