Mohammed Abouzaid
Given a smooth projective toric variety X, we construct an A-infinity category of Lagrangians with boundary on a level set of the Landau-Ginzburg mirror of X. We prove that this category is quasi-equivalent to the DG category of line bundles on X. This establishes part of the Homological Mirror Conjecture for toric varieties.
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We introduce the notion of a tropical coamoeba which gives a combinatorial description of the Fukaya category of the mirror of a toric Fano stack. We show that the polyhedral decomposition of a real n-torus into (n + 1) permutohedra gives a tropical coamoeba for the mirror of the projective space, and prove a torus-equivariant version of homological mirror symmetry for the projective space. As a corollary, we obtain homological mirror symmetry for toric orbifolds of the projective space.
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。
Homological mirror symmetry of toric Fano surfaces via Morse homotopy
应用SYZ构造于环面Fano曲面，通过考虑矩多面体的Morse同伦而非Fukaya范畴来讨论同调镜像对称。
This is a sequel to our paper arXiv:2008.13462, where we proposed a definition of the Morse homotopy of the moment polytope of toric manifolds. Using this as the substitute of the Fukaya category of the toric manifolds, we proved a version of homological mirror symmetry for the projective spaces and their products via Strominger-Yau-Zaslow construction of the mirror dual Landau-Ginzburg model. In this paper we go this way further and extend our previous result to the case of the Hirzebruch surface $\mathbb{F}_1$.
Morse-Bott cohomology from homological perturbation theory
该研究在最小横截性假设下基于同调扰动理论构建Morse-Bott理论的上链复形，讨论不同构造的关联，完成等变情形构造并验证性质，在闭流形情形验证其恢复正则上同调，还提出与polyfold理论结合的计划。
We study homological mirror symmetry for toric varieties, exploring the relationship between various Fukaya-Seidel categories which have been employed for constructing the mirror to a toric variety. In particular, we realize tropical Lagrangian sections as objects of a partially wrapped category and construct a Lagrangian correspondence mirror to the inclusion of a toric divisor. As a corollary, we prove that tropical sections generate the Fukaya-Seidel category, completing a Floer-theoretic proof of homological mirror symmetry for projective toric varieties. In the course of the proof, we dev…
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}…
SYZ mirror of Hirzebruch surface $\mathbb{F}_k$ and Morse homotopy
本文利用Strominger-Yau-Zaslow镜像对构造和Morse同伦，将Hirzebruch曲面 $\mathbb{F}_1$ 的同调镜像对称性推广到 $\mathbb{F}_k$。
We introduce tropical corals, balanced trees in a half-space, and show that they correspond to holomorphic polygons capturing the product rule in Lagrangian Floer theory for the elliptic curve. We then prove a correspondence theorem equating counts of tropical corals to punctured log Gromov--Witten invariants of the Tate curve. This implies that the homogeneous coordinate ring of the mirror to the Tate curve is isomorphic to the degree-zero part of symplectic homology, confirming a prediction of homological mirror symmetry.
On intrinsic homological mirror symmetry for toric degenerations
本文针对Calabi--Yau流形的极大单能退化，利用固定点Floer上同调构造候选镜像族，并在热带拉格朗日截面存在时建立导出范畴到Fukaya范畴的完全忠实嵌入，验证了若干环带退化情形。
Rigidity and Realizability for Tropical Curves in Dimension 3
本研究利用伪全纯圆盘边界的$H_1(L)$类提出拉格朗日三维流形的无障碍判据，证明具裤对分解的刚性热带曲线的拉格朗日提升无障碍，并在同调镜像对称成立的前提下推出镜像阿贝尔三维流形中存在对应实现。
Smoothing, scattering, and a conjecture of Fukaya
本文证明了Fukaya关于镜对称的猜想的一个重构，用热带几何替代多值Morse理论，通过热带化预dgBV代数并与半平坦形变理论比较，从Maurer-Cartan解提取散射图。
We prove Homological Mirror Symmetry for a smooth d-dimensional Calabi-Yau hypersurface in projective space, for any d > 2 (for example, d = 3 is the quintic three-fold). The main techniques involved in the proof are: the construction of an immersed Lagrangian sphere in the `d-dimensional pair of pants'; the introduction of the `relative Fukaya category', and an understanding of its grading structure; a description of the behaviour of this category with respect to branched covers (via an `orbifold' Fukaya category); a Morse-Bott model for the relative Fukaya category that allows one to make ex…
Morse superpotentials and blow-ups of surfaces
本文基于热带几何框架，研究了环面曲面吹胀的 Landau-Ginzburg 镜像，提出识别超势临界点位置的方法并证明其非退化性，进而通过临界点数量与上同调秩相等导出闭弦镜像对称。
In this paper we discuss two major conjectures in Mirror Symmetry: Strominger-Yau-Zaslow conjecture about torus fibrations, and the homological mirror conjecture (about an equivalence of the Fukaya category of a Calabi-Yau manifold and the derived category of coherent sheaves on the dual Calabi-Yau manifold). Our point of view on the origin of torus fibrations is based on the standard differential-geometric picture of collapsing Riemannian manifolds as well as analogous considerations for Conformal Field Theories. It seems to give a description of mirror manifolds much more transparent than th…
Realizability in tropical geometry and unobstructedness of Lagrangian submanifolds
本文提出热带子簇的 $B$-可实现性概念，证明若拉格朗日子流形提升具有定义良好的 Floer 上同调且可通过界定上链变为无障碍，则该热带子簇是 $B$-可实现的，并应用于证明亏格零热带曲线及热带阿贝尔曲面内热带曲线的可实现性。
Tropical Lagrangian coamoebae and free resolutions
本文研究 $(\mathbb{C}^\times)^n$ 的拉格朗日子流形的余阿米巴，通过将模的极小自由分解关联到单纯复形（热带拉格朗日余阿米巴），揭示了镜像相干层的同调代数，并推广了二聚体模型理论。
As an explicit example of an $A_\infty$-structure associated to geometry, we construct an $A_\infty$-structure for a Fukaya category of finitely many lines (Lagrangians) in $\R^2$, ie., we define also {\em non-transversal} $A_\infty$-products. This construction is motivated by homological mirror symmetry of (two-)tori, where $\R^2$ is the covering space of a two-torus. The strategy is based on an algebraic reformulation of Morse homotopy theory through homological perturbation theory (HPT) as discussed by Kontsevich and Soibelman in math.SG/0011041, where we introduce a special DG category whi…
Microlocal Morse theory of wrapped Fukaya categories
本文推广 Nadler–Zaslow 对应，用 Reeb 轨迹与范畴紧的无限秩层刻画无穷远态射，证明停止的部分包裹 Fukaya 范畴等价于微支撑受限层的紧对象范畴，并描述稳定极化 Weinstein 扇形。
Cusped hyperbolic Lagrangians as mirrors to lines in three-space
本文构造了三维环面余切丛中带孤立锥形奇点的尖点双曲拉格朗日子流形及对应浸入拉格朗日，证明其经局部系统修饰后可作为射影三维空间中直线的同调镜像，同时构造了适配浸入对象的包裹Fukaya范畴。
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