Derived categories, stability conditions and geometric applications.

派生类别、稳定性条件和几何应用。

• 批准号: EP/T018658/1
• 负责人: Soheyla Feyzbakhsh
• 金额: $ 51.65万
• 依托单位: Imperial College London
• 依托单位国家: 英国
• 项目类别: Fellowship
• 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%2FT018658%2F1
• 关键词: Derived; categories; stability; conditions; geometric

Geometry studies higher-dimensional curved spaces. We can describe these spaces by equations, but the only case where we have any hope to use them for calculation is when the equations are polynomials. The resulting spaces are the objects of algebraic geometry, which are called varieties. Although these objects have been studied for a long time, there are still lots of crucial open problems: If we are given a variety, can we embed it in other well-known varieties? For instance, can we find a "nice'' surface which contains a given curve? If yes, how many such surfaces exist, and can we characterise them via some of the geometrical properties of the curve? The geometric information of varieties can be encoded in algebraic objects, known as derived categories. Inspired by ideas in string theory, Bridgeland introduced the notion of stability conditions on derived categories. This topic has been highly studied due to its connections to various fields in mathematics and physics, and lots of ideas and techniques have been developed in the area. Now is the time to employ the whole spectrum of modern tools in derived categories and stability conditions to solve so far intractable geometrical problems. My recent work proves that deformation of stability conditions and varying stability status of an object (wall-crossing phenomenon) are powerful new techniques for solving long-standing geometrical problems, that do not appear to involve derived categories. Surprisingly, stability conditions and wall-crossing truly provide the right context for studying those problems. The main goal of this research programme is to draw on ideas and tools in algebra, geometry and mathematical physics to describe some outstanding geometrical problems in terms of derived categories and stability conditions, and then apply wall-crossing techniques to solve them.

几何学研究高维的弯曲空间。我们可以用方程来描述这些空间，但我们唯一希望用它们来计算的情况是当方程是多项式时。由此产生的空间是代数几何的对象，称为簇。虽然这些对象已经被研究了很长时间，但仍然存在许多关键的开放问题：如果我们被赋予一个品种，我们能否将其嵌入其他知名品种？例如，我们能找到一个包含给定曲线的“好”曲面吗？如果是，有多少这样的曲面存在，我们可以通过曲线的一些几何性质来描述它们吗？簇的几何信息可以编码在代数对象中，称为派生范畴。受弦论思想的启发，布里杰兰引入了导出范畴的稳定性条件的概念。由于它与数学和物理学的各个领域都有联系，因此这个主题得到了高度的研究，并且在该领域已经开发了许多想法和技术。现在是时候采用整个频谱的现代工具在派生类别和稳定性条件，以解决迄今棘手的几何问题。我最近的工作证明，变形的稳定性条件和变化的稳定性状态的对象（跨壁现象）是强大的新技术，解决长期存在的几何问题，似乎不涉及派生类别。令人惊讶的是，稳定性条件和穿墙确实为研究这些问题提供了正确的背景。 该研究计划的主要目标是利用代数，几何和数学物理中的思想和工具来描述一些突出的几何问题的导出类别和稳定性条件，然后应用跨壁技术来解决这些问题。

项目成果

Higher rank Clifford indices of curves on a K3 surface

K3 曲面上曲线的高阶 Clifford 指数

• DOI: 10.1007/s00029-021-00664-z
• 发表时间: 2021
• 期刊: Selecta Mathematica
• 作者: Feyzbakhsh S

New perspectives on categorical Torelli theorems for del Pezzo threefolds

德尔佩佐三重分类托雷利定理的新视角

• DOI: 10.48550/arxiv.2304.01321
• 发表时间: 2023
• 作者: Feyzbakhsh S

Serre-invariant stability conditions and Ulrich bundles on cubic threefolds

Serre 不变稳定性条件和三次三次上的 Ulrich 丛

• DOI: 10.46298/epiga.2022.9611
• 发表时间: 2023
• 期刊: Épijournal de Géométrie Algébrique
• 作者: Feyzbakhsh S

Curve counting and S-duality

曲线计数和 S 对偶性

• DOI: 10.46298/epiga.2023.volume7.9818
• 发表时间: 2023
• 期刊: Épijournal de Géométrie Algébrique
• 作者: Feyzbakhsh S

The desingularization of the theta divisor of a cubic threefold as a moduli space

三次三次的 theta 除数作为模空间的去奇异化

• 期刊: Geometry & Topology
• 影响因子: 2
• 作者: A. Bayer, S. Beentjes, S. Feyzbakhsh, G. Hein, D. Martinelli, F. Rezaee, B. Schmidt.