Higher type adjunction inequalities in Seiberg-Witten theory. (English) Zbl 1028.57031
Let \(X\) be a smooth, closed, oriented \(4\)-manifold with \(b^{+}_{2}>1\). For any Spin\(_{C}\)-structure \(s\), there is a Seiberg-Witten invariant \(SW_{X,s}(z_{1}\cdots z_{r})\), associated to homology classes \(z_{1}, \ldots, z_{r}\) in \(H_{0}(X)\) and \(H_{1}(X)\), graded so that the class of the point \(x\in H_{0}(X)\) has degree \(2\) and the elements in \(H_{1}(X)\) have degree \(2\). The Seiberg-Witten invariant is only non-zero when the total degree equals \(d(s)=(c_{1}(s)^{2} - (2\chi(X) + 3\sigma(X)))/4\), the dimension of the moduli space of Seiberg-Witten solutions. \(X\) is of simple type if the Seiberg-Witten invariants are zero whenever \(d(s)>0\) and some \(z_i\) is the class of the point \(x\).
This paper deals with an strengthened version of the adjunction inequality for non-simple type manifolds. For any smoothly embedded surface \(\Sigma \subset X\) with genus \(g>0\) and representing a non-torsion homology class with self-intersection \([\Sigma]^{2}\geq 0\), we have an adjunction inequality \(|\langle c_{1}(s),[\Sigma]\rangle |+[\Sigma]^{2} + 2d(s) \leq 2g-2\). This is an improvement of the classical adjunction inequality, in which the term \(2d(s)\) does not appear. Unfortunately, no examples of non-simple type manifolds with \(b^{+}_{2}>1\) are yet known.
There is a similar statement when \(b^{+}_{2}=1\), when the Seiberg-Witten invariants depend on chambers. All these results are proved by doing a partial study of a Seiberg-Witten-Floer theory for circle-bundles over \(\Sigma\).
This work is reminiscent of P. Oszváth and Z. Szabó [Ann. Math. 151, 93-124 (2000; Zbl 0967.53052)] where the adjunction inequality is proved in full generality for symplectic \(4\)-manifolds (recall that such manifolds are always of simple type). An alternative proof of the results of this paper, but using Seiberg-Witten-Floer homology of \(\Sigma \times S^{1}\), appears in [V. Muñoz and B-L. Wang, Seiberg-Witten-Floer homology of a surface times a circle for non-torsion spinc-structures, ArXiv:math.DG/9905050]. There is also a version of the higher type adjunction inequality for Donaldson invariants [V. Muñoz, Trans. Am. Math. Soc. 353, 2635-2654 (2001; Zbl 0969.57025)].
This paper deals with an strengthened version of the adjunction inequality for non-simple type manifolds. For any smoothly embedded surface \(\Sigma \subset X\) with genus \(g>0\) and representing a non-torsion homology class with self-intersection \([\Sigma]^{2}\geq 0\), we have an adjunction inequality \(|\langle c_{1}(s),[\Sigma]\rangle |+[\Sigma]^{2} + 2d(s) \leq 2g-2\). This is an improvement of the classical adjunction inequality, in which the term \(2d(s)\) does not appear. Unfortunately, no examples of non-simple type manifolds with \(b^{+}_{2}>1\) are yet known.
There is a similar statement when \(b^{+}_{2}=1\), when the Seiberg-Witten invariants depend on chambers. All these results are proved by doing a partial study of a Seiberg-Witten-Floer theory for circle-bundles over \(\Sigma\).
This work is reminiscent of P. Oszváth and Z. Szabó [Ann. Math. 151, 93-124 (2000; Zbl 0967.53052)] where the adjunction inequality is proved in full generality for symplectic \(4\)-manifolds (recall that such manifolds are always of simple type). An alternative proof of the results of this paper, but using Seiberg-Witten-Floer homology of \(\Sigma \times S^{1}\), appears in [V. Muñoz and B-L. Wang, Seiberg-Witten-Floer homology of a surface times a circle for non-torsion spinc-structures, ArXiv:math.DG/9905050]. There is also a version of the higher type adjunction inequality for Donaldson invariants [V. Muñoz, Trans. Am. Math. Soc. 353, 2635-2654 (2001; Zbl 0969.57025)].
Reviewer: Vicente Muñoz (Madrid)
MSC:
| 57R57 | Applications of global analysis to structures on manifolds |
| 53D35 | Global theory of symplectic and contact manifolds |
| 57R58 | Floer homology |
