Lagrangian spheres, symplectic surfaces and the symplectic mapping class group

TL;DR

This paper investigates the properties of Lagrangian spheres in symplectic 4-manifolds with specific Kodaira dimensions, establishing results on their uniqueness, existence, and implications for the symplectic mapping class group.

Contribution

It provides new criteria for the existence and uniqueness of Lagrangian spheres and characterizes classes represented by them, impacting the understanding of the symplectic mapping class group.

Findings

Homologous Lagrangian spheres are smoothly isotopic in certain rational manifolds.

Generalization of Hamiltonian uniqueness for spheres with Euler number less than 8.

Characterization of classes represented by Lagrangian spheres when Kodaira dimension is -infinity.

Abstract

Given a Lagrangian sphere in a symplectic 4-manifold $(M, \omega)$ with $b^+=1$, we find embedded symplectic surfaces intersecting it minimally. When the Kodaira dimension $\kappa$ of $(M, \omega)$ 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 and ternary, 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.