Homotopy Colimits of DG Categories and Fukaya Categories

加权射影空间的同调镜像对称与Morse同伦

Homological mirror symmetry for weighted projective spaces and Morse Homotopy

本文将环面流形上的加权Morse同伦范畴与凝聚层导出范畴的同调镜像对称研究推广至环面轨形，讨论了加权射影空间的对应版本同调镜像对称。

通过稳定化因子的Rakaya代数

Fukaya algebras via stabilizing divisors

利用Cieliebak-Mohnke引入的稳定因子技术，构造了紧致辛流形中具有有理辛类的紧致、定向、相对自旋的Lagrange算子的有限维、严格酉的代数.同伦类型的代数和模空间的弱Maurer-Cartan方程的解决方案是独立的扰动数据的选择。Floer上同调是一个复杂的向量丛在空间上的弱Maurer-Cartan方程的解决方案的上同调，并被证明是独立的扰动数据的选择规范等价。

Homological mirror symmetry of $\mathbb{C}P^n$ and their products via Morse homotopy

We propose a way of understanding homological mirror symmetry when a complex manifold is a smooth compact toric manifold. So far, in many example, the derived category $D^b(coh(X))$ of coherent sheaves on a toric manifold $X$ is compared with the Fukaya-Seidel category of the Milnor fiber of the corresponding Landau-Ginzburg potential. We instead consider the dual torus fibration $π:M \to B$ of the complement of the toric divisors in $X$, where $\bar{B}$ is the dual polytope of the toric manifold $X$. A natural formulation of homological mirror symmetry in this set-up is to define $Fuk(\bar{M}…

Global Fukaya category I

Let $Ham (M,ω) $ denote the Frechet Lie group of Hamiltonian symplectomorphisms of a monotone symplectic manifold $(M, ω) $. Let $NFuk (M, ω)$ be the $A _{\infty} $-nerve of the Fukaya category $Fuk (M, ω)$, and let $(|\mathbb{S}|, NFuk (M, ω))$ denote the $NFuk (M, ω)$ component of the ``space of $\infty$-categories'' $|\mathbb{S}| $. Using Floer-Fukaya theory for a monotone $(M, ω)$ we construct a natural up to homotopy classifying map \begin{equation*} BHam (M, ω) \to (|\mathbb{S}|, NFuk (M, ω)). \end{equation*} This verifies one sense of a conjecture of Teleman on existence of action of $H…

邻近拉格朗日量的谱等价性

Spectral equivalence of nearby Lagrangians

该研究构造了稳定极化刘维尔扇区的系数取于交换环谱$R$的包裹Donaldson–Fukaya范畴，证明闭流形余切丛中闭恰当拉格朗日$R$-膜同构于零截面的$R$-膜结构，推广了整数环上的经典结果。

包裹范畴的辛上同调与对偶

Symplectic cohomology and duality for the wrapped Fukaya category

考虑刘维尔流形中的一个精确拉格朗日集合的包裹的范畴W。在一个非退化的条件下，意味着存在足够的拉格朗日，我们表明，自然几何映射从霍赫希尔德同调的W辛上同调和从辛上同调的Hochschild上同调的W是同构，在某种程度上兼容的环和模结构。这是一个更一般的对偶性的结果，对于包裹的Brackaya范畴，它应该被认为是一个非紧版本的Calabi-Yau结构。新的成分是：（1）通过全纯面组的包裹形式的W的Fourier-Mukai理论，（2）新的几何运算，来自具有两个负穿孔和任意多个正穿孔的圆盘，（3）Cardy条件的推广，以及（4）使用同伦单位和A-无穷混洗乘积将非退化性与对角线的分解联系起来。

Derived equivalences of gentle algebras via Fukaya categories

Following the approach of Haiden-Katzarkov-Kontsevich arXiv:1409.8611, to any homologically smooth graded gentle algebra $A$ we associate a triple $(Σ_A, Λ_A; η_A)$, where $Σ_A$ is an oriented smooth surface with non-empty boundary, $Λ_A$ is a set of stops on $\partial Σ_A$ and $η_A$ is a line field on $Σ_A$, such that the derived category of perfect dg-modules of $A$ is equivalent to the partially wrapped Fukaya category of $(Σ_A, Λ_A ;η_A)$. Modifying arguments of Johnson and Kawazumi, we classify the orbit decomposition of the action of the (symplectic) mapping class group of $Σ_A$ on the h…

曲A-无穷范畴：附加与同伦

Curved A-infinity-categories: adjunction and homotopy

我们围绕其模范畴的等价性发展了一个弯曲A无穷范畴的理论。这使得弯曲和非弯曲的A-无穷范畴的统一处理，推广了经典理论的非弯曲的A-无穷代数。此外，该理论是足够的一般性，以处理两个范畴和范畴的矩阵因子分解，以及提供一个上下文，其中统一和范畴化的前范畴可以进行。我们的理论是围绕两个函子建立的：伴随代数函子U_e和函子Q_*。大部分的文件是致力于证明关键的附加和同伦定理这些函子。此外，我们还研究了模范畴的非零性，给出了一个精确的陈述，并证明了一个称为“Positselski-Kontsevich消失”的结果。

K-theoretic invariants of Hamiltonian fibrations

We introduce new invariants of Hamiltonian fibrations with values in the suitably twisted K-theory of the base. Inspired by techniques of geometric quantization, our invariants arise from the family analytic index of a family of natural $Spin^c$-Dirac operators. As an application we give new examples of non-trivial Hamiltonian fibrations, that have not been previously detected by other methods. As one crucial ingredient we construct a potentially new homotopy equivalence map, with a certain naturality property, from $BU$ to the space of index $0$ Fredholm operators on a Hilbert space, using el…

Singularity categories of normal crossings surfaces, descent, and mirror symmetry

Given a smooth 3-fold $Y$, a line bundle $L \to Y$, and a section $s$ of $L$ such that the vanishing locus of $s$ is a normal crossings surface $X$ with graph-like singular locus, we present a way to reconstruct the singularity category of $X$ as a homotopy limit of several copies of the category of matrix factorizations of $xyz : \mathbb{A}^{3} \to \mathbb{A}^{1}$ (the mirror to the Fukaya category of the pair of pants). This extends our previous result for the case where $L$ is trivialized. The key technique is the classification of non-two-periodic autoequivalences of the category of matrix…

团簇复合物的源空间和扰动

Source Spaces and Perturbations for Cluster Complexes

我们定义了由度量线段连接的标记复圆盘构成的对象，并构造了这些对象的非对称和对称模空间。这允许选择相干扰动的相应版本的弗洛尔轨迹提出的Cornea和Lalonde。这些扰动的目的是导致另一种描述的（受阻）$A_\infty$ -结构的研究，谷，哦，太田和小野。给定一个具有极小Maslov数$N_L \geq 2$的$Pin_{\pm}$单调拉格朗日子流形$L \subset (M,ω)$，我们定义了一个$A_\infty$ -代数（分别为.微分分次代数）结构的一般莫尔斯函数的临界点$L$。它被写为一个cochain（resp.链）复合物扩展了Oh引入的珍珠复合物，并由Biran和Cornea进一步解释，配备了其量子产物。证明了该构造是同伦不变的，定义了一个从$Pin_{\pm}$单调拉格朗日子流形$h\mathcal{L}^{mono, \pm}(M,ω)$的同伦范畴到上链的同伦范畴的函子.链）络合物$hK(Λ\text{-mod})$，其中$Λ$是系数在$\mathbb{Z}$中的Novikov环。