  The authors compute the support varieties of all irreducible modules for the
small quantum group $u_\zeta(\mathfrak{g})$, where $\mathfrak{g}$ is a simple
complex Lie algebra, and $\zeta$ is a primitive $\ell$-th root of unity with
$\ell$ larger than the Coxeter number of $\mathfrak{g}$. The calculation
employs the prior calculations and techniques of Ostrik and of
Nakano--Parshall--Vella, as well as deep results involving the validity of the
Lusztig character formula for quantum groups and the positivity of parabolic
Kazhdan-Lusztig polynomials for the affine Weyl group. Analogous support
variety calculations are provided for the first Frobenius kernel $G_1$ of a
reductive algebraic group scheme $G$ defined over the prime field
$\mathbb{F}_p$.
