Algebraic Geometry Seminar: "Symplectic embeddings via algebraic capacities"

Abstract: A central question in modern symplectic geometry is “when does one symplectic manifold embed in another?”, which has been seen to be highly subtle from landmark results such as Gromov’s nonsqueezing theorem and more recently McDuff’s solution of the Hofer conjecture. The latter shows that certain obstructions coming from optimisation problems in a symplectic homology theory completely characterise when embeddings can occur. Connections with algebraic geometry began to emerge from work of Biran and McDuff—Polterovich describing tight connections between symplectic embeddings of disjoint unions of balls (i.e. ball packing problems) and the algebraic geometry of blowups of P^2. This talk will describe work over the last few years continuing in this vein of bringing algebraic techniques to bear on symplectic embedding problems. We describe a sequence of invariants of a polarised algebraic surface — algebraic capacities — that serve to obstruct symplectic embeddings, in many interesting cases sharply. Using this perspective we resolve two conjectures on the asymptotics of certain invariants in symplectic geometry, and prove a combinatorial bound on the Gromov width of toric surfaces conjectured by Averkov—Nill—Hofscheier.

Host: Matt Kerr