Simply connected minimal symplectic 4-manifolds with signature less than --1

TL;DR

This paper constructs new examples of minimal, simply connected symplectic 4-manifolds with specified Euler characteristics and signatures, expanding the known landscape of such manifolds especially with negative signatures.

Contribution

It provides explicit constructions of minimal, simply connected symplectic 4-manifolds with a wide range of Euler characteristics and signatures, including new examples with signature less than -1.

Findings

Constructed symplectic 4-manifolds for all admissible pairs (e,σ) with specified conditions.

Produced infinite families of manifolds with signature zero and -1 for large Euler characteristics.

Extended the known existence results for minimal, simply connected symplectic 4-manifolds.

Abstract

For each pair $(e,\sigma)$ of integers satisfying $2e+3\sigma\ge 0$, $\sigma\leq -2$, and $e+\sigma\equiv 0\pmod{4}$, with four exceptions, we construct a minimal, simply connected symplectic 4-manifold with Euler characteristic $e$ and signature $\sigma$. We also produce simply connected, minimal symplectic 4-manifolds with signature zero (resp. signature -1) with Euler characteristic $4k$ (resp. $4k+1$) for all $k\ge 46$ (resp. $k\ge 49$).