Enumerative geometry of Hilbert schemes

希尔伯特格式的枚举几何

• 批准号: 1001609
• 负责人: Alexei Oblomkov
• 金额: $ 12.39万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-15 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001609&HistoricalAwards=false
• 关键词: Enumerative; geometry; Hilbert; schemes

The main objective of this project is to study the geometry of moduli spaces of sheaves on low dimensional varieties and reveal possible connections of the subject with other fields of mathematics. The celebrated Gromov-Witten/Donaldson-Thomas correspondence is a conjectural correspondence between the integrals of characteristic classes of the universal ideal-sheaf over the Hilbert scheme of curves in the threefold and integrals over the moduli space of maps into the threefold. The conjecture is proven for toric threefolds and the lowest degree nontrivial characteristic classes. The project proposed is intended to extend the known results beyond the toric case and study the case of higher degree characteristic classes. Another part of the project is devoted to studying the topology of the Hilbert scheme of points on singular planar curves. A conjectural formula was proposed by the PI and V. Shende for the Poincare polynomials of the Hilbert scheme in terms of link invariants of the links of the singularities of the curve. To prove the formula is one of the goals of the project.Moduli spaces of sheaves are mathematical models for the field theories.Mathematical physics predictions, such as duality between the field theories and string theory or path integral formulas for knot invariants, can be translated into fascinating mathematical conjectures. The goal of this project is to find a proof of some of these conjectures.

该项目的主要目标是研究低维变量上滑轮模空间的几何形状，并揭示该学科与其他数学领域的可能联系。著名的格罗莫夫-维滕/唐纳森-托马斯对应关系是三重曲线希尔伯特方案上的通用理想束的特征类积分与三重映射模空间上的积分之间的猜想对应。该猜想对于环面三重和最低度非平凡特征类得到了证明。提出的项目旨在将已知结果扩展到复曲面案例之外，并研究更高程度的特征类别的案例。该项目的另一部分致力于研究奇异平面曲线上点的希尔伯特方案的拓扑。 PI和V. Shende根据曲线奇点链接的链接不变量提出了Hilbert格式的庞加莱多项式的猜想公式。证明该公式是该项目的目标之一。滑轮的模空间是场论的数学模型。数学物理预测，例如场论和弦理论之间的对偶性或结不变量的路径积分公式，可以翻译变成令人着迷的数学猜想。该项目的目标是找到其中一些猜想的证明。