J-holomorphic curves in a nef class

Taubes established fundamental properties of J-holomorphic subvarieties in dimension 4 in [8]. In this paper, we further investigate properties of reducible J-holomorphic subvarieties. We offer an upper bound of the total genus of a subvariety when the class of the subvariety is J-nef. For a spherical class, it has particularly strong consequences. It is shown that, for any tamed J, each irreducible component is a smooth rational curve. It might be even new when J is integrable. We also completely classify configurations of maximal dimension. To prove these results, we treat subvarieties as weighted graphs and introduce several combinatorial moves.