  Over a local ring $R$, the theory of cohomological support varieties attaches
to any bounded complex $M$ of finitely generated $R$-modules an algebraic
variety $V_R(M)$ that encodes homological properties of $M$. We give lower
bounds for the dimension of $V_R(M)$ in terms of classical invariants of $R$.
In particular, when $R$ is Cohen-Macaulay and not complete intersection we find
that there are always varieties that cannot be realized as the cohomological
support of any complex. When $M$ has finite projective dimension, we also give
an upper bound for $ \dim V_R(M)$ in terms of the dimension of the radical of
the homotopy Lie algebra of $R$. This leads to an improvement of a bound due to
Avramov, Buchweitz, Iyengar, and Miller on the Loewy lengths of finite free
complexes. Finally, we completely classify the varieties that can occur as the
cohomological support of a complex over a Golod ring.
