Bridgeland Moduli of Derived Objects on Algebraic Surfaces

连贯的轮子是代数几何的基本要素。它们是向量丛到一个闭于核和余核之下的范畴的自然扩张。它们的自然性和有用性最初是在Serre的里程碑式的论文(FAC)中探索的。传统上，光滑射影簇上的凝聚层是根据支撑度和“稳定性”(几何不变量理论)来分解的。然而，最近在弦理论方面的工作指出了向量丛(物理学文献中的D-膜)的复形范畴上的完整的稳定性条件。这些看起来像倒立的捆，和倒立的捆一样，似乎有非常好的特性。在与Daniele Arcara的合作中，PI将所有复杂曲面的稳定性条件建立在严格的数学基础上，在当前的建议中，他将探索这一新理论在代数几何中的经典问题上的应用。代数几何是研究多变量多项式方程组的解集的形状的学科。这项研究的一个重要工具是构造不变量，即允许人们区分不同形状的辅助结构。令人惊讶的是，弦理论家近年来对代数几何做出了非常重要的贡献。在与这个项目相关的工作中，他们提出了‘D-膜’的‘’稳定性流形‘’的存在，这似乎是一个非常强大的新工具，既可以区分不同的形状，也可以回答代数几何中的经典问题(例如，一个人需要多少变量才能嵌入一个特定的形状？)PI将在他之前解释二维情况的工作的基础上开发这一新工具。

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.