Study of Invariants of Calabi-Yau manifolds via Representation Theory of The Moduli of Stable Sheaves

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The aim of the present project was to clarify the relations between various invariants of Calabi-Yau manifolds from the representation theoretic point of view of the noduli pace of stable sheaves on them. Although the part concerning representation theory has not been established yet, we have found important relations between the invariants, as exlained below.First, we have foud a method for constructing stable bundles on general Calabi-Yau manifolds which are not necessarily elliptic. The method consisits in constructing stable bundles by means of elementary transformations, which utilizes extensions of tivial sheaves and coherent sheaves of rank 0 with support on some divisors, hence reveals a relation between BPS invariants and holomorphic Casson invariants which has not been explored before. Using our method, we could give explicit examples of stable bundles on many Calabi-Yau manifolds, which have not been treated before by the usual method of spectral covers due to Friedman-Morga … More n-Witten.Secondly, we have shown that the moduli of stable rank-2 bundles on Calabi-Yau manifolds, which are complete intersections in the projective bundles associated to vector bundles on curves, are isomorphic to projective spaces. In particular, it follows that when the base manifolds have dimension larger than two, the moduli spaces do not necessarily admit symplectic structures. We applied this result to the computation of holomorphic Casson invariants. We expect that the method utilized in the proof of this result, which consists in representing stable bundles as extensions of two line bundles, admit applications to other varieties.Finally, we have generalized the Brill-Noether theory which have been already established for Calabi-Yau manifolds.. More concretely, wehave generalized the Brill-Noether duality to arbitrary nonsingular projective manifolds and further given a criterion for Brill-Noether loci to be open subsets of the moduli space. By means of these results, we could clarify the birational geometry of the moduli spaces. As an application, we have proven that the moduli of stable sheaves on Calabi-Yau threefolds with elliptic fibraitions are birational to the projective bundles on Hilbert schemes of 0-dimensional bschemes of the base surface, which suggests another relation between Gromov-Witten invariants and holomorpic Casson invariants, since our result shows that elliptic curves on Calabi-Yau manifolds give rise to stable sheaves on them. Less