  For the moduli spaces of Abelian differentials, the Euler characteristic is
one of the most basic intrinsic topological invariants. We give a formula for
the Euler characteristic that relies on intersection theory on the smooth
compactification by multi-scale differentials. It is a consequence of a formula
for the full Chern polynomial of the cotangent bundle of the compactification.
  The main new technical tools are an Euler sequence for the cotangent bundle
of the moduli space of Abelian differentials and computational tools in the
Chow ring, such as normal bundles to boundary divisors.
