Symplectic Torelli groups of rational surfaces

Abstract

We call a symplectic rational surface $(X,ω)$ \textit{positive} if $c_1(X)\cdot[ω]>0$. 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.