Gromov's K-area and symplectic rigidity (Q1924206)

The author presents several results about a new symplectic invariant introduced by M. Gromov. The ingredients are: a closed symplectic manifold \((M^{2n},\Omega)\), a complex vector bundle \(E\to M\) with Hermitian metric \(h\), an \(h\)-preserving connection \(\nabla\) and a \(\Omega\)-compatible almost complex structure \(J\) on \(M\). The latter means that \(g(\xi,\eta)= \Omega(\xi,J\eta)\) is a Riemannian metric. Gromov defined the quantity \[ K\text{-area} (M,\Omega)= \sup_E k^E(M,\Omega) \] where \(E\) ranges over the (homologically significant) vector bundles. Here \[ k^E(M,\Omega)= \sup_{J,h,\nabla}|R^\nabla |^{-1}_{J,H} \quad\text{and}quad |R^\nabla|_{J,h}= \sup_{(\xi,\eta),v}|h(R^\nabla (\xi,\eta)v,v)| \] with \(v\in E_{(\cdot)}\) \(h\)-unit vector and \((\xi,\eta)\) in \(T_{(\cdot)}M\) \(g\)-orthonormal. The author discusses symplectic rigidity type results for the \(K\)-area with respect to individual vector bundles.