  For a one parameter family of Calabi-Yau threefolds, Green, Griffiths and
Kerr have expressed the total singularities in terms of the degrees of Hodge
bundles and Euler number of the general fiber. In this paper, we show that the
total singularities can be expressed by the sum of asymptotic values of BCOV
invariants, studied by Fang, Lu and Yoshikawa. On the other hand, by using
Yau's Schwarz lemma, we prove Arakelov type inequalities and Euler number bound
for Calabi-Yau family over a compact Riemann surface.
