  Let $\zeta$ be a complex $\ell$th root of unity for an odd integer $\ell>1$.
For any complex simple Lie algebra $\mathfrak g$, let
$u_\zeta=u_\zeta({\mathfrak g})$ be the associated "small" quantum enveloping
algebra. In general, little is known about the representation theory of quantum
groups (resp., algebraic groups) when $l$ (resp., $p$) is smaller than the
Coxeter number $h$ of the underlying root system. For example, Lusztig's
conjecture concerning the characters of the rational irreducible $G$-modules
stipulates that $p \geq h$. The main result in this paper provides a
surprisingly uniform answer for the cohomology algebra
$\opH^\bullet(u_\zeta,{\mathbb C})$ of the small quantum group. When $\ell>h$,
this cohomology algebra has been calculated by Ginzburg and Kumar \cite{GK}.
Our result requires powerful tools from complex geometry and a detailed
knowledge of the geometry of the nullcone of $\mathfrak g$. In this way, the
methods point out difficulties present in obtaining similar results for the
restricted enveloping algebra $u$ in small characteristics, though they do
provide some clarification of known results there also. Finally, we establish
that if $M$ is a finite dimensional $u_\zeta$-module, then
$\opH^\bullet(u_\zeta,M)$ is a finitely generated $\opH^\bullet(u_\zeta,\mathbb
C)$-module, and we obtain new results on the theory of support varieties for
$u_\zeta$.
