Stability and Existence of Surfaces in Symplectic 4-Manifolds with $b^+=1$

TL;DR

This paper investigates the stability and existence of symplectic surfaces in 4-manifolds with specific topological properties, introducing new techniques to classify and construct such surfaces.

Contribution

It establishes stability results for symplectic surfaces in 4-manifolds with $b^+=1$ and introduces the tilted transport method for constructing spheres.

Findings

Existence of Lagrangian ADE-configurations in symplectic 4-manifolds.

Classification of negative symplectic spheres in manifolds with $oxed{- ext{infinity}}$ Kodaira dimension.

Introduction of the tilted transport construction technique.

Abstract

We establish various stability results for symplectic surfaces in symplectic $4-$manifolds with $b^+=1$. These results are then applied to prove the existence of representatives of Lagrangian ADE-configurations as well as to classify negative symplectic spheres in symplectic $4-$manifolds with $\kappa=-\infty$. This involves the explicit construction of spheres in rational manifolds via a new construction technique called the tilted transport.