叶状结构与稳定映射
Foliations and stable maps
AI总结：
本文利用稳定映射导出栈研究全局向量场的代数轨迹，构造可复合的有理轨迹范畴结构，并提出计数周期轨道的代数 Ruelle Zeta 函数。
摘要：
本文是正在进行的一系列通过导出代数几何研究代数簇上叶状结构的工作的一部分。我们在此聚焦于全局定义的向量场这一特定情形，以及其代数积分曲线的全局行为。对于带有全局向量场 $ν$ 的光滑真簇 $X$，我们考虑在到 $X$ 的、具有 $n$ 个标记点的亏格 $g$ 稳定映射导出栈上的诱导向量场 $ν_{g,n}$。当 $(g,n)$ 为 $(0,2)$ 或 $(1,0)$ 时，$ν_{g,n}$ 的零点导出栈定义了 $ν$ 的真代数轨迹模空间。当 $(g,n)=(0,2)$ 时，代数轨迹的行为非常类似于从 $ν$ 的一个零点到另一个零点的有理代数路径，尤其它们可以复合。这种复合由 Gromov-Witten 理论中通常的粘合映射表示，我们利用它给出三个不同范畴层次的范畴构造，在某种意义上通过去范畴化相互关联。为做到这一点，我们尤其必须处理非拟光滑导出栈的虚拟基本类。当 $(g,n)=(1,0)$ 时，$ν_{1,0}$ 的零点可被视为光滑实流形上向量场周期轨道的代数类似物。特别地，我们提出一个计数 $ν_{1,0}$ 零点的 Zeta 函数，我们愿将其视为 Ruelle 动力 Zeta 函数的代数版本。本文最后简要说明如何通过考虑 $F$-等变稳定映射的导出栈，将这些结果推广到一般一维叶状结构 $F$ 的情形。
英文摘要：
This paper is part of an ongoing series of works on the study of foliations on algebraic varieties via derived algebraic geometry. We focus here on the specific case of globally defined vector fields and the global behaviour of their algebraic integral curves. For a smooth and proper variety $X$ with a global vector field $ν$, we consider the induced vector field $ν_{g,n}$ on the derived stack of stable maps, of genus $g$ with $n$ marked points, to $X$. When $(g,n)$ is either $(0,2)$ or $(1,0)$, the derived stack of zeros of $ν_{g,n}$ defines a proper \emph{moduli of algebraic trajectories} of $ν$. When $(g,n)=(0,2)$ algebraic trajectories behave very much like rational algebraic paths from one zero of $ν$ to another, and in particular they can be composed. This composition is represented by the usual gluing maps in Gromov-Witten theory, and we use it give three categorical constructions, of different categorical levels, related, in a certain sense, by decategorification. In order to do this, in particular, we have to deal with virtual fundamental classes of non-quasi-smooth derived stacks. When $(g,n)=(1,0)$, zeros of $ν_{1,0}$ might be thought as algebraic analogues of periodic orbits of vector fields on smooth real manifolds. In particular, we propose a Zeta function counting the zeros of $ν_{1,0}$, that we like to think of as an algebraic version of Ruelle's dynamical Zeta function. We conclude the paper with a brief indication on how to extend these results to the case of general one dimensional foliation $F$, by considering the derived stack of $F$-equivariant stable maps.
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