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We give here some extensions of Gromov's and Polterovich's theorems on$\karea$ of $ \mathbb{CP} ^{n}$, particularly in the symplectic and Hamiltoniancontext. Our main methods involve Gromov-Witten theory, and some connectionswith Bott periodicity, and loop groups. The argument is closely connected withstudy of jumping curves in $ \mathbb{CP} ^{n}$, and as an upshot we prove a newsymplectic geometric theorem on these jumping curves.

Gromov K-area and jumping curves in CP^n