Bridgeland Moduli of Derived Objects on Algebraic Surfaces

代数曲面上派生对象的布里奇兰模

• 批准号: 0901128
• 负责人: Aaron Bertram
• 金额: $ 22.29万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-15 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901128&HistoricalAwards=false
• 关键词: Bridgeland; Moduli; Derived; Objects; Algebraic

Coherent sheaves are the bread and butter of algebraic geometry. They are the natural extension of vector bundles to a category that is closed under kernels and cokernels. Their naturality and usefulness was first explored in Serre's landmark paper (FAC). Traditionally, the coherent sheaves on a smooth projective variety are broken down in terms of dimension of support and ``stability'' (Geometric Invariant Theory). However, recent work in string theory points to an entire manifold of stability conditions on categories of complexes of vector bundles (D-branes in the physics literature). These resemble perverse sheaves, and like perverse sheaves seem to have extremely nice properties. In joint work with Daniele Arcara, the PI put stability conditions on a rigorous mathematical footing for all complex surfaces, and in the current proposal he will explore the applications of this new theory to ``classical'' problems in algebraic geometry.Algebraic geometry is the study of the shapes of solution sets of systems of polynomial equations in many variables. One crucial tool in this study is the construction of invariants, i.e. auxiliary structures that allow one to distinguish among the different shapes. Rather surprisingly, string theorists have made very significant contributions to algebraic geometry in recent years. In work relevant to this project, they have proposed the existence of a ``stability manifold'' for ``D-branes,'' which seems to be a very powerful new tool for both distinguishing different shapes and for answering classical questions in algebraic geometry (e.g. How many variables does one need in order to embed a particular shape?) The PI will develop this new tool, building on his previous work explaining the two-dimensional case.

连贯的滑轮是代数几何形状的面包和黄油。它们是向量捆绑包的自然扩展到在内核和焦点下关闭的类别。它们的自然性和实用性首先在Serre的地标纸（FAC）中探索。传统上，光滑的投影型品种上的连贯滑轮在支撑和``稳定性''（几何不变理论）方面被分解。然而，弦理论中的最新工作表明了稳定条件的整个多种形式，这是矢量束的复合物类别（物理文献中的D-溴）。这些类似于不正当的滑轮，并且像不正正的滑轮似乎具有非常好的特性。在与Daniele Arcara的联合合作中，PI将稳定条件置于所有复杂表面的严格数学基础上，在当前的建议中，他将探讨该新理论在代数几何形状中``经典''问题的应用。algebebraic几何形状是对多元素型系统的溶液集合的研究。这项研究中的一种关键工具是构造不变性，即辅助结构，可以区分不同形状。令人惊讶的是，近年来，弦理论家为代数几何形状做出了非常重要的贡献。在与该项目相关的工作中，他们提出了``d-branes''的存在``稳定性歧管''，这似乎是一个非常有力的新工具，可以区分不同的形状，并且可以在代数几何形状中区分经典问题，例如，在代数几何形状中，有多少次数可以启动per new the emper new suff the empers emper new shew shew shew shew shew shew shew there new nek and and emper nek and and emper nek nek and efors nek nek and efor new the nook？） 案件。