Kodaira纤维与环绕Floer上同调
Kodaira fibres and wrapped Floer cohomology
浏览论文内容
中文总结 AI 辅助
本文证明奇异椭圆纤维邻域为Weinstein域，计算其环绕Floer上同调集中于零次，并建立与显式代数的范畴等价，应用于镜像对称。
中文摘要 AI 辅助
设 $F$ 是相对极小的复椭圆纤维化中具有光滑全空间的一个奇异纤维，并设 $\Omega$ 为 $F$ 附近的一个非退化全纯二形式。我们证明 $F$ 的一个小邻域是对于 $\mathrm{Re}\\,\Omega$ 的Weinstein域，其完备化是Legendre手术，其余核（cocores）通过完成与 $F$ 的分量横截的全纯圆盘获得。对于每个系数域和每个乘法bulk类，我们计算这些余核的环绕Floer上同调，并证明它集中在零次。这些余核生成，因此bulk形变的环绕Fukaya范畴等价于一个显式代数的完美模范畴：对于法交叉纤维是仿射型的乘法预投射代数，对于类型 $II$、$III$ 和 $IV$ 是带关系的箭图代数。应用包括仿射管道dg代数的形式性、在平凡bulk类下与解消仿射曲面的镜像等价，以及在根单位bulk类下与代数环面的商栈的镜像等价（针对四个星形纤维）。
英文摘要
Let $F$ be a singular fibre of a relatively minimal complex elliptic fibration with smooth total space, and let $Ω$ be a nonvanishing holomorphic two-form near $F$. We show that a small neighbourhood of $F$ is a Weinstein domain for $\mathrm{Re}\,Ω$ whose completion is a Legendrian surgery, with cocores obtained by completing holomorphic disks transverse to the components of $F$. For every coefficient field and every multiplicative bulk class, we compute the wrapped Floer cohomology of these cocores and prove that it is concentrated in degree zero. The cocores generate, so the bulk-deformed wrapped Fukaya category is equivalent to the category of perfect modules over an explicit algebra: a multiplicative preprojective algebra of affine type for the normal crossing fibres, and a quiver algebra with relations for types $II$, $III$ and $IV$. Applications include formality of the affine plumbing dg algebras; mirrors given by resolved affine surfaces at every classical bulk class for affine D, E and types II, III, IV; and mirrors given by quotient stacks of algebraic tori at root-of-unity bulk classes for the four affine stars.
