This expository article considers a circle of Lie-theoretic ideas involving Hessenberg varieties, Poisson geometry, and wonderful compactifications. In more detail, one may associate a symplectic Hamiltonian
G
G
-variety
μ
:
G
×
S
⟶
g
\mu :G\times \mathcal {S}\longrightarrow \mathfrak {g}
to each complex semisimple Lie algebra
g
\mathfrak {g}
with adjoint group
G
G
and fixed Kostant section
S
⊆
g
\mathcal {S}\subseteq \mathfrak {g}
. This variety is one of Bielawski’s hyperkähler slices, and it is central to Moore and Tachikawa’s work on topological quantum field theories. It also bears a close relation to two log symplectic Hamiltonian
G
G
-varieties
μ
¯
S
:
G
×
S
¯
⟶
g
\overline {\mu }_{\mathcal {S}}:\overline {G\times \mathcal {S}}\longrightarrow \mathfrak {g}
and
ν
:
H
e
s
s
⟶
g
\nu :\mathrm {Hess}\longrightarrow \mathfrak {g}
. The former is a Poisson transversal in the log cotangent bundle of the wonderful compactification
G
¯
\overline {G}
, while the latter is the standard family of Hessenberg varieties. Each of
μ
¯
\overline {\mu }
and
ν
\nu
is known to be a fibrewise compactification of
μ
\mu
.
We exploit the theory of Poisson slices to relate the fibrewise compactifications mentioned above. Our work is shown to be compatible with a Poisson isomorphism obtained by Bălibanu.