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J-holomorphic curves in a nef class
Taubes established fundamental properties of holomorphic subvarieties in
dimension 4 in \cite{T1}. In this paper, we further investigate properties of
reducible holomorphic subvarieties. We offer an upper bound of the total
genus of a subvariety when the class of the subvariety is nef. For a
spherical class, it has particularly strong consequences. It is shown that, for
any tamed , each irreducible component is a smooth rational curve. We also
completely classify configurations of maximal dimension. To prove these results
we treat subvarieties as weighted graphs and introduce several combinatorial
moves.Comment: 30 pages. v2 Section 4.3 revised; minor changes elsewhere. v3
mistakes correcte
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