AI 中文总结
本文提出微局部perverse schobers的范畴化定义，证明其在Radon变换下对曲线$y^m=x^n$不变，并关联Fourier变换、周期SODs与球面单子。
AI 中文摘要
Perverse schobers是perverse sheaves的一种范畴化，最初由Kapranov和Schechtman提出。本文旨在启动perverse schobers的微局部研究。我们首先对$\operatorname{Perv}(\mathbb{C},R)/\operatorname{Loc}(\mathbb{C})$进行范畴化，这是复直线上以$R$为奇点的perverse sheaves的范畴模去局部系统。我们利用这一点提出了一个关于微局部perverse schobers的一般定义，这些schobers支撑在超曲面芽的开余法丛上，并猜想该定义在Radon变换下不变。在此我们做出关键观察：虽然perverse schobers的类似商$\mathsf{2Perv}(\mathbb{C},R)/\mathsf{2Loc}(\mathbb{C})$是一个合理的范畴化，但由此产生的理论在Radon变换下并非不变。事实上，我们提出的范畴化可以通过以普遍方式修正这一失败的一个实例来恢复。当我们的超曲面是$\mathbb{C}^2_{x,y}$中的曲线$y^m=x^n$时，我们证明了Radon不变性。在此过程中，我们解释了我们的理论如何与perverse schobers的Fourier变换、周期半正交分解（periodic SODs）以及球面单子（spherical monads）相关联。
英文摘要
Perverse schobers are a categorification of perverse sheaves, originally proposed by Kapranov and Schechtman. The purpose of this paper is to initiate a microlocal study of perverse schobers. We first categorify $\operatorname{Perv}(\mathbb{C},R)/\operatorname{Loc}(\mathbb{C})$, the category of perverse sheaves on a complex line with singular points at $R$, modulo local systems. We use this to propose a general definition for microlocal perverse schobers supported on the open conormal to a germ of a hypersurface, and we conjecture that this is invariant under Radon transform. Here we make the key observation that while the analogous quotient of perverse schobers $\mathsf{2Perv}(\mathbb{C},R)/\mathsf{2Loc}(\mathbb{C})$ is a reasonable categorification, the resulting theory fails to be invariant under Radon transform. In fact, our proposed categorification can be recovered by correcting an instance of this failure in a universal manner. We prove Radon invariance when our hypersurface is the curve $y^m=x^n$ in $\mathbb{C}^2_{x,y}$. Along the way, we explain how our theory relates to Fourier transforms of perverse schobers, periodic SODs, and spherical monads.
Comments80 pages, comments welcome
