Lagrangian spheres, symplectic surfaces and the symplectic mapping class group

Li, Tian-Jun¹ ; Wu, Weiwei¹

¹ School of Mathematics, University of Minnesota, 206 Church Street, Minneapolis MN 55455, USA

Geometry & topology, Tome 16 (2012) no. 2, pp. 1121-1169.

Résumé

Given a Lagrangian sphere in a symplectic $4$–manifold $\left(M,\omega \right)$ with $b^{+}=1$, we find embedded symplectic surfaces intersecting it minimally. When the Kodaira dimension $\kappa$ of $\left(M,\omega \right)$ is $-\infty$, this minimal intersection property turns out to be very powerful for both the uniqueness and existence problems of Lagrangian spheres. On the uniqueness side, for a symplectic rational manifold and any class which is not characteristic, we show that homologous Lagrangian spheres are smoothly isotopic, and when the Euler number is less than 8, we generalize Hind and Evans’ Hamiltonian uniqueness in the monotone case. On the existence side, when $\kappa =-\infty$, we give a characterization of classes represented by Lagrangian spheres, which enables us to describe the non-Torelli part of the symplectic mapping class group.

Détail

DOI : 10.2140/gt.2012.16.1121

Classification : 53D05, 53D12, 53D42
Keywords: Lagrangian sphere, symplectomorphism group

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