Stable Rank-2 Bundles on Calabi-yau Manifolds

Recently there is a surge of research interest in the construction of stable vector bundles on Calabi-Yau manifolds motivated by questions from string theory. An interesting aspect of the moduli spaces of stable sheaves on Calabi-Yau manifolds is their relation to the higher dimensional gauge theory studied by Donaldson, R. Thomas and Tian et al. [D-T, Tho, Tia]. A holomorphic Casson invariant for Calabi-Yau three-fold has been defined in [D-T, Tho]. This invariant requires the full description of the moduli spaces. Comparing with the work of stable sheaves on surfaces, our knowledge about stable sheaves on Calabi-Yau manifolds is very limited. Examples of full moduli spaces are scarce. However, some of sucsessful methods for stable bundles on surfaces have been extended to CalabiYau manifolds, for instance, the work of Friedman, Morgan and Witten [FMW] on elliptic fibrations, the work of Bridgeland and Maciocia [B-M] on Fourier-Mukai transform for elliptic and K3-fibrations, and the work of Thomas [Tho] where full examples were computed by using K3-fibrations and the Serre construction, among others. The technique of chamber structures has been used quite often in the theory of stable vector bundles on surfaces. In this paper, we make use of chamber structures to construct full moduli spaces of stable sheaves with certain Chern classes on some Calabi-Yau manifolds. As an application, we will compute some holomorphic Casson invariants. To introduce our results, we recall the definitions of stabilities. Let L be an ample line bundle on a smooth projective variety X of dimension n, and V be a rank-r torsion-free sheaf on X. We say that V is (slope) L-stable if c1(F ) · c1(L) rank(F ) < c1(V ) · c1(L) r