Derived Categories and Other Invariants of Algebraic Varieties

代数簇的派生范畴和其他不变量

• 批准号: 1902251
• 负责人: Martin Olsson
• 金额: $ 17万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-15 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1902251&HistoricalAwards=false
• 关键词: Derived; Categories; Other; Invariants; Algebraic

The project is focused on a variety of questions in algebraic geometry, loosely centered around the study of invariants, both cohomological and categorical. Algebraic geometry is concerned with the study of the geometric spaces which locally can be modeled as the set of solutions of systems of polynomials in several variables, typically over a field but also over arithmetically interesting rings such as the integers. Such spaces are ubiquitous in mathematics, arising in many other areas such as representation theory, combinatorics and mathematical physics. They also arise naturally in a variety of applications in other disciplines such as computer science and engineering. A common theme in mathematics when studying complex objects, such as algebraic varieties, is to consider various features or invariants which are more tractable yet rich enough to capture meaningful information. The project will advance our understanding of several such invariants and study applications in algebraic and arithmetic geometry.More specifically the project is concerned with the following topics. Most of the work will be carried out with collaborators and students. (1) Derived categories of coherent sheaves on smooth projective algebraic varieties over a field and understanding additional cohomological features of equivalences between such categories. The goal is understand to what extent the derived category of coherent sheaves, together with additional structure, determines the isomorphism class, or possibly the birational equivalence class, of an algebraic variety. (2) Log coherent sheaves and Hochschild and topological Hochschild homology for log schemes. The objective is to understand a suitable dg category of coherent sheaves associated to a morphism of log schemes. The investigation of such a category also has broader implications for the study of degeneration phenomena for coherent sheaves. (3) Reconstruction theorems for algebraic varieties. In particular, the PI will investigate new approaches to, and generalizations of, reconstruction results previously studied using model theory. (4) Homotopy groups of log schemes and applications to the study of crystalline fundamental groups. In addition to these four projects, the PI will investigate several other projects together with graduate students.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目的重点是代数几何中的各种问题，松散地围绕着研究不变量，包括上同调和范畴。代数几何学关注的是研究几何空间，这些几何空间可以局部地建模为多个变量的多项式系统的解的集合，通常在域上，但也在算术有趣的环上，例如整数。这样的空间在数学中无处不在，出现在许多其他领域，如表示论，组合学和数学物理。它们也自然地出现在其他学科的各种应用中，如计算机科学和工程。在数学中，当研究复杂对象（如代数簇）时，一个常见的主题是考虑各种特征或不变量，这些特征或不变量更易于处理，但足够丰富，可以捕获有意义的信息。本计画将增进我们对数个不变量的了解，并研究其在代数与算术几何中的应用。大部分工作将与合作者和学生一起进行。(1)导出域上光滑射影代数簇上的凝聚层范畴，并理解这些范畴之间等价的附加上同调特征。我们的目标是了解在何种程度上的派生范畴的相干层，连同额外的结构，确定同构类，或可能的双有理等价类，代数簇。(2)对数凝聚层与对数格式的Hochschild同调和拓扑Hochschild同调。我们的目标是了解一个合适的dg类相干层相关的一个态射的日志计划。对这样一个范畴的研究对于相干层的退化现象的研究也有更广泛的意义。(3)代数簇的重构定理。特别是，PI将研究新的方法，并推广，重建结果以前使用模型理论研究。(4)对数格式的同伦群及其在晶体基本群研究中的应用。除了这四个项目外，PI还将与研究生一起调查其他几个项目。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Points with large stabilizer groups and sections of vector bundles

具有大稳定组的点和向量束的部分

• DOI: 10.1016/j.aim.2022.108628
• 发表时间: 2022
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Olsson, Martin

A geometric construction of semistable extensions of crystalline representations

晶体表示的半稳定延伸的几何构造

• DOI: 10.2140/tunis.2021.3.207
• 发表时间: 2021
• 期刊: Tunisian Journal of Mathematics
• 影响因子: 0.9
• 作者: Olsson, Martin

Deformation theory of perfect complexes and traces

完美复合体和迹线的变形理论

• DOI: 10.2140/akt.2022.7.651
• 发表时间: 2022
• 期刊: Annals of K-Theory
• 影响因子: 0.6
• 作者: Lieblich, Max;Olsson, Martin
• 通讯作者: Olsson, Martin

KUMMER COVERINGS AND SPECIALISATION

KUMMER 覆盖物和专业化

• DOI: 10.1017/s1474748020000511
• 发表时间: 2021
• 期刊: Journal of the Institute of Mathematics of Jussieu
• 影响因子: 0.9
• 作者: Olsson, Martin