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DERIVED CATEGORIES AND ALGEBRAIC K-THEORY OF SINGULARITIES

奇点的派生范畴和代数 K 理论

基本信息

批准号：
EP/T019379/1
负责人：
Evgeny Shinder
金额：
$ 42.4万
依托单位：
University of Sheffield
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2020
资助国家：
英国
起止时间：
2020 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FT019379%2F1
关键词：
DERIVED CATEGORIES ALGEBRAIC THEORY SINGULARITIES

项目摘要

Singularities are ubiquitous in mathematics and physics. In the realm of the physical world a good example of a singularity is a black hole. In mathematical terms having a singularity usually means that a denominator is becoming zero in a coefficient or in a solution to a differential equation. This project is about singularities in more abstract area of mathematics: Algebraic Geometry. Here the non-singular (that is smooth) objects are much better understood than singular ones, and yet singularities play a crucial role in the modern Algebraic Geometry such as in the Minimal Model Program (Caucher Birkar Fields Medal 2018). One way to study geometric objects and shapes in Algebraic Geometry (called algebraic varieties) is to attach algebraic invariants to them, such as numbers, rings, or categories. One of the central such modern invariants is the so-called derived category of coherent sheaves. Derived categories of coherent sheaves are much better understood for non-singular varieties, than for singular ones, for the basic reason that singularities provide a new layer of complications to deal with. In this proposal I suggest a systematic study of derived categories of singular algebraic varieties, their decomposition into simpler pieces (semiorthogonal decompositions), their numerical properties (algebraic K-theory) and the relationship between derived categories of singular varieties and their nonsingular replacements (resolutions of singularities). The study is connected to several areas of modern pure mathematics: Algebra, Algebraic Geometry, Homological Algebra, Category Theory, Algebraic K-theory, and mixes these in new and meaningful ways in order to enhance our understanding of singularities.

奇点在数学和物理中无处不在。在物理世界中，奇点的一个很好的例子是黑洞。在数学术语中，具有奇点通常意味着在系数或微分方程的解中分母变为零。这个项目是关于奇点在更抽象的数学领域：代数几何。在这里，非奇异（即光滑）对象比奇异对象更容易理解，但奇异性在现代代数几何中起着至关重要的作用，例如在最小模型程序（Caucher Birkar Fields奖2018）中。研究代数几何中的几何对象和形状（称为代数簇）的一种方法是将代数不变量附加到它们上，例如数字，环或类别。这种现代不变量的核心之一是所谓的凝聚层的导出范畴。相干层的导出范畴对于非奇异簇比对于奇异簇更容易理解，其基本原因是奇异性提供了一层新的复杂性来处理。在这个建议中，我建议系统地研究奇异代数簇的衍生范畴，它们分解成更简单的部分（半正交分解），它们的数值性质（代数K理论）以及奇异簇的衍生范畴与它们的非奇异替换（奇异性的分解）之间的关系。这项研究与现代纯数学的几个领域有关：代数，代数几何，同调代数，范畴论，代数K理论，并以新的和有意义的方式混合这些，以提高我们对奇点的理解。