The curve cone of almost complex 4-manifolds

In this paper, we study the curve cone of an almost complex $4$-manifold which is tamed by a symplectic form. In particular, we prove the cone theorem as in Mori theory for all such manifolds using the Seiberg-Witten theory. For small rational surfaces and minimal ruled surfaces, we study the configuration of negative curves. We define abstract configuration of negative curves, which records the homological and intersection information of curves. Combinatorial blowdown is the main tool to study these configurations. As an application of our investigation of the curve cone, we prove the Nakai-Moishezon type duality for all almost K\"ahler structures on $\mathbb CP^2\#k\overline{\mathbb CP^2}$ with $k\le 9$ and minimal ruled surfaces with a negative curve. This is proved using a version of Gram-Schmidt orthogonalization process for the $J$-tamed symplectic inflation.