Model projective twists and generalised lantern relations

Abstract

We use Picard-Lefschetz theory to introduce a new local model for the planar projective twists $\tau_{\mathbb{A}\mathbb{P}^2} \in \mathrm{Symp}_{ct}(T^*\mathbb{A}\mathbb{P}^2), \ \mathbb{A} \in \{ \mathbb{R}, \mathbb{C} \}$. In each case, we construct an exact Lefschetz fibration $\pi\colon T^*\mathbb{A}\mathbb{P}^2\to \mathbb{C}$ with three singular fibres, and define a compactly supported symplectomorphism $\varphi \in \mathrm{Symp}_{ct}(T^*\mathbb{A}\mathbb{P}^2)$ on the total space. Given two disjoint Lefschetz thimbles $\Delta_{\alpha},\Delta_{\beta} \subset T^*\mathbb{A}\mathbb{P}^2$, we compute the Floer cohomology groups $\mathrm{HF}(\varphi^k(\Delta_{\alpha}), \Delta_{\beta}; \mathbb{Z}/2\mathbb{Z})$ and verify (partially for $\mathbb{C}\mathbb{P}^2$) that $\varphi$ is indeed isotopic to (a power of) the projective twist in its local model. The constructions we present are governed by generalised lantern relations, which provide an isotopy between the total monodromy of a Lefschetz fibration and a fibred twist along an $S^1$-fibred coisotropic submanifold of the smooth fibre. We also use these relations to generate non-exact fillings for the contact manifolds $(ST^*\mathbb{C}\mathbb{P}^2, \xi_{std}), (ST^*\mathbb{R}\mathbb{P}^3,\xi_{std})$, and to study two classes of monotone Lagrangian submanifolds of $(T^*\mathbb{C}\mathbb{P}^2, d\lambda_{\mathbb{C}\mathbb{P}^2})$.