Serre-Taubes duality for pseudoholomorphic curves

Details Serre-Taubes duality for pseudoholomorphic curves Serre-Taubes duality for pseudoholomorphic curves

2001-06-26

arxiv.org/abs/math/0106220v2

Mathematics

Symplectic Geometry

52 pages, no figures; Section 5 has been re-written to include some additional motivation for the main conjecture (cf. Theorem

Scientific paper

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.

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