Geometry and topology of fine compactified universal Jacobians

该项目将研究紧致泛雅可比矩阵作为向量空间的上同调，与稳定尖曲线Mbargn的模空间相同。计算上同调的一个成功的方法是通过Deligne的“重量瑜伽”。其思想是，通常的整数值（紧支撑的）欧拉特征在拓扑空间分解为开子集及其补集的情况下是可加的。对于代数簇，这种可加性对于“包含权重”的欧拉特征线的改进版本仍然有效，例如对于在混合霍奇结构类别中取值的欧拉特征线。如果所分析的簇是光滑和紧致的，那么通过霍奇结构的纯粹性，后者的欧拉特征的知识相当于知道贝蒂数。稳定尖曲线的模空间允许分层，其层次由具有较低亏格和点的模空间组成。这种分层已经被广泛地用于计算曲线的模空间的上同调，从低亏格开始，使用前面段落中概述的方法。我们提出了一个类似的方法，通用的紧雅可比矩阵：一个光滑的和紧凑的空间纤维的模空间的稳定曲线，其纤维在每个光滑曲线是雅可比品种的曲线。我们建议研究：（a）低亏格的显式计算，从亏格2开始（Pagani-Tommasi最近的论文www.example.com的主要结果之一涵盖了亏格1的情况https://arxiv.org/abs/2012.09142）。(b)试图发现一般结构，如格茨勒-卡普拉诺夫的“模运算”理论。(Fine紧化的泛雅可比算子可以被解释为乘法群C\{0} G的“可容许G覆盖”，并且当G是有限的情况在Jarvis-Kauffman-Kimura的工作中被解决了https：//arxiv.org/abs/math/0302316和Petersen https：//arxiv.org/pdf/1205.0420.pdf。

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.)