Widths of Complements of Skeleta

TL;DR

This paper investigates the symplectic geometry of complements of divisors in rationally connected varieties, providing conditions for Lagrangian skeletons to act as barriers and establishing bounds on their Gromov width, especially for hyperplane arrangements.

Contribution

It introduces new sufficient conditions for Lagrangian skeletons to serve as barriers and derives bounds on Gromov width, with specific results for hyperplane arrangements in projective space.

Findings

Lagrangian skeletons can act as barriers under certain conditions

Bounds on Gromov width are established for complements of divisors

Tight bounds are obtained for hyperplane arrangements in two dimensions

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.