Geometry and topology of fine compactified universal Jacobians

精细紧致通用雅可比行列式的几何和拓扑

• 批准号: 2669914
• 金额: --
• 依托单位: University of Liverpool
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2669914
• 关键词: Geometry; topology; fine; compactified; universal

The project will be to study the cohomology of compactified universal Jacobians as a vector space, in the same way as was done for the moduli spaces of stable pointed curves Mbargn. A successful method to calculate the cohomology is via Deligne's "Yoga of weights". The idea is that the usual, integer-valued (compactly supported) Euler characteristic is additive under a decomposition of a topological space into an open subset and its complement. For algebraic varieties, this additivity remains valid for refined versions of the Euler characteristics that "incorporate the weights", for example for the Euler characteristic that takes values in the category of Mixed Hodge structures. If the variety under analysis is smooth and compact, then by purity of the Hodge structures, the knowledge of the latter Euler characteristic is equivalent to knowing the Betti numbers. The moduli spaces of stable pointed curves admit a stratification whose strata consist of moduli spaces with lower genus and points. This stratification has been extensively used to calculate the cohomology of the moduli spaces of curves "inductively" starting from low genus, using the method outlined in the previous paragraph. We propose a similar approach for the universal compactified Jacobian: a smooth and compact space fibered over the moduli space of stable curves, whose fiber over each smooth curve is the Jacobian variety of that curve. We propose looking at: (a) Explicit calculations in low genus, starting in genus 2 (the genus 1 case is covered by one of the main results of the recent paper https://arxiv.org/abs/2012.09142 by Pagani-Tommasi). (b) Trying to detect general structure, as in the theory of "Modular Operads" by Getzler-Kapranov. (Fine compactified universal Jacobians can be interpreted as "admissible G-covers" for G the multiplicative group C\{0}, and the case when G is finite was addressed in work by Jarvis-Kauffman-Kimura https://arxiv.org/abs/math/0302316, and Petersen https://arxiv.org/pdf/1205.0420.pdf.)

该项目将是将压缩通用雅各布人作为矢量空间的同时研究，就像稳定尖曲线MBARGN的模量空间相同的方式。一种成功计算共同体的方法是通过Deligne的“权重瑜伽”。这个想法是，在拓扑空间分解为开放子集及其补体的情况下，通常的，整数值值（紧凑）欧拉的特征是加性的。对于代数品种，这种添加性对于“结合权重”的Euler特征的精制版本仍然有效，例如，对于欧拉（Euler）特征，该特征将值列入混合霍奇结构的类别中。如果所分析的多样性是平滑而紧凑的，则通过霍奇结构的纯度，对后者的特征的知识等同于了解贝蒂数字。稳定尖曲线的模量空间接受了一个分层，其地层由属和点较低的模量空间组成。该分层已广泛用于使用上一段中概述的方法从低属开始，从低属开始计算曲线的模量空间的共同体。我们为通用压实的雅各布式提出了类似的方法：在稳定曲线的模量空间上纤维纤维的平滑而紧凑的空间，其在每条平滑曲线上的纤维是该曲线的雅各布品种。我们建议查看：（a）低属中的显式计算，从属2开始（属1属的情况涵盖了最近的论文https://arxiv.org/abs/2012.09142的主要结果之一，由Pagani-tommasi）。 （b）试图检测一般结构，如Getzler-Kapranov的“模块化”理论。 （精细的压实通用雅各布人可以将其解释为G乘型组C \ {0}的“可允许的g-cobs”，并且Jarvis-Kauffman-Kimura https://arxiv.org/arxiv.org/arxiv.org/arxiv.org/math/math/math/math/0302316，petersens and petersens，以及Petersens，以及Petersens，以及Petersens，以及Petersens，and Petersens，and Petersens，and Petersens，and Petersens，以及Petersen，以及https://arxiv.org/pdf/1205.0420.pdf。）