Moduli spaces of sheaves on Hermitian manifolds

目标。本文的主要目的是研究紧(广义)复流形上稳定丛和稳定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，这类对象的模空间通常继承了基流形的有趣结构，从而产生了具有特定几何结构的空间的新的、更高维的例子。此外，如果流形上的复结构与广义复结构有关，则还可以考虑广义全纯丛，在这种情况下，广义全纯丛由全纯向量丛的对和丛上的额外算子组成。联合希格斯丛和全纯泊松模给出了适合这一广义设置的例子，仅举两例。这项研究还旨在通过考察广义全纯丛的模空间的显例来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。这些例子将反过来对基础复流形上稳定丛和层的模空间的几何产生新的见解。这项研究的第二个目标是为一种名为Born几何的新几何奠定一些基础，该几何最近被引入作为称为亚弦理论的对偶对称弦理论公式的几何背景。提案中考虑的复杂流形在数学和物理的许多领域中扮演着基本的角色。因此，这项研究的结果将提供许多明确的空间例子，这些例子将引起广泛的几何学家、物理学家、规范理论家和可积系统领域的研究人员的兴趣。

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.