Bridgeland Moduli of Derived Objects on Algebraic Surfaces

凝聚层是代数几何的基础。它们是向量丛到在核和上核下闭的范畴的自然扩展。它们的自然性和有用性首先在Serre的里程碑式论文（FAC）中进行了探索。传统上，光滑射影簇上的相干层是根据支撑维数和“稳定性”（几何不变理论）来分解的。然而，最近弦理论的工作指出了向量丛（物理学文献中的D-膜）复形范畴的稳定性条件的整个流形。这些类似于反常层，并且像反常层一样，似乎具有非常好的性质。在与Daniele Arcara的合作中，PI将稳定性条件置于所有复杂曲面的严格数学基础上，在当前的提案中，他将探索这一新理论在代数几何中的“经典”问题中的应用。代数几何是对多变量多项式方程组的解集形状的研究。这项研究的一个关键工具是构建不变量，即允许人们区分不同形状的辅助结构。相当令人惊讶的是，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，为了嵌入一个特定的形状，需要多少个变量？）PI将开发这个新工具，建立在他以前解释二维情况的工作基础上。

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.