Stability and existence of surfaces in symplectic 4-manifolds with \(b^+=1\) (Q1668310)

The main result of this article is about the stability of simple normal crossings (or, positively intersecting transverse) symplectic surfaces (real dimension 2) in symplectic 4-folds with $b^+=1$; e.g., rational 4-manifolds. Other conclusions, including similar statements for configurations of Lagrangian surfaces, have been driven from the main result and previous works of the authors. The results are proved using standard techniques in $J$-holomorphic curves, Seiberg Witten theory, and degeneration/smoothing of symplectic manifolds. \par More precisely, (by a slight change of notation and terminology) they consider a positively intersecting transverse configuration $V=\bigcup_{v\in G} V_v$ of oriented surfaces with a dual graph $G$ in an oriented $4$-manifold $X$. Each edge $e$ in $G$ corresponds to an intersection point of two components $V_v$ and $V_{v'}$ and is weighted by the intersection number at that intersection point (which is a positive integer). $G$ is called simple if all the weights are $1$. This is called a curve configuration if $X$ has a symplectic structure $\omega$, $V_v$ are all symplectic, and there exists an almost complex structure $J$ compatible with $\omega$ making each $V$ a (nodal) $J$-holomorphic curve. \par A curve configuration $V$ in $(X,\omega)$ is called stable if for every other symplectic structure deformation $\widetilde\omega$ equivalent to $\omega$ satisfying $\widetilde\omega(V_v)>0$, for all $v\in G$, there exists an associated curve configuration $\widetilde{V}$ with the same dual graph $G$ and the same homology classes $[\widetilde{V}_v]=[V_v]$. \par The main theorem (Theorem 1.3) says that if $X$ has $b^+=1$, then $V$ is stable. Moreover, $\widetilde{V}$ can be chosen to be smoothly isotopic to $V$. \par As a corollary of this, using a surgery that replaces a symplectic configuration with a Lagrangian configuration (and vice versa), they obtain a similar result for Lagrangian ADE-type configurations in rational or ruled 4-manifolds. \par Finally, they use the results to classify $-4$-spheres (smooth and symplectic) in rational 4-manifolds. They construct symplectic $-4$-spheres using ``tilted transport''. This adds to the previously obtained classifications of $-1,-2,-3$ spheres in rational $4$-manifolds.