Moduli spaces of sheaves on non-Kaehler Hermitian manifolds

这一建议的目的是研究紧致复流形上不允许Kaehler度量的稳定丛和层的模空间的几何性质。复流形研究中的一个重要问题是对这些流形上的对象进行分类，如向量丛和层。极化射影簇上的全纯向量丛已经得到了广泛的研究，它们的几何到目前为止已经被很好地理解了；然而，对于非射影情形，我们所知的要少得多。这是因为许多用于射影簇的技巧只给出了一般紧致复解析流形的部分结果。本研究的目的是通过考察非Kaehler紧复流形等非射影流形来更好地理解非射影图景。研究非Kaehler复流形上的丛的另一个动机来自弦理论。这个理论特别感兴趣的是具有全纯平凡正则丛的复流形，因为它们是弦紧化的潜在背景--流形上的某些向量丛决定了这些紧化。直到最近，物理文献中出现的大多数模型都是在配备了Kaehler度量的椭圆Calabi-Yau流形上紧凑的弦。然而，现在人们对非Kaehler流形作为弦紧化的其他可能背景的兴趣越来越大。在该提案中考虑的一些非Kaehler复流形在许多数学和物理领域中起着基本的作用。因此，这项研究的结果将提供许多明确的模空间的例子，这些例子将引起广泛的几何学家、物理学家、规范理论家、可积系统领域的研究人员和模型理论家的兴趣。

The aim of this proposal is to study the geometric properties of moduli spaces of stable bundles and sheaves on compact complex manifolds that do not admit Kaehler metrics. An important problem in the study of complex manifolds is the classification of objects on these manifolds such as vector bundles and sheaves. Holomorphic vector bundles on polarized projective varieties have been extensively studied and their geometry is by now well understood; however, far less is known in the non-projective case. This is due to the fact that many of the techniques used for projective varieties only give partial results for general compact complex analytic manifolds. The goal of this research is to gain a better understanding of the non-projective picture by examining non-projective varieties such as non-Kaehler compact complex manifolds.Another motivation for the study of bundles on non-Kaehler complex manifolds comes from string theory. Of particular interest in this theory are complex manifolds with holomorphically trivial canonical bundles as they are potential backgrounds for compactifications of strings - certain vector bundles on the manifold determine these compactifications. Most of the models appearing in the physics literature have, until recently, been of strings compactified on elliptic Calabi-Yau manifolds equipped with a Kaehler metric. Nevertheless, there is now growing interest in non-Kaehler manifolds as other possible backgrounds for string compactifications.Some of the non-Kaehler complex manifolds considered in the proposal play a fundamental role in many areas of mathematics and physics. The results of this research will therefore provide many explicit examples of moduli spaces that will be of interest to a wide range of geometers, physicists, gauge theorists, researchers in the field of integrable systems, and model theorists.