Lefschetz pencils and the canonical class for symplectic 4-manifolds

Details Lefschetz pencils and the canonical class for symplectic 4-manifolds Lefschetz pencils and the canonical class for symplectic 4-manifolds

2000-12-10

arxiv.org/abs/math/0012067v2

Mathematics

Symplectic Geometry

47 pages. The role of the transversality argument in Section 7 has been clarified, and the treatments of smoothing nodal surfa

Scientific paper

We present a new proof of a result due to Taubes: if X is a closed symplectic four-manifold with b_+(X) > 1+b_1(X) and with some positive multiple of the symplectic form a rational class, then the Poincare dual of the canonical class of X may be represented by an embedded symplectic submanifold. The result builds on the existence of Lefschetz pencils on symplectic four-manifolds. We approach the topological problem of constructing submanifolds with locally positive intersections via almost complex geometry. The crux of the argument is that a Gromov invariant counting pseudoholomorphic sections of an associated bundle of symmetric products is non-zero.

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