Lagrangian spheres, symplectic surfaces and the symplectic mapping class group

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.

References in corpus (5)

• Quantum Structures for Lagrangian Submanifolds
• The Space of Symplectic Structures on Closed 4-Manifolds
• Lagrangian isotopies in Stein manifolds
• Secondary Stiefel-Whitney class and diffeomorphisms of rational and ruled symplectic 4-manifolds
• Tamed to compatible: Symplectic forms via moduli space integration

Cited by in corpus (24)

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• $S^1$-invariant symplectic hypersurfaces in dimension $6$ and the Fano condition
• Stability and Existence of Surfaces in Symplectic 4-Manifolds with $b^{+}=1$
• Moduli space of $J$-holomorphic subvarieties
• A Chiang-type Lagrangian in CP2
• Generalizations of planar contact manifolds to higher dimensions
• Exotically knotted disks and complex curves
• On the topology of real Lagrangians in toric symplectic manifolds
• Kodaira Dimension in Low Dimensional Topology
• Hamiltonian unknottedness of certain monotone Lagrangian tori in $S^2\times S^2$
• Projective twists and the Hopf correspondence
• On a class of symplectic $4$-orbifolds with vanishing canonical class
• Topology of symplectomorphism groups and ball-swappings
• Homology supported in Lagrangian submanifolds in mirror quintic threefolds
• Uniqueness of real Lagrangians up to cobordism
• Finite group actions on symplectic Calabi-Yau $4$-manifolds with $b_1>0$
• The symplectic mapping class group of $\CC P^2 n{\bar{\CC P^2}}$ with $n\le 4$
• Exact Lagrangians in $A_n$-surface singularities
• The quantum Johnson homomorphism and symplectomorphism of 3-folds
• Symplectic rational $G$-surfaces and equivariant symplectic cones
• Non-orientable Lagrangian surfaces in rational 4-manifolds