Moduli spaces of sheaves on Hermitian manifolds

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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.