Lefschetz pencils and the canonical class for symplectic four-manifolds

By Simon Donaldson, I. Smith. 2000

80 citations recorded by the source corpus.

We present a new proof of a result due to Taubes: if (X,ω) is a closed symplectic four-manifold with b+(X)>1+b1(X) and λ[ω]∈H2(X;Q) for some λ∈R+, then the Poincare dual of KX 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f68adbfd09f36bb53b79389c70b28b0c981edfda

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In Geometry and Topology

1. I. Smith
2. Michael Hutchings
3. Peter S. Ozsv'ath
4. Zolt'an Szab'o
5. P. Seidel
6. Timothy Perutz
7. Ciprian Manolescu
8. A. Stipsicz

In Mathematical Physics

1. Peter S. Ozsv'ath
2. Zolt'an Szab'o
3. I. Smith
4. Michael Hutchings
5. Timothy Perutz
6. Ciprian Manolescu
7. A. Stipsicz
8. Jacob Rasmussen

Citations in the corpus

• Wrinkled fibrations on near-symplectic manifolds (2007, background): “In [6], Donaldson and Smith define an invariant of a symplectic manifold X by counting holomorphic sections of a relative Hilbert scheme that is constructed from a Lefschetz fibration on a blow-up of X .”
• The symplectic vortex equations and invariants of Hamiltonian group actions (2001, methodology): “If this program can be carried out then, combined with the work of Donaldson–Smith in [10], it might lead to an alternative proof of Taubes’ theorem [33, 34, 35] about the relation between the Seiberg–Witten and the Gromov invariants.”
• Lefschetz pencils and divisors in moduli space (2000, methodology): “5) Remark In [5] and [22], symplectic submanifolds are constructed by studying the Gromov invariants of fibrations of symmetric products associated to a Lefschetz pencil.”
• Holomorphic disks and topological invariants for closed three-manifolds (2001, background): “f : S(X) −→ D, whose fiber over t is identified as S(X)t ∼= Sym(Xt) (see also [6] for a symplectic construction of this object).”
• Lagrangian matching invariants for fibred four-manifolds: I (2006, background): “In the presence of a symplectic Lefschetz fibration, the Donaldson–Smith invariant [5; 26] mediates between the gauge-theoretic and symplectic viewpoints.”