Comprehensive Study of Stable Bundles on Calabi-Yau Manifolds

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In this research project we planned to solve the existence problem of stable bundles on Calabi-Yau manifolds and to clarify the geometric structure of their moduli spaces. We obtained the following results concerning these subjects.As to the existence problem, we proved that the extension sheaf of two stable sheaves is again stable, under certain minimality assumption on their first Chern class. As a result, one may construct stable sheaves inductively from sheaves of lower rank. In particular, the method yields stable bundles by means of elementary transformation from globally generated line bundles on a divisor. Until recently, methods of explicit construction of stable bundles has been known only for elliptic Calabi-Yau manifolds, while our work enables us to find stable bundles on arbitrary Calabi-Yau manifolds in principle.Concerning the geometry of moduli space, we determined their birational structure in many cases. More explicity, we proved that the reflection functor defines an isomorphism between the Brill-Noether loci of the moduli spaces with different Mukai vectors (the Brill-Noether duality), which is a higher dimensional generalization of the result due to Markman-Yoshioka in the case of K3 surface. Exploiting the Brill-Noether duality, one deduces that moduli spaces have a component which is birational to the Grassmann bundle over moduli space of sheaves of lower rank. Our method applies to other varieties which are not necessarily Calabi-Yau. For example, we determined the birational structure of the moduli space of stable sheaves on certain threefolds with Del-Pezzo fibrations.