Lefschetz pencils and the canonical class for symplectic 4-manifolds
Abstract
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.
Keywords
Cite
@article{arxiv.math/0012067,
title = {Lefschetz pencils and the canonical class for symplectic 4-manifolds},
author = {Simon Donaldson and Ivan Smith},
journal= {arXiv preprint arXiv:math/0012067},
year = {2007}
}
Comments
47 pages. The role of the transversality argument in Section 7 has been clarified, and the treatments of smoothing nodal surfaces controlling bubbling have been revised. Various other minor changes