On possible values of the signature of flat symplectic bundles over surfaces with boundary

Abstract

We show that every integer in the interval $[2p\chi (\Sigma ),-2p\chi (\Sigma )]$ is achieved by the signature of a rank $2p$ flat symplectic bundle over a surface with boundary $\Sigma$. When $p=1$, one can prescribe the type (elliptic, parabolic, hyperbolic) of the holonomy along the boundary.