Geography of symplectic Lefschetz fibrations and rational blowdowns

Abstract

We produce simply connected, minimal, symplectic Lefschetz fibrations realizing all the lattice points in the symplectic geography plane below the Noether line. This provides a symplectic extension of the classical works populating the complex geography plane with holomorphic Lefschetz fibrations. Our examples are obtained by rationally blowing down Lefschetz fibrations with clustered nodal fibers, the total spaces of which are potentially new homotopy elliptic surfaces. Similarly, clustering nodal fibers on higher genera Lefschetz fibrations on standard rational surfaces, we get rational blowdown configurations that yield new constructions of small symplectic exotic $4$-manifolds. We present an example of a construction of a minimal symplectic exotic $\mathbb{CP} \# 5\,\overline{\mathbb{CP}}$ through this procedure applied to a genus-$3$ fibration.