  We provide formulas for the Chern classes of linear submanifolds of the
moduli spaces of Abelian differentials and hence for their Euler
characteristic. This includes as special case the moduli spaces of
k-differentials, for which we set up the full intersection theory package and
implement it in the sage-program diffstrata.
  As an application, we give an algebraic proof of the theorems of
Deligne-Mostow and Thurston that suitable compactifications of moduli spaces of
k-differentials on the 5-punctured projective line with weights satisfying the
INT-condition are quotients of the complex two-ball.
