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.

arXiv comment47 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

References in corpus (4)

• Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds
• Lefschetz pencils and divisors in moduli space
• Lefschetz fibrations and the Hodge bundle
• Serre-Taubes duality for pseudoholomorphic curves

Cited by in corpus (5)

• Clustering of critical points in Lefschetz fibrations and the symplectic Szpiro inequality
• Topology of broken Lefschetz fibrations and near-symplectic 4-manifolds
• Serre-Taubes duality for pseudoholomorphic curves
• A note on Hilbert schemes of nodal curves
• Monodromy, vanishing cycles, knots and the adjoint quotient