Given an anticanonical divisor in a projective variety, one naturally obtains a monotone Kähler manifold, and the divisor complement is naturally a Liouville manifold. For certain kinds of singular divisors, we will outline a result obtaining rigid (in particular, superheavy) neighbourhoods of the Lagrangian skeleton of the complement, of prescribed volume dependent on the divisor, and illustrate this with some interesting examples where the skeleton itself is superheavy.