Moduli spaces of sheaves on Hermitian manifolds

目标。本文的主要目的是研究紧（广义）复流形上稳定束和束的模空间的几何性质。复流形研究中的一个重要问题是流形上物体的分类问题，如全纯矢量束和矢量束。事实上，这些对象的模空间通常从基流形中继承有趣的结构，从而产生具有特定几何结构的新的高维空间示例。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束，在这种情况下，广义全纯束由一个全纯向量束的对和一个额外的算子组成。同希格斯束和全纯泊松模就是符合这种广义设定的例子，这里仅举两个例子。本研究还旨在通过检查广义全纯束的模空间的显式示例，并确定这些空间允许的几何结构类型，从而更好地理解广义全纯束的几何结构。这些例子将反过来对复杂流形上稳定束和束的模空间的几何结构产生新的见解。这项研究的第二个目标是为一种称为玻恩几何的新几何奠定一些基础，这是最近作为一种称为元弦理论的弦理论的对偶对称公式的几何背景引入的。******影响。本文所讨论的复流形在数学和物理的许多领域中起着重要的作用。因此，这项研究的结果将提供许多空间的明确例子，这些例子将引起广泛的几何学者、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.