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（公法111-5）资助的。当一个光滑的复曲面退化为一个奇异曲面，并具有一个普通的双点时，存在一个所谓的消失环：一个二维球面坍缩为一个点。然而，在理论的模量的表面，更复杂的退化自然出现，没有消失的循环。在某些情况下，哈金描述了一个刚性的全纯向量丛的光滑纤维，这是类似于一个消失的周期。Hacking将使用这个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，空间的形状可以随时间连续变化;一个很好的例子是一个大肥皂泡的表面。空间的几何形状也会发生突变，或者？退化？例如，一个气泡可以分成两个更小的气泡。Hacking将研究给定空间在几何性质方面的可能退化。以前的工作在这个问题上使用的概念？消失周期 - 给定空间的一个子集，它收缩到退化中的一个点。Hacking将使用给定空间上的线性空间的丛，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。

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.