Cohomology Jumping Loci

本项目从多个角度探讨空间的拓扑与几何、基本群的代数结构、相关上同调跳跃轨迹的代数几何之间深刻而复杂的联系。 这项研究是在这样一种背景下进行的：它抽象出两个关键特征，这两个特征使得拓扑和代数之间的词典对于超平面安排的补充工作得很好：形式性（这导致了切锥定理，与特征和共振变体有关）和准投射性（这确保了特征变体由可能翻译的子环面组成）。这两个性质一起对共振簇的几何性质施加了强有力的限制，导致了对可实现为光滑（准）射影复簇的基本群的非线性群的强有力的阻碍。最近，出现了新的联系，将空间的上同调跳跃轨迹与其自由阿贝尔覆盖的同调有限性联系起来，从而得出Dwyer-Fried和Bieri-Neumann-Strebel-Renz不变量的结构。经典跳跃轨迹的推广--从秩一局部系统到非阿贝尔设置，从上同调环到微分分次代数--扩展了研究范围，在这个项目中使用的拓扑和群论方法揭示了在各种背景下发生的物体的组合和几何结构，允许不同领域之间的交叉授粉，起源于给定区域的想法在新的环境中得到了卓有成效的应用。 上同调跳跃轨迹理论影响了一系列空间和群的研究，包括复曲面复形和矩角复形;直角Artin和Coxeter群;真实的和复拟复曲面流形;Kaehler和拟Kaehler流形;超平面排列的Milnor纤维;以及构形空间和模空间的紧化。对这些对象的研究，以及它们与奇点理论的多重联系，图论和低维拓扑学提供了代数、拓扑学和组合学之间丰富的相互作用，产生了从拓扑机器人学到理论计算机科学等领域的应用。这项调查是与几位合作者以及研究生和本科生一起进行的。 参加密集的研究期和研讨会是为了向新一代的学生和年轻的研究人员介绍一个非常活跃的跨学科研究领域。

This project explores from various angles the deep andintricate connections between the topology and geometryof a space, the algebraic structure of its fundamentalgroup, and the algebraic geometry of the associatedcohomology jumping loci. The study is done in a contextwhich abstracts two of the crucial features that makethe dictionary between topology and algebra work sowell for complements of hyperplane arrangements:formality (which leads to the Tangent Cone Theorem,relating the characteristic and resonance varieties), andquasi-projectivity (which insures that the characteristicvarieties consist of possibly translated subtori). Together,these two properties put strong constraints on the geometryof the resonance varieties, leading to powerful obstructionsto finitely presented groups being realizable as fundamentalgroups of smooth, (quasi-) projective complex varieties.Recently, new connections have emerged, relating the cohomologyjumping loci of a space to the homological finiteness propertiesof its free abelian covers, and thereby to the structure of theDwyer-Fried and Bieri-Neumann-Strebel-Renz invariants.Generalizations of the classical jump loci---from rank onelocal systems to the non-abelian setting, and from cohomologyrings to differential graded algebras---extend the scope ofthe investigation, and broaden the range of its applicability.The topological and group-theoretic methods utilizedin this project shed new light on the combinatorial andgeometric structure of objects occurring in a varietyof contexts, allowing for cross-pollination betweendifferent fields, with ideas originating in a given areabeing fruitfully applied in new settings. The theory ofcohomology jumping loci impacts the study of a widearray of spaces and groups, including toric complexesand moment-angle complexes; right-angled Artin andCoxeter groups; real and complex quasi-toric manifolds;Kaehler and quasi-Kaehler manifolds; Milnor fibrationsof hyperplane arrangements; as well as configurationspaces and compactifications of moduli spaces.The study of these objects, with their multipleconnections to the theory of singularities, graphtheory, and low-dimensional topology, provides arich interplay between algebra, topology, andcombinatorics, yielding applications to areasranging from topological robotics to theoreticalcomputer science. The investigation is being conductedtogether with several collaborators, as well as graduateand undergraduate students. Participation in intensiveresearch periods and workshops is meant to introduce anew generation of students and young researchers to avery active, interdisciplinary area of study.