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

本项目的目的是澄清各种不变量之间的关系的Calabi-Yau流形从表示论的角度来看，节空间的稳定层。虽然关于表示论的部分还没有建立起来，但我们已经发现了不变量之间的重要关系，如下所示：首先，我们找到了一般Calabi-Yau流形上不一定是椭圆形的稳定丛的构造方法。该方法是通过初等变换构造稳定丛，利用秩为0的序层和凝聚层在某些因子上的扩张，从而揭示了BPS不变量与全纯Casson不变量之间的关系.利用我们的方法，我们可以给出许多Calabi-Yau流形上的稳定丛的显式例子，这些例子以前没有用Friedman-Morga的谱覆盖方法处理过 ...更多信息 n-Witten.其次，我们证明了Calabi-Yau流形上的稳定秩-2丛的模同构于射影空间，这些稳定秩-2丛是与曲线上向量丛相关联的射影丛的完全交.特别是，当基流形的维度大于二时，模空间不一定允许辛结构。我们将这个结果应用于全纯Casson不变量的计算。我们希望，在证明这一结果，其中包括在表示稳定的两个线丛的扩展，承认其他品种的应用程序中使用的方法。最后，我们推广了Brill-Noether理论已经建立的Calabi-Yau流形。更具体地说，我们将Brill-Noether对偶推广到任意非奇异射影流形，并进一步给出Brill-Noether轨迹是模空间开子集的一个判据。利用这些结果，我们可以阐明模空间的双有理几何。作为应用，我们证明了具有椭圆弯曲的Calabi-Yau三重流形上的稳定层的模对于基曲面的0维b格式的Hilbert格式上的投射丛是双有理的，这表明了Gromov-Witten不变量与全态Casson不变量之间的另一种关系，因为我们的结果表明Calabi-Yau流形上的椭圆曲线产生其上的稳定层.少

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