乘法预投射代数的几何

Fukaya categories of plumbings and multiplicative preprojective algebras

Given an arbitrary graph $Γ$ and non-negative integers $g_v$ for each vertex $v$ of $Γ$, let $X_Γ$ be the Weinstein $4$-manifold obtained by plumbing copies of $T^*Σ_v$ according to this graph, where $Σ_v$ is a surface of genus $g_v$. We compute the wrapped Fukaya category of $X_Γ$ (with bulk parameters) using Legendrian surgery extending our previous work arXiv:1502.07922 where it was assumed that $g_v=0$ for all $v$ and $Γ$ was a tree. The resulting algebra is recognized as the (derived) multiplicative preprojective algebra (and its higher genus version) defined by Crawley-Boevey and Shaw ar…

关于某些野特征簇的非交换泊松几何

On the noncommutative Poisson geometry of certain wild character varieties

本文猜想并部分证明了着色乘法箭图簇的自然泊松结构可通过 Kontsevich-Rosenberg 原理由底层裂变代数上的非交换泊松结构诱导，并在区间和三角形两种情形下给出了证明。

型$\tilde{A}_n$预投射代数的几何实现与镜像对称

Geometric realisations of type $\tilde{A}_n$ preprojective algebras in homological mirror symmetry

研究通过几何方法实现型$\tilde{A}_n$预投射代数的同调镜像对称性，展示其与Landau-Ginzburg势能的奇点范畴的关系。

Symplectic resolutions for multiplicative quiver varieties and character varieties for punctured surfaces

We study the algebraic symplectic geometry of multiplicative quiver varieties, which are moduli spaces of representations of certain quiver algebras, introduced by Crawley-Boevey and Shaw, called multiplicative preprojective algebras. They are multiplicative analogues of Nakajima quiver varieties. They include character varieties of (open) Riemann surfaces fixing conjugacy class closures of the monodromies around punctures, when the quiver is "crab-shaped". We prove that, under suitable hypotheses on the dimension vector of the representations, or the conjugacy classes of monodromies in the ch…

微局部层与微局部簇

Microlocal sheaves and quiver varieties

我们将Nakajima箭图簇（或者更确切地说，它们的乘法形式）与反常层的模空间联系起来。更确切地说，我们考虑了反常层概念的推广：节点曲线X上的微局部层。它们被定义为X的正规化上的反常层，在每个节点附近具有傅里叶变换条件，并形成一个阿贝尔范畴M（X）。一个是类似的三角范畴DM（X）的微局部复形。对于紧X，我们证明了DM（X）是2维的Calabi-Yau。当X的所有分支都是有理数时，M（X）等价于与X的交图相关的乘法预投射代数的表示范畴。在适当的意义上箭图品种得到的微局部层的模空间与消失圈的框架在奇点。当X的分支有更高的亏格时，会导致预投射代数和X簇有趣的推广。我们从伪Hamilton约化和群值矩映射的角度分析了它们。

Birational geometry of symplectic quotient singularities

For a finite subgroup $Γ\subset \mathrm{SL}(2,\mathbb{C})$ and for $n\geq 1$, we use variation of GIT quotient for Nakajima quiver varieties to study the birational geometry of the Hilbert scheme of $n$ points on the minimal resolution $S$ of the Kleinian singularity $\mathbb{C}^2/Γ$. It is well known that $X:=\mathrm{Hilb}^{[n]}(S)$ is a projective, crepant resolution of the symplectic singularity $\mathbb{C}^{2n}/Γ_n$, where $Γ_n=Γ\wr\mathfrak{S}_n$ is the wreath product. We prove that every projective, crepant resolution of $\mathbb{C}^{2n}/Γ_n$ can be realised as the fine moduli space of $…