Symplectic submanifolds in dimension 6 from hyperelliptic Lefschetz fibrations

Abstract

We provide a closed, simply connected, symplectic $6$-manifold having infinitely many codimension $2$ symplectic submanifolds. These are mutually homologous but homotopy inequivalent, and furthermore, they cannot admit complex structures. The key ingredient for the construction is hyperelliptic Lefschetz fibrations on $4$-manifolds. As a corollary, we present a similar result on symplectic submanifolds of codimension $2$ in higher dimensions. In the appendix, we give a proof of the well-known fact that all symplectic submanifolds of codimension $2$ in $({\mathbb{CP}}^3,{\omega }_{\mathrm{FS}})$ of a fixed degree $\le 3$ are mutually diffeomorphic.