Over a local ring
R
R
, the theory of cohomological support varieties attaches to any bounded complex
M
M
of finitely generated
R
R
-modules an algebraic variety
V
R
(
M
)
{\mathrm {V}}_R(M)
that encodes homological properties of
M
M
. We give lower bounds for the dimension of
V
R
(
M
)
{\mathrm {V}}_R(M)
in terms of classical invariants of
R
R
. In particular, when
R
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
M
has finite projective dimension, we also give an upper bound for
dim
V
R
(
M
)
\dim {\mathrm {V}}_R(M)
in terms of the dimension of the radical of the homotopy Lie algebra of
R
R
. This leads to an improvement of a bound due to Avramov, Buchweitz, Iyengar, and Miller on the Loewy lengths of finite free complexes, and it recovers a result of Avramov and Halperin on the homotopy Lie algebra of
R
R
. Finally, we completely classify the varieties that can occur as the cohomological support of a complex over a Golod ring.