Transverse hilbert schemes, bi-hamiltonian systems and hyperkähler geometry

Dedicated to the memory of Sir Michael Francis Atiyah (1929-2019) We give a characterization of Atiyah’s and Hitchin’s transverse Hilbert schemes of points on a symplectic surface in terms of bi-Poisson structures. Furthermore, we describe the geometry of hyperkähler manifolds arising from the transverse Hilbert scheme construction, with particular attention paid to the monopole moduli spaces.