2001/06/26 by Ivan Smith, Smith, Ivan
Mathematics · #53D35 #57R17 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG) #math.AG #math.SG #msc:53D35 #msc:57R17
paper · pdf · doi:10.48550/arxiv.math/0106220
52 pages, no figures; Section 5 has been re-written to include some additional motivation for the main conjecture (cf. Theorem 1.2)
openalex publication_date 2001/06/26 · arxiv created 2001/07/13 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 b1 = 0. This reproves another theorem due to Taubes: two symplectic homology projective planes with negative canonical class and equal volume are symplectomorphic.