Exceptional vector bundles and degenerations of surfaces

该奖项是根据2009年《美国复苏和再投资法案》(Public Law 111-5)资助的。当一个光滑的复杂曲面退化为具有普通双点的奇异曲面时，存在一个所谓的消失循环：一个塌缩为一点的二维球体。然而，在曲面的模理论中，自然会出现更复杂的退化，对于这些退化，不存在消失循环。在其中一些情况下，海辛描述了光滑纤维上的刚性全纯向量丛，这类似于一个消失的循环。Hking将使用这种构造来研究曲面的模空间的边界和曲面上稳定向量丛的分类。在自然界中，空间的形状可以随着时间连续变化；一个很好的例子是大型肥皂泡的表面。空间的几何形状也可能发生突变或“退化”，例如，一个气泡可以分成两个较小的气泡。黑客将根据给定空间的几何性质来研究其可能的退化。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，它收缩到退化中的一个点。黑客将在给定空间上使用线性空间丛，这些空间的结构在退化后变得更简单。这种方法适用于旧技术不提供任何信息的重要例子。

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.