Abstract Motivated by observations in physics, mirror symmetry is the concept that certain manifolds come in pairs X and Y such that the complex geometry on X mirrors the symplectic geometry on Y. It allows one to deduce symplectic information about Y from known complex properties of X. Strominger-Yau-Zaslow [61] described how such pairs arise geometrically as torus fibrations with the same base and related fibers, known as SYZ mirror symmetry. Kontsevich [43] conjectured that a complex invariant on X (the bounded derived category of coherent sheaves) should be equivalent to a symplectic invariant of Y (the Fukaya category, see [9] , [29] , [49] , [1] ). This is known as homological mirror symmetry. In this project, we first use the construction of “generalized SYZ mirrors” for hypersurfaces in toric varieties following Abouzaid-Auroux-Katzarkov [6] , in order to obtain X and Y as manifolds. The complex manifold is the genus 2 curve Σ 2 (so of general type c 1 0 ) as a hypersurface in its Jacobian torus. Its generalized SYZ mirror is a Landau-Ginzburg model ( Y , v 0 ) equipped with a holomorphic function v 0 : Y → C which we put the structure of a symplectic fibration on. We then describe an embedding of a full subcategory of D b C o h ( Σ 2 ) into a cohomological Fukaya-Seidel category of Y as a symplectic fibration. While our fibration is one of the first nonexact, non-Lefschetz fibrations to be equipped with a Fukaya category, the main geometric idea in defining it is the same as in Seidel's construction for Fukaya categories of Lefschetz fibrations in [55] and in Abouzaid-Seidel [3] .

中文翻译：

---

复数属 2 曲线上同调的分类镜对称性

摘要 受物理学观察的启发，镜像对称是这样一种概念，即某些流形成对出现 X 和 Y，使得 X 上的复几何反映 Y 上的辛几何。它允许人们从 X 的已知复性质推导出关于 Y 的辛信息. Strominger-Yau-Zaslow [61] 描述了这些对如何在几何上以具有相同基础和相关纤维的环面纤维形式出现，称为 SYZ 镜像对称。Kontsevich [43] 推测 X 上的复不变量（相干滑轮的有界派生范畴）应该等价于 Y 的辛不变量（Fukaya 范畴，见 [9] 、 [29] 、 [49] 、 [1] ）。这被称为同调镜像对称。在这个项目中，我们首先在 Abouzaid-Auroux-Katzarkov [6] 之后对复曲面变体中的超曲面使用“广义 SYZ 镜”的构造，以获得 X 和 Y 作为流形。复流形是属 2 曲线 Σ 2 （一般类型为 c 1 0 ）作为其雅可比环面中的超曲面。它的广义 SYZ 镜像是一个 Landau-Ginzburg 模型 ( Y , v 0 )，它配备了一个全纯函数 v 0 : Y → C，我们将其置于辛纤维化结构上。然后，我们将 D b C oh ( Σ 2 ) 的完整子类别嵌入到 Y 的上同调 Fukaya-Seidel 类别中作为辛纤维化。虽然我们的纤维化是第一个配备 Fukaya 范畴的非精确、非 Lefschetz 纤维化之一，但定义它的主要几何思想与 Seidel 中的相同