论文标题
模型投影曲折和广义灯笼关系
Model projective twists and generalised lantern relations
论文作者
论文摘要
我们使用picard-lefschetz理论为平面投射曲折介绍了一个新的本地模型$τ_ {\ mathbb {a} \ Mathbb {p}^2} \ in \ mathrm {symp} _ {ct} _ {ct} _ {ct} \ Mathbb {a} \ in \ {\ Mathbb {r}，\ Mathbb {C} \} $。在每种情况下，我们构造一个精确的lefschetz纤维化$π\ colon t^*\ mathbb {a} \ mathbb {p}^2 \ to \ mathbb {c} $，带有三个单数纤维，并定义一个紧凑的支持的符号符号$φ\ IN \ Mathrm {symp} _ {ct}（T^*\ Mathbb {a} \ Mathbb {p}^2）$在总空间上。给定两个不相交的Lefschetz Thimbles $δ_α，Δ_β\ subset t^*\ mathbb {a} \ mathbb {p}^2 $，我们计算了浮动群群体$ \ mathrm {hf}（hf}（φ^k（Δ__α），δ__β; \ Mathbb {Z}/2 \ Mathbb {Z}）$并验证（部分用于$ \ Mathbb {C} \ Mathbb {p}^2 $），$φ$的确是（在其本地模型中的投射扭转）同位素的同位素。 我们提出的结构受广义灯笼关系的控制，该关系提供了LEFSCHETZ纤维的总单构和沿$ s^1 $ fibred coisotropic submanifold的纤维旋转之间的同位素。我们还使用这些关系来生成非外观填充物的触点$（ST^*\ Mathbb {C} \ Mathbb {p}^2，ξ_{std}），（st^*\ Mathbb {r} \ Mathbb {r} \ Mathbb {p} $（T^*\ Mathbb {C} \ Mathbb {p}^2，dλ_{\ Mathbb {c} \ Mathbb {p}^2}）$。
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 $(T^*\mathbb{C}\mathbb{P}^2, dλ_{\mathbb{C}\mathbb{P}^2})$.
