On Legendrian curves in $\mathbb P^3$

Legendrian cone structures and contact prolongations

We study a cone structure ${\mathcal C} \subset {\mathbb P} D$ on a holomorphic contact manifold $(M, D \subset T_M)$ such that each fiber ${\mathcal C}_x \subset {\mathbb P} D_x$ is isomorphic to a Legendrian submanifold of fixed isomorphism type. By characterizing subadjoint varieties among Legendrian submanifolds in terms of contact prolongations, we prove that the canonical distribution on the associated contact G-structure admits a holomorphic horizontal splitting.

平移接触栈的导出勒让德范畴

A Derived Legendrian Category for Shifted Contact Stacks

本文为n-平移接触导出阿廷栈X构造了导出勒让德范畴F_c(X)及勒让德对应的(∞,2)-范畴Leg_n，并应用于模理论。

Lagrangian Skeleta and Plane Curve Singularities

We construct closed arboreal Lagrangian skeleta associated to links of isolated plane curve singularities. This yields closed arboreal Lagrangian skeleta for Weinstein pairs and Weinstein 4-manifolds associated to max-tb Legendrian representatives of algebraic links. We provide computations of Legendrian and Weinstein invariants, and discuss the symplectic topological nature of the Fomin-Pylyavskyy-Shustin-Thurston cluster algebra associated to a singularity. Finally, we present a conjectural ADE-classification for Lagrangian fillings of certain Legendrian links and list some related problems.

Distributions, first integrals and Legendrian foliations

We study germs of holomorphic distributions with "separated variables'. In codimension one, a well know example of this kind of distribution is given by the canonical contact structure on $\mathbb{P}^{2m+1}$ . Another example is the Darboux distribution, which gives the normal local form of any contact structure. Given a germ $D$ of holomorphic distribution with separated variables in $(\mathbb{C}^n,0)$, we show that there exists , for some $κ\in \mathbb{Z}_{\geq 0}$ related to the Taylor coefficients of $D$, a holomorphic submersion $H_{D}: (\mathbb{C}^n,0) \rightarrow (\mathbb{C}^κ,0)$ such …

导出辛空间的等变商和 Legendre 相交定理

Equivariant Quotients of Derived Symplectic Spaces and Legendrian Intersection Theorem

本文将经典接触几何的横截性引理推广到导出代数几何中，证明了导出 Legendre 相交定理，并应用于模空间理论。

通过环面作用枚举有理切触曲线

Enumeration of rational contact curves via torus actions

本文利用环面作用和Bott公式证明了任意切触复射影空间中有理切触曲线的Gromov-Witten数具有枚举性，并计算了低次情形下的精确值。

On the Nash modification of a germ of complex analytic singularity

For a germ $(X,0) \subset (\mathbb{C}^n,0)$ of reduced, equidimensional complex analytic singularity its Nash modification can be constructed as an analytic subvariety $ Z \subset \mathbb{C}^n \times G(k,n)$. We give a characterization of the subvarieties of $\mathbb{C}^n \times G(k,n)$ that are the Nash modification of its image under the projection to $\mathbb{C}^n$. This result generalizes the characterization of conormal varieties as Legendrian subvarieties of $\mathbb{C}^n \times \check{\mathbb{P}}^{n-1}$ with its canonical contact structure. As a by-product we define the $d$-conormal spa…

Legendrian链环与簇代数的微局部理论

Microlocal Theory of Legendrian Links and Cluster Algebras

本文通过微局部平行输运和辛拓扑方法，为网格plabic图相关的模栈构造了簇结构，并引入新概念证明了簇对偶性。

谱曲线的镜像计数

Mirror Counts of Spectral Curves

本文证明凸格多边形定义的环面簇上结点曲线计数与勒让德链环的rulings计数等价，并猜想ruling分解恢复精细热带不变量。

A combinatorial calculation of the Landau-Ginzburg model $M= \mathbb C^3, W=z_1 z_2 z_3$

The aim of this paper is to apply ideas from the study of Legendrian singularities to a specific example of interest within mirror symmetry. We calculate the Landau-Ginzburg $A$-model with $M= \mathbb C^3, W=z_1 z_2 z_3$ in its guise as microlocal sheaves along the natural singular Lagrangian thimble $L = {\mathit Cone}(T^2)\subset M$. The description we obtain is immediately equivalent to the $B$-model of the pair-of-pants $\mathbb P^1 \setminus \{0, 1, \infty\}$ as predicted by mirror symmetry.

Varieties of minimal rational tangents of unbendable rational curves subordinate to contact structures

A nonsingular rational curve $C$ in a complex manifold $X$ whose normal bundle is isomorphic to $${\mathcal O}_{{\mathbb P}^1}(1)^{\oplus p} \oplus {\mathcal O}_{{\mathbb P}^1}^{\oplus q}$$ for some nonnegative integers $p$ and $q$ is called an unbendable rational curve on $X$. Associated with it is the variety of minimal rational tangents (VMRT) at a point $x \in C,$ which is the germ of submanifolds ${\mathcal C}^C_x \subset {\mathbb P} T_x X$ consisting of tangent directions of small deformations of $C$ fixing $x$. Assuming that there exists a distribution $D \subset TX$ such that all small…