Serre-Taubes duality for pseudoholomorphic curves

TL;DR

This paper extends Taubes' duality for Gromov invariants in symplectic four-manifolds by interpreting it through Serre duality on Lefschetz pencil fibers, providing new proofs and insights into symplectic surface existence and classification.

Contribution

It interprets Taubes' duality via Serre duality on Lefschetz pencils and offers new proofs for symplectic surface existence and symplectomorphism classification in four-manifolds.

Findings

Established Serre duality symmetry for invariants counting sections of symmetric product bundles.

Provided a new proof for the existence of symplectic surfaces in certain four-manifolds.

Reproved that symplectic homology projective planes with specific conditions are symplectomorphic.

Abstract

According to Taubes, the Gromov invariants of a symplectic four-manifold X with b_+ > 1 satisfy the duality Gr(A) = +/- Gr(K-A), where K is Poincare dual to the canonical class. Extending joint work with Simon Donaldson in math.SG/0012067, we interpret this result in terms of Serre duality on the fibres of a Lefschetz pencil, by proving an analogous symmetry for invariants counting sections of associated bundles of symmetric products. Using similar methods we give a new proof of an existence theorem for symplectic surfaces in four-manifolds with b_+ = 1 and b_1 = 0. This reproves another theorem due to Taubes: two symplectic homology projective planes with negative canonical class and equal volume are symplectomorphic.