  Let $C$ be a smooth complex projective curve and $G$ a connected complex
reductive group. We prove that if the center $Z(G)$ of $G$ is disconnected,
then the Kirwan map $H^*\big(\operatorname{Bun}(G,C),\mathbb{Q}\big)\rightarrow
H^*\big(\mathcal{M}_{\operatorname{Higgs}}^{\operatorname{ss}},\mathbb{Q}\big)$
from the cohomology of the moduli stack of $G$-bundles to the moduli stack of
semistable $G$-Higgs bundles, fails to be surjective: more precisely, the
"variant cohomology" (and variant intersection cohomology) of the stack
$\mathcal{M}_{\operatorname{Higgs}}^{\operatorname{ss}}$ of semistable
$G$-Higgs bundles, is always nontrivial. We also show that the image of the
pullback map
$H^*\big(M_{\operatorname{Higgs}}^{\operatorname{ss}},\mathbb{Q}\big)\rightarrow
H^*\big(\mathcal{M}_{\operatorname{Higgs}}^{\operatorname{ss}},\mathbb{Q}\big)$,
from the cohomology of the moduli space of semistable $G$-Higgs bundles to the
stack of semistable $G$-Higgs bundles, cannot be contained in the image of the
Kirwan map. The proof uses a Borel-Quillen--style localization result for
equivariant cohomology of stacks to reduce to an explicit construction and
calculation.
