Symplectic 4-Manifolds on the Noether Line and between the Noether and Half Noether Lines

TL;DR

This paper constructs new simply connected symplectic 4-manifolds with exotic smooth structures on the Noether line and between the Noether and half Noether lines, using star surgeries and complex singularities.

Contribution

It introduces novel constructions of symplectic 4-manifolds with specific topological properties via star surgeries and complex singularity configurations.

Findings

Constructed simply connected minimal symplectic 4-manifolds with one Seiberg-Witten basic class.

Developed methods to embed star-shaped plumbings in elliptic surfaces.

Provided geometric constructions of complex singularities without monodromy arguments.

Abstract

We construct simply connected, minimal, symplectic 4-manifolds with exotic smooth structures and each with one Seiberg-Witten basic class up to sign, on the Noether line and between the Noether and half Noether lines by star surgeries introduced by Karakurt and Starkston, and by using complex singularities. We also construct certain configurations of complex singularities in the rational elliptic surfaces geometrically, without using any monodromy arguments. By using these configurations, we give symplectic embeddings of star shaped plumbings inside (some blow-ups of) elliptic surfaces.