A complex Fano projective variety M (for example, projective space) is canonically a monotone symplectic manifold. Given an effective anticanonical divisor D, we obtain a distinguished (and symplectically, very special) subset L, the skeleton, onto which the complement M-D retracts, and an increasing family of compact neighbourhoods of L which exhaust M-D.
In a symplectic manifold, we can ask if a subset is rigid: that is, can it be displaced from itself by a Hamiltonian isotopy? In the above setting, we can ask a quantitative refinement of this question: what is the smallest neighbourhood of L in our family which is rigid in M?
I will discuss a variation of this question involving spectral invariants, and how it can be answered for kinds of singular (i.e. possibly not SNC) divisors D, and how this answer depends on properties of D, and give some examples of interesting rigid isotropic cell complexes L obtained in this manner.