Symplectic submanifolds of \({\varvec{D^2\times Y^4}}\)

Let \(\pi : Y^4 \rightarrow D^2\) $\pi :Y^4\to D^2$ be a symplectic Lefschetz fibration and \(\Theta \) $\mathrm{\Theta }$ be a certain symplectic structure on \(D^2\times Y\) $D^2\times Y$ , where \(D^2\) $D^2$ is a 2-disc and Y is a compact oriented 4-manifold. In this article, we describe a method to construct a class of symplectic submanifolds of \((D^2 \times Y, \Theta )\) $(D^2\times Y,\mathrm{\Theta })$ such that the projection of \(D^2 \times Y\) $D^2\times Y$ on its first factor restricted to each of these submanifolds is a symplectic Lefschetz fibration. In order to achieve this, given a Lefschetz fibration \(\pi : Y^4 \rightarrow D^2\) $\pi :Y^4\to D^2$ , we use the notion of a pull back Lefschetz fibration \(\pi ^\prime :f^*(Y, \pi )\subset D \times Y\rightarrow D^2\) ${\pi }^{\prime }:f^{\ast }(Y,\pi )\subset D\times Y\to D^2$ of the fibration \(\pi \) $\pi$ by a smooth map \(f: D^2 \rightarrow D^2\) $f:D^2\to D^2$ satisfying certain properties. This notion was introduced by Zuddas. As an application of this, we prove that every allowable bounded Lefschetz fibration with the fiber having the connected boundary and genus \(g\ge 3\) $g\ge 3$ can be realized as a proper symplectic submanifold of \((D^2\times \mathcal {D}E(n), \Theta )\) $(D^2\times \mathcal{D}E(n),\mathrm{\Theta })$ possibly with corners for \(n=-3, -4\) $n=-3,-4$ , where \(\mathcal {D}E(n)\) $\mathcal{D}E(n)$ is the unit 2–disc bundle over \(S^2\) $S^2$ with the Euler number n and \(\Theta \) $\mathrm{\Theta }$ is a certain fixed symplectic structure on \(D^2\times \mathcal {D}E(n)\) $D^2\times \mathcal{D}E(n)$ .