模型射影扭曲和广义灯笼关系
Model projective twists and generalised lantern relations
摘要：
本文利用Picard-Lefschetz理论，提出了一种新的平面射影扭曲$τ_{\mathbb{A}\mathbb{P}^2} \in \mathrm{Symp}_{ct}(T^*\mathbb{A}\mathbb{P}^2), \ \mathbb{A} \in \{ \mathbb{R}, \mathbb{C} \}$的局部模型。在每一种情况下，我们构造了一个确切的Lefschetz纤维化$π\colon T^*\mathbb{A}\mathbb{P}^2\to \mathbb{C}$与三个奇异纤维，并定义了一个紧支撑的全空间上的辛同构$φ\in \mathrm{Symp}_{ct}(T^*\mathbb{A}\mathbb{P}^2)$。给定两个不相交的Lefschetz顶针$Δ_α,Δ_β \subset T^*\mathbb{A}\mathbb{P}^2$，我们计算的Floer上同调群$\mathrm{HF}(φ^k(Δ_α), Δ_β; \mathbb{Z}/2\mathbb{Z})$和验证（部分为$\mathbb{C}\mathbb{P}^2$），$φ$确实是同位素的（一个权力）在其本地模型的投影扭曲。我们提出的结构是由广义灯笼关系，它提供了一个全单值的Lefschetz纤维化和一个沿$S^1$ -双螺旋的共各向同性子流形的光滑纤维的双螺旋扭曲之间的合痕。我们也利用这些关系生成接触流形$(ST^*\mathbb{C}\mathbb{P}^2, ξ_{std}), (ST^*\mathbb{R}\mathbb{P}^3,ξ_{std})$的非精确填充，并研究了$(T^*\mathbb{C}\mathbb{P}^2, dλ_{\mathbb{C}\mathbb{P}^2})$的两类单调拉格朗日子流形。
英文摘要：
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 $(T^*\mathbb{C}\mathbb{P}^2, dλ_{\mathbb{C}\mathbb{P}^2})$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
