  The tau function on the moduli space of generic holomorphic 1-differentials
on complex algebraic curves is interpreted as a section of a line bundle on the
projectivized Hodge bundle over the moduli space of stable curves. The
asymptotics of the tau function near the boundary of the moduli space of
1-differentials is computed, and an explicit expression for the pullback of the
Hodge class on the projectivized Hodge bundle in terms of the tautological
class and the classes of boundary divisors is derived. This expression is used
to clarify the geometric meaning of the Kontsevich-Zorich formula for the sum
of the Lyapunov exponents associated with the Teichm\"uller flow on the Hodge
bundle.
