  Affine varieties among all algebraic varieties have simple structures. For
example, an affine variety does not contain any complete algebraic curve. In
this paper we study affine related properties of strata of $k$-differentials on
smooth curves which parameterize sections of the $k$-th power of the canonical
line bundle with prescribed orders of zeros and poles. We show that if there is
a prescribed pole of order at least $k$, then the corresponding stratum does
not contain any complete curve. Moreover, we explore the amusing question
whether affine invariant manifolds arising from Teichm\"uller dynamics are
affine varieties, and confirm the answer for Teichm\"uller curves, Hurwitz
spaces of torus coverings, hyperelliptic strata as well as some low genus
strata.
