[论文解读] A theory of dormant opers on pointed stable curves -- a proof of Joshi's conjecture

Yasuhiro Wakabayashi|arXiv (Cornell University)|Nov 5, 2014

Algebraic Geometry and Number Theory参考文献 47被引用 16

一句话总结

本文建立了关于休眠主丛——在正特征下指向稳定曲线的特殊联络——的全面理论，统一了主丛、对数联络与模空间。通过计算Quot-概形的Gromov-Witten不变量，证明了Joshi的猜想，揭示了$ p $-进Teichmüller理论与正特征下枚举几何之间的深刻联系。关键结果是给出了休眠主丛模空间度数的显式公式，该公式以单位根之和表示，并与Vafa-Intriligator型不变量相关联。

ABSTRACT

This manuscript presents a detailed and original account of the theory of opers defined on pointed stable curves in arbitrary characteristic and their moduli. In particular, it includes the development of the study of dormant opers, which are opers of a certain sort in positive characteristic. The theory of dormant opers (or more generally, opers in positive characteristic) on pointed stable curves, which has proved to be rather rich and deep, was born in the work of S. Mochizuki, who developed the theory for $\mathfrak{sl}_2$-opers and used it to establish $p$-adic Teichmüller theory. Some parts of Mochizuki's work were later extended in the case of proper smooth curves by K. Joshi, C. Pauly, and other mathematicians. This manuscript represents an advance in the theory of opers that takes the subject beyond the work of Mochizuki, Joshi, and Pauly. In particular, we provide general unified formulations and the basics of principal bundles and connections defined on families of pointed stable curves. The notion of an oper is accordingly introduced in the context of logarithmic algebraic geometry. Some of the results can be regarded as generalizations of results obtained in the fundamental work on the geometric Langlands program developed by A. Beilinson and V. Drinfeld. We also describe various properties and assertions about (dormant) opers, such as duality, comparison with differential operators, and compactification of the moduli space. Our goal is to give an explicit formula, conjectured by Joshi, for the generic number of dormant $\mathfrak{sl}_n$-opers. We do so by obtaining a detailed understanding of the moduli space of dormant opers and computing the Gromov-Witten invariants for Quot-schemes in characteristic zero. This formula reveals an interaction between studies in $p$-adic Teichmüller theory and certain areas of mathematics, including Gromov-Witten theory.

研究动机与目标

• 在任意特征下，为指向稳定曲线的主丛，特别是休眠主丛，发展统一的理论。
• 将Mochizuki和Joshi–Pauly在$p$-进Teichmüller理论方面的工作推广至高阶李代数和任意指向稳定曲线。
• 计算在正特征下，一般亏格$g$的曲线上休眠$\mathrm{sl}_n$-主丛的通用数量，验证Joshi的猜想。
• 在特征零下，建立休眠主丛模空间与Quot-概形Gromov-Witten不变量之间的精确联系。
• 为正特征下主丛的融合规则与对偶结构提供几何与代数基础。

提出的方法

• 通过对数代数几何与一族指向稳定曲线的$\hbar$-平坦联络，形式化主丛。
• 引入$(g, \hbar)$-主丛及其模空间的概念，利用Hitchin-Mochizuki映射与$p$-曲率消失条件。
• 利用特征零下Quot-概形的理论，计算控制休眠主丛模空间度数的Gromov-Witten不变量。
• 应用形变理论与上同调技巧，分析休眠主丛的无穷小形变，并证明模堆栈的通用平坦性。
• 通过伪融合环构造，建立正交与辛李代数主丛之间的对偶性。
• 将计数问题约化为涉及单位根与伯努利数的留数计算，借助Vafa-Intriligator公式。

实验结果

研究问题

• RQ1 在正特征下，一般指向稳定曲线的亏格$g$曲线上，休眠$\mathrm{sl}_n$-主丛的通用数量是多少？
• RQ2 休眠主丛的模空间与特征零下Quot-概形的Gromov-Witten不变量有何关系？
• RQ3 休眠主丛模堆栈的几何与代数结构是什么？它是否在曲线上模空间上通用平坦？
• RQ4 在正特征下，主丛的对偶性与融合规则如何体现，特别是对经典李代数而言？
• RQ5 Vafa-Intriligator公式能否被调整，以通过留数方法计算休眠主丛模空间的度数？

主要发现

• 在正特征下，一般亏格$g$的曲线上，休眠$\mathrm{sl}_n$-主丛的通用数量由$ n $元组的$ p $次单位根（互异）之和给出，其值等于$ p^{(n-1)(g-1)-1}/n! $乘以单位根上的对称有理函数。
• 当$ n=3 $时，度数$ d_{3,g} = \deg(\mathcal{M}_{\mathrm{dorm}, \mathrm{sl}_3}^g) $是$ p $的$ 2g-2 $次多项式，已显式计算至$ g=6 $。
• 休眠$\mathrm{sl}_n$-主丛模空间度数的公式，与特征零下$\mathrm{Quot}_{n,0}^\Theta$在$\overline{\mathcal{M}}_{g,0}$上的Gromov-Witten不变量的Vafa-Intriligator公式完全一致。
• 休眠主丛的模堆栈在曲线上模空间上是通用平坦的，这是证明计数有限且可计算的关键步骤。
• $\mathrm{so}_{2l+1}$与$\mathrm{sp}_{2m}$主丛之间的对偶性诱导其模空间之间的同构，推广了表示理论中的经典对偶性。
• 度数的计算可约化为涉及$\sin(\pi \theta / p)$项的多重留数，这些项与$\cot(\pi \theta / p)$的洛朗展开及伯努利数相关。