# Symplectic submanifolds in dimension $6$ from hyperelliptic Lefschetz fibrations

## TL;DR

This paper constructs a simply connected symplectic 6-manifold with infinitely many homologous but homotopy-inequivalent symplectic submanifolds using hyperelliptic Lefschetz fibrations, showing they cannot admit complex structures.

## Contribution

It introduces a novel construction of symplectic submanifolds in dimension 6 via hyperelliptic Lefschetz fibrations, expanding understanding of symplectic topology in higher dimensions.

## Key findings

- Existence of infinitely many non-homotopy equivalent symplectic submanifolds
- Submanifolds are homologous but cannot admit complex structures
- Extension of results to higher-dimensional symplectic submanifolds

## 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 $\leq 3$ are mutually diffeomorphic.