Widths of Complements of Skeleta
Elliot Gathercole
TL;DR
The work addresses quantitative obstructions to ball embeddings in symplectic manifolds arising as affine complements of divisors, by bounding the Gromov width of the complement of a Lagrangian skeleton associated to the divisor. It develops a stratified symplectic framework with commuting Hamiltonians to model hyperplane arrangements and constructs a neck-stretching argument that yields a holomorphic building connecting to a reflected region away from the skeleton. The key contributions include explicit width bounds in terms of algebro-geometric data, a general barrier criterion for the skeleton, sharp results in projective space for generic hyperplane arrangements in low dimensions, and a local ball-embedding criterion from orthogonal crossing models. These results connect Gromov–Witten-type data with symplectic width and provide tools for understanding rigidity phenomena beyond smooth divisors.
Abstract
We establish some sufficient conditions for the Lagrangian skeleton of the affine complement of an effective ample Q-divisor in a smooth rationally connected projective variety to be a Lagrangian barrier in the sense of Biran, and establish bounds on the Gromov width of the complement of the skeleton. We particularly focus on hyperplane arrangements in projective space, where we obtain tight bounds in two dimensions when the divisor is a generic collection of at least three lines.