Widths of Complements of Skeleta

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.

💡 Research Summary

The paper investigates the relationship between Lagrangian skeleta arising from affine complements of effective ample ℚ‑divisors and the notion of a Lagrangian barrier introduced by Biran. A Lagrangian barrier is a closed subset B of a symplectic manifold (M, ω) such that the Gromov width of the complement M \ B is strictly smaller than the Gromov width of M. Biran’s original result applied to smooth complex hypersurfaces D in a Kähler manifold: for a divisor representing k