J-holomorphic curves in a nef class
Abstract
Taubes established fundamental properties of $J-$holomorphic subvarieties in dimension 4 in \cite{T1}. 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. We also completely classify configurations of maximal dimension. To prove these results we treat subvarieties as weighted graphs and introduce several combinatorial moves.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2012
- DOI:
- 10.48550/arXiv.1210.3337
- arXiv:
- arXiv:1210.3337
- Bibcode:
- 2012arXiv1210.3337L
- Keywords:
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- Mathematics - Symplectic Geometry;
- Mathematics - Algebraic Geometry;
- Mathematics - Complex Variables;
- Mathematics - Geometric Topology
- E-Print:
- 30 pages. v2 Section 4.3 revised