  We prove that the nonvarying strata of abelian and quadratic differentials in
[CM1, CM2] have trivial tautological rings and are affine varieties. We also
prove that strata of $k$-differentials of infinite area are affine varieties
for all $k$. Vanishing of homology in degree higher than the complex dimension
follows as a consequence for these affine strata. Moreover we prove that the
stratification of the Hodge bundle for abelian and quadratic differentials of
finite area is extremal in the sense that merging two zeros in each stratum
leads to an extremal effective divisor in the boundary. A common feature
throughout these results is a relation of divisor classes in strata of
differentials as well as its incarnation in Teichm\"uller dynamics.
