Symplectic Torelli groups of rational surfaces

TL;DR

This paper investigates the structure of symplectic Torelli groups of positive rational surfaces, establishing their triviality or isomorphism to sphere braid groups based on root system type, and confirms several conjectures in symplectic topology.

Contribution

It proves the symplectic Torelli group is trivial or a sphere braid group depending on the root system type, and confirms Lagrangian spherical Dehn twists generate the Torelli group for positive rational surfaces.

Findings

Symplectic Torelli group is trivial for type A surfaces.

Symplectic Torelli group is a sphere braid group for type D surfaces.

All symplectic toric surfaces have trivial Torelli groups.

Abstract

We call a symplectic rational surface $(X,\omega)$ \textit{positive} if $c_1(X)\cdot[\omega]>0$. The positivity condition of a rational surface is equivalent to the existence of a divisor $D\subset X$, such that $(X, D)$ is a log Calabi-Yau surface. The cohomology class of a symplectic form can be endowed with a \textit{type} using the root system associated to its Lagrangian spherical classes. In this paper, we prove that the symplectic Torelli group of a positive rational surface is trivial if it is of type $\mathbb{A}$, and is a sphere braid group if it is of type $\mathbb{D}$. As an application, we answer affirmatively a long-term open question that Lagrangian spherical Dehn twists generate the symplectic Torelli group $Symp_h(X)$ when $X$ is a positive rational surface. We also prove that all symplectic toric surfaces have trivial symplectic Torelli groups. Lastly, we verify that Chiang-Kessler's symplectic involution is Hamiltonian, answering a question of Kedra positively. Our key new input is the recent study of almost complex subvarieties due to Li-Zhang and Zhang. Inspired by these works, we define a new \textit{coarse stratification} for the almost complex structures for positive rational surfaces. We also combined symplectic field theory and the parametrized Gromov-Witten theories for our applications.