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 VR(M) that encodes homological properties of M. We give lower
bounds for the dimension of VR(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 VR(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, and it recovers a result of Avramov and Halperin on the homotopy
Lie algebra of R. Finally, we completely classify the varieties that can occur
as the cohomological support of a complex over a Golod ring.