Gromov $K\mkern-4mu$–area and jumping curves in $\mathbb{CP}^n$

Abstract

We give here some extensions of Gromov’s and Polterovich’s theorems on $k$–area of ${ℂ\mathbb{P}}^n$, particularly in the symplectic and Hamiltonian context. Our main methods involve Gromov–Witten theory, and some connections with Bott periodicity and the theory of loop groups. The argument is closely connected with the study of jumping curves in ${ℂ\mathbb{P}}^n$, and as an upshot we prove a new symplectic-geometric theorem on these jumping curves.