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

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.