The Gromov width of complex Grassmannians

Abstract

We show that the Gromov width of the Grassmannian of complex $k$–planes in ${ℂ}^n$ is equal to one when the symplectic form is normalized so that it generates the integral cohomology in degree 2. We deduce the lower bound from more general results. For example, if a compact manifold $N$ with an integral symplectic form $\omega$ admits a Hamiltonian circle action with a fixed point $p$ such that all the isotropy weights at $p$ are equal to one, then the Gromov width of $\left(N,\omega \right)$ is at least one. We use holomorphic techniques to prove the upper bound.