Geometry and topology of fine compactified universal Jacobians

本课题将研究紧化泛雅可比矩阵作为向量空间的上同调性，就像研究稳定点曲线的模空间一样。计算上同调的一个成功方法是通过Deligne的“重量瑜伽”。其思想是，通常的整数值（紧支持）欧拉特征在拓扑空间分解为开放子集及其补时是可加的。对于代数变量，这种可加性对于“合并权重”的欧拉特征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。如果所分析的变化是光滑和紧凑的，那么通过霍奇结构的纯度，了解后者的欧拉特征相当于知道了贝蒂数。稳定点曲线的模空间允许分层，其地层由具有下格和下点的模空间组成。这种分层已被广泛用于计算曲线模空间的上同调，从低属开始“归纳”，使用前一段概述的方法。对于普遍紧化雅可比矩阵，我们提出了类似的方法：在稳定曲线的模空间上有一个光滑紧化的空间纤维，其在每条光滑曲线上的纤维是该曲线的雅可比变分。我们建议考虑：(a)低属的显式计算，从属2开始（属1的情况由Pagani-Tommasi最近的论文https://arxiv.org/abs/2012.09142的主要结果之一所涵盖）。(b)试图探测一般结构，如盖兹勒-卡普拉诺夫的“模操作”理论。（精细紧化全称雅可比矩阵可以解释为G的“可容许G-盖”，对于G的乘法群C\{0}，当G是有限的情况在Jarvis-Kauffman-Kimura https://arxiv.org/abs/math/0302316和Petersen https://arxiv.org/pdf/1205.0420.pdf的工作中得到了解决。）

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.)