Abstract

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.