Brunella Charlotte Torricelli

arXiv2022

Model projective twists and generalised lantern relations

We use Picard-Lefschetz theory to introduce a new local model for the planar projective twists $τ_{\mathbb{A}\mathbb{P}^2} \in \mathrm{Symp}_{ct}(T^*\mathbb{A}\mathbb{P}^2), \ \mathbb{A} \in \{ \mathbb{R}, \mathbb{C} \}$. In each case, we construct an exact Lefschetz fibration $π\colon T^*\mathbb{A}\mathbb{P}^2\to \mathbb{C}$ with three singular fibres, and define a compactly supported symplectomorphism $φ\in \mathrm{Symp}_{ct}(T^*\mathbb{A}\mathbb{P}^2)$ on the total space. Given two disjoint Lefschetz thimbles $Δ_α,Δ_β \subset T^*\mathbb{A}\mathbb{P}^2$, we compute the Floer cohomology groups $\mathrm{HF}(φ^k(Δ_α), Δ_β; \mathbb{Z}/2\mathbb{Z})$ and verify (partially for $\mathbb{C}\mathbb{P}^2$) that $φ$ 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 $(ST^*\mathbb{C}\mathbb{P}^2, ξ_{std}), (ST^*\mathbb{R}\mathbb{P}^3,ξ_{std})$, and to study two classes of monotone Lagrangian submanifolds of $(T^*\mathbb{C}\mathbb{P}^2, dλ_{\mathbb{C}\mathbb{P}^2})$.

math.SG↗

arXiv2022

Projective twists and the Hopf correspondence

Given Lagrangian (real, complex) projective spaces $K_1, \dots , K_m$ in a Liouville manifold $(X, ω)$ satisfying a certain cohomological condition, we show there is a Lagrangian correspondence that assigns a Lagrangian sphere $L_i \subset K$ of another Liouville manifold $(Y, Ω)$ to any given projective Lagrangian $K_i \subset X$, $i=1, \dots m$. We use the Hopf correspondence to study \emph{projective twists}, a class of symplectomorphisms akin to Dehn twists, but defined starting from Lagrangian projective spaces. When this correspondence can be established, we show that it intertwines the autoequivalences of the compact Fukaya category $\mathcal{F}uk(X)$ induced by the (real, complex, quaternionic) projective twists $τ_{K_i} \in π_{0}(\mathrm{Symp}_{ct}(X))$ with the corresponding autoequivalences of $\mathcal{F}uk(Y)$ induced by the Dehn twists $τ_{L_i} \in π_{0}(\mathrm{Symp}_{ct}(Y))$, for $i=1, \dots m$. Using the Hopf correspondence, we obtain a free generation result for projective twists in a \emph{clean plumbing} of projective spaces and various results about products of positive powers of Dehn/projective twists in Liouville manifolds. The same techniques are also used to show that the Hamiltonian isotopy class of the projective twist (along the zero section in $T^*\mathbb{CP}$) in $\mathrm{Symp}_{ct}(T^*\mathbb{CP}^n)$ does depend on a choice of framing, for $n\geq19$. Another application of the Hopf correspondence delivers two examples of smooth homotopy complex projective spaces $K\simeq \mathbb{CP}^n$ that do not admit Lagrangian embeddings into $(T^*\mathbb{CP}^n, dλ_{\mathbb{CP}^n})$, for $n=4,7$.

math.SG↗