Exceptional vector bundles and degenerations of surfaces

特殊的向量束和表面的退化

• 批准号: 0855760
• 负责人: Paul Hacking
• 金额: $ 12万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2009-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0855760&HistoricalAwards=false
• 关键词: Exceptional; vector; bundles; degenerations; surfaces

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).When a smooth complex surface degenerates to a singular surface with an ordinary double point there is a so called vanishing cycle: a two dimensional sphere which collapses to a point. However, in the theory of moduli of surfaces, more complicated degenerations naturally arise for which there are no vanishing cycles. In some of these cases Hacking has described a rigid holomorphic vector bundle on the smooth fibre which is analogous to a vanishing cycle. Hacking will use this construction to study the boundary of the moduli space of surfaces and the classification of stable vector bundles on surfaces.In nature, the shape of a space can vary continuously with time; a good example is the surface of a large soap bubble. The geometry of a space can also undergo an abrupt change or ?degeneration?, for example, a bubble can divide into two smaller bubbles. Hacking will study the possible degenerations of a given space in terms of its geometric properties. Previous work on this problem used the notion of a ?vanishing cycle? - a subset of the given space which shrinks to a point in the degeneration. Hacking will use bundles of linear spaces over the given space whose structure becomes simpler under the degeneration. This method applies to important examples where the old techniques do not provide any information.

该奖项是根据 2009 年美国复苏和再投资法案（公法 111-5）资助的。当一个光滑的复杂表面退化为具有普通双点的奇异表面时，就会出现所谓的消失循环：二维球体塌缩成一个点。然而，在曲面模理论中，自然会出现更复杂的退化，其中不存在消失循环。在其中一些情况下，哈金描述了光滑纤维上的刚性全纯矢量丛，类似于消失循环。 Hacking将利用这种构造来研究曲面模空间的边界以及曲面上稳定向量丛的分类。在自然界中，空间的形状可以随时间连续变化；一个很好的例子是大肥皂泡的表面。空间的几何形状也可能发生突然的变化或“退化”，例如，一个气泡可以分成两个更小的气泡。黑客将研究给定空间在几何特性方面可能的退化。以前关于这个问题的工作使用了“消失循环”的概念。 - 给定空间的一个子集，它缩小到退化中的一个点。 Hacking 将在给定空间上使用线性空间束，其结构在退化下变得更简单。此方法适用于旧技术无法提供任何信息的重要示例。