J-holomorphic curves in a nef class

Tian Jun Li, Weiyi Zhang

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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.

Original languageEnglish (US)
Pages (from-to)12070-12104
Number of pages35
JournalInternational Mathematics Research Notices
Issue number22
StatePublished - 2015

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