[Paper Review] Lefschetz pencils and the canonical class for symplectic 4-manifolds

Simon Donaldson, Ivan Smith|ArXiv.org|Dec 10, 2000

Geometric and Algebraic Topology30 references15 citations

TL;DR

This paper presents a new, symplectic-geometric proof of Taubes' result that the canonical class of a symplectic 4-manifold with $ b_+ > 1 + b_1 $ and rational symplectic class can be represented by an embedded symplectic surface. Using Lefschetz pencils and pseudoholomorphic sections of symmetric product bundles, the authors establish a non-vanishing Gromov invariant, avoiding Seiberg-Witten theory and translating complex algebraic geometry techniques into the symplectic setting.

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.

Motivation & Objective

• To provide a new proof of Taubes' result on the representability of the canonical class by an embedded symplectic surface in symplectic 4-manifolds.
• To avoid reliance on Seiberg-Witten theory by using almost-complex geometry and pseudoholomorphic curve techniques.
• To extend algebraic geometry arguments—based on holomorphic 2-forms and their zero-sets—into the non-integrable symplectic setting.
• To establish the existence of a non-vanishing Gromov invariant counting pseudoholomorphic sections in symmetric product fibrations.
• To demonstrate that the canonical class can be realized as a symplectic submanifold under the condition that a multiple of $[ω]$ is rational.

Proposed method

• Utilize Lefschetz pencils on symplectic 4-manifolds, obtained via blow-up, to construct a fibration $\pi: X' \to S^2$ with Riemann surface fibers.
• Construct a bundle $X_r(f)$ of symmetric products over the fibration, with $r = 2g - 2$, to model the geometry of holomorphic sections.
• Define a Gromov invariant counting pseudoholomorphic sections of $X_r(f)$ using almost-complex structures compatible with the symplectic form.
• Apply index theory and elliptic PDE methods to ensure the moduli space of such sections is non-empty and regular.
• Use the Hilbert scheme and relative ampleness to embed symmetric products into projective bundles and analyze singular fibers via normalization and blow-ups.
• Prove that bubble components in singular fibers realize only homology classes equivalent to multiples of the fiber class $h$, ruling out spurious contributions.

Experimental results

Research questions

• RQ1 Can the canonical class of a symplectic 4-manifold be represented by an embedded symplectic surface without using Seiberg-Witten theory?
• RQ2 How can the classical algebraic geometry argument for Kähler surfaces—using the zero-set of a holomorphic 2-form—be adapted to non-integrable almost-complex structures?
• RQ3 What role do Lefschetz pencils play in realizing canonical classes via pseudoholomorphic sections of symmetric product fibrations?
• RQ4 Under what conditions does a non-vanishing Gromov invariant for sections of $X_r(f)$ imply the existence of a symplectic representative of $K_X$?
• RQ5 Why does the rationality condition on $[\omega]$ not obstruct the existence of such a representative, and can it be removed?

Key findings

• The Poincaré dual of the canonical class $K_X$ is represented by a smooth embedded symplectic surface in $X$ under the hypotheses $b_+ > 1 + b_1$ and rationality of a multiple of $[\omega]$.
• A non-vanishing Gromov invariant counting pseudoholomorphic sections of the symmetric product bundle $X_r(f)$ is established, ensuring the existence of such sections.
• Bubbles in singular fibers of the fibration $X_r(f)$ contribute only homology classes that are multiples of the fiber class $h$, preventing spurious topological obstructions.
• The proof avoids the Seiberg-Witten equations entirely, relying instead on pseudoholomorphic curve theory and almost-complex geometry.
• The rationality condition on $[\omega]$ is necessary for the current method, though it is not a significant restriction in practice due to deformation equivalence to rational forms.
• The construction generalizes the classical algebraic argument: in the Kähler case, the zero-set of a holomorphic 2-form gives $K_X$, and here, pseudoholomorphic sections play the analogous role.