Moduli spaces of sheaves on Hermitian manifolds

目标.这个提议的主要目标是研究紧致（广义）复流形上稳定丛和层的某些模空间的几何性质。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，如全纯向量丛和层。事实上，这类对象的模空间经常从基流形继承有趣的结构，从而产生新的、高维的具有特定几何结构的空间例子。此外，如果流形上的复结构与广义复结构相关，那么也可以考虑广义全纯丛，在这种情况下，它由全纯向量丛和丛上的额外算子组成。适合这种广义设置的例子是co-Higgs丛和全纯泊松模，仅举两例。本研究还旨在通过考察广义全纯丛模空间的显式例子，更好地理解广义全纯丛的几何，并确定这些空间允许什么类型的几何结构。这些例子将反过来产生新的见解的几何模空间的稳定丛和层的基础复杂的流形。这项研究的第二个目标是为一种新的几何学--玻恩几何奠定基础，玻恩几何学是最近引入的，作为弦理论对偶对称公式--元弦理论的几何背景。该提案中考虑的复流形在数学和物理的许多领域中发挥着重要作用。因此，这项研究的结果将提供许多明确的例子，空间，将感兴趣的范围广泛的几何学家，物理学家，规范理论家和研究人员在该领域的可积系统。

Objectives. The main goal of this proposal is to study the geometric properties of certain moduli spaces of stable bundles and sheaves on compact (generalized) complex manifolds. An important problem in the study of complex manifolds is the classification of objects on these manifolds such as holomorphic vector bundles and sheaves. In fact, moduli spaces of such objects often inherit interesting structures from the base manifold, thus giving rise to new, higher dimensional examples of spaces with specific geometric structures. Moreover, if the complex structure on the manifold is related to a generalized complex structure, then one may also consider generalized holomorphic bundles, which consist, in this case, of pairs of a holomorphic vector bundle together with an extra operator on the bundle. Examples that fit into this generalized setting are given by co-Higgs bundles and holomorphic Poisson modules, to name just two. This research also aims at gaining a better understanding of the geometry of generalized holomorphic bundles by examining explicit examples of their moduli spaces, and determining what types of geometric structures these spaces admit. These examples will in turn yield new insights into the geometry of moduli spaces of stable bundles and sheaves on the underlying complex manifolds. A secondary goal of this research is to lay some of the foundations for a new geometry called Born geometry, which was very recently introduced as a geometric background for a duality symmetric formulation of string theory called metastring theory.Impact. The 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 spaces that will be of interest to a wide range of geometers, physicists, gauge theorists, and researchers in the field of integrable systems.