Moduli spaces of sheaves on non-Kaehler Hermitian manifolds

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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.