Lagrangian spheres, symplectic surfaces and the symplectic mapping class group
It is known that a Lagrangian sphere \(L\) in a symplectic Del Pezzo surface with Euler number at most 7 can be displaced from certain symplectic spheres with positive Chern number up to Hamiltonian isotopy [\textit{J. D. Evans}, J. Topol. 3, No. 1, 181--227 (2010; Zbl 1235.53084)]. This paper generalizes this result in the following two ways.NEWLINENEWLINE Theorem 1.1. Let \(L\) be a Lagrangian spere in a symplectic 4-manifold \((M,\omega)\), and \(A\in H_2(M;\mathbb{Z})\) with \(A^2\geq -1\). Suppose \(A\) is represented by a symplectic sphere \(C\). Then \(C\) can be isotoped symplectically to another representative of \(A\) which intersects \(L\) minimally.NEWLINENEWLINE Theorem 1.2. Suppose \((M,\omega)\) is a symplectic 4-manifold with \(b^+= 1\) and \(L\) is a Lagrangian sphere. Assume \(A\in H^2(M,\mathbb{Z})\) satisfies \(\omega(A)> 0\), \(A^2> 0\) and \(A\cdot E\geq 0\) for all \(\omega\)-exceptional \(E\). Then there exists a symplectic surface in the class \(nA\) intersecting \(L\) minimally for large \(n\in\mathbb{N}\).NEWLINENEWLINE Here, two closed surfaces are said to intersect minimally if they intersect transversally at \(|k|\) points, where \(k\) is the homological intersection number, and \(E\in H_2(M,\mathbb{Z})\) is said to be of \(\omega\)-exceptional class if \(E\) is represented as an \(\omega\)-symplectic \((-1)\) sphere.NEWLINENEWLINE To show these theorems, first embedded connected pseudoholomorphic submanifolds for a class of compatible almost complex structures suitable for applying symplectic field theory are produced by using symplectic Seiberg-Witten theory and Taubes' \(\text{SW}\Rightarrow\text{GT}\) [\textit{C. H. Taubes}, J. Differ. Geom. 52, No. 3, 453--609 (1999; Zbl 1040.53096), Prop. 3.5]. Then, via neck stretching (explained in \S2.2), the symplectic field theory (SFT; a brief review is given in \S2), especially a compactness result in the SFT [\textit{F. Bourgeois} et al., Geom. Topol. 7, 799--888 (2003; Zbl 1131.53312), Th.2.7] and a finiteness result on the limit of punctures of compact connected pseudoholomorophic submanifolds of \(M\) (Lemma 3.8), produces in the limit the desired symplectric surface which intersects \(L\) minimally.NEWLINENEWLINE This minimal intersection property is powerful when the Kodaira dimension \(\kappa\) of \((M,\omega)\) is \(-\infty\) (this means \(M\) is either symplectically rational or ruled), and the following theorems are proved.NEWLINENEWLINE Theorem 1.4. Let \((M,\omega)\) be a symplectic 4-manifold with \(\kappa=-\infty\). The class \(\xi\in H_2(M,\mathbb{Z})\) is represented by a Lagrangian sphere if \(\xi\) is \(K_\omega\)-null spherical and \(\omega(\xi)= 0\).NEWLINENEWLINE Here, by \(K_\omega\)-null spherical class, a class \(\xi\) is meant such that \(\xi^2= -2\), \(K_\omega(\xi)= 0\), \(K_\omega\) is the symplectic canonical class, and it is represented by a smooth sphere.NEWLINENEWLINE Theorem 1.5. Let \((M,\omega)\) be a symplectic rational manifold with Euler number \(\chi\leq 7\), and \(\xi\) a \(K_\omega\)-null spherical class with \(\omega(\xi)= 0\). If \(\xi\) is not characteristic when \(\chi= 6\), then Lagrangian spheres in \(M\) are unique up to Hamiltonian isotopy.NEWLINENEWLINE Theorem 1.6. Let \((M,\omega)\) be a symplectic rational manifold and \(\xi\) a \(K_\omega\)-null spherical class with \(\omega(\xi)= 0\). If \(\xi\) is not characteristic when \(\chi= 6\), then Lagrangian spheres in \(\xi\) are unique up to smooth isotopy.NEWLINENEWLINE The authors also conjecture that the Torelli part \(\text{Symp}_h(M,\omega)\), which is a subgroup of \(\text{Symp}(M, \omega)\) acting trivially on the homology, should also act transitively on the space of Lagrangian spheres in a fixed homology class (Conj. 1.7). The following symplectic version of M. Noether's Theorem on birational automorphisms of \(\mathbb{C} P^2\) [\textit{M. Alberich-Carramiñana}, Geometry of the plane Cremona maps. Berlin: Springer (2002; Zbl 0991.14008)] is also proved.NEWLINENEWLINE Theorem 1.8. Let \((M,\omega)\) be a symplectic 4-manifold with \(\kappa=-\infty\). Then the homological action of \(\text{Symp}(M,\omega)\) is generated by Lagrangian Dehn twists.NEWLINENEWLINE To show these theorems, first, \(K\)-null spherical classes are studied when \(\kappa=-\infty\) and the generators of the geometric automorphism group of \(M\), etc., are shown to be \(K\)-twists, etc. (Prop. 4.7, Prop. 4.16). By using these results and Theorems 1.1 and 1.2, Theorems 1.4 and 1.8 are proved in \S5. Then after reviewing Hind's uniqueness theorem of Lagrangian spheres in \(S^2\times S^2\) (Theorem 6.1. [\textit{R. Hind}, Geom. Funct. Anal. 14, No. 2, 303--318 (2004; Zbl 1066.53129)]), Theorems 1.5 and 1.6 are proved in \S6, the last section.NEWLINENEWLINE The authors remark that completely different proofs of Theorems 1.4 and 1.8 are given in [\textit{V. Shevehishin}, Secondary Seifert-Whitney class and diffeomorphisms of rational and ruled symplectic 4-manifolds, preprint (2009), \url{arXiv:0904.0283}].
scientific article; zbMATH DE number 6068622
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