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 which are mutually homologous but homotopy inequivalent and moreover, which 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 \(\leq 3\) are mutually diffeomorphic.