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Lefschetz pencils and the canonical class for symplectic four-manifolds. (English) Zbl 1012.57040

This paper gives a new proof of a result due to Taubes [C. H. Taubes, Math. Res. Lett. 2, No. 2, 221-238 (1995; Zbl 0894.57020)]. The main theorem proved here is the following: if \((X,\omega)\) is a closed symplectic four-manifold with \(b_{+}>1+b_{1}\) and \([\omega]\) is a multiple of a rational cohomology class, then the Poincaré dual of the canonical class \(K_{X}\) may be represented by an embedded symplectic submanifold.
This new proof is simpler and more geometric than the original one since it does not use the deep results of Taubes that relate Gromov-Witten and Seiberg-Witten invariants of a symplectic four-manifold.
The outline of the proof is as follows: for the symplectic manifold \(X\) there exists a symplectic Lefschetz pencil [S. K. Donaldson, J. Differ. Geom. 53, No. 2, 205-236 (1999; Zbl 1040.53094)]. Then \(X\) is blown up to get a symplectic Lefschetz fibration \(f:\widetilde{X}\rightarrow \mathbb{C} P^{1}\). This is endowed with an almost complex structure that makes the fibers and the projection holomorphic. Then one searches for symplectic submanifolds representing \(K_{X}\) by looking for pseudo-holomorphic surfaces intersecting in \(r=2g-2\) positive points the generic fiber. For this the authors construct the fibration \({X_r}\rightarrow \mathbb{C} P^{1}\) of \(r\)-symmetric products of the fibers of \(f\), and look for sections of this fibration which intersect transversely the big diagonal. The crux of the argument is that a Gromov invariant counting pseudo-holomorphic sections of this fibration is non-zero. For proving this it is necessary to deform the almost-complex structure.

MSC:

57R17 Symplectic and contact topology in high or arbitrary dimension
57R95 Realizing cycles by submanifolds
14H15 Families, moduli of curves (analytic)
53D35 Global theory of symplectic and contact manifolds
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