Geography of symplectic Lefschetz fibrations and rational blowdowns

TL;DR

This paper constructs new simply connected minimal symplectic Lefschetz fibrations that realize all lattice points below the Noether line, using rational blowdowns, leading to novel symplectic 4-manifolds and exotic structures.

Contribution

It extends symplectic geography by producing all lattice points below the Noether line via rational blowdowns of Lefschetz fibrations, introducing potentially new homotopy elliptic surfaces.

Findings

Constructed all lattice points below the Noether line with symplectic Lefschetz fibrations.

Produced new homotopy elliptic surfaces through rational blowdowns.

Built a minimal symplectic exotic  with genus-3 fibration.

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.