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

基于稳定轮模表示论的Calabi-Yau流形不变量研究

• 批准号: 15540039
• 负责人: NAKASHIMA Tohru
• 金额: $ 1.6万
• 依托单位: Tokyo Metropolitan University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540039/
• 关键词: Calabi-Yau manifolds; Stable sheaves; Moduli spaces; 安定ベクトル束

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

本课题的目的是从稳定层的节距表示理论的角度来阐明Calabi-Yau流形的各种不变量之间的关系。虽然关于表示理论的部分还没有建立，但我们已经发现了不变量之间的重要关系，如下所述：首先，我们已经找到了构造一般Calabi-Yau流形上不一定是椭圆的稳定丛的方法。该方法是利用初等变换构造稳定丛，利用零阶凝聚层和零阶凝聚层在某些因子上的支集来构造稳定丛，从而揭示了BPS不变量与全纯Casson不变量之间的一种前人没有研究过的关系。利用我们的方法，我们可以给出许多Calabi-Yau流形上稳定丛的显式例子，这些例子以前没有用Friedman-Morga…的谱覆盖方法处理过其次，证明了Calabi-Yau流形上稳定的秩2丛的模同构于射影空间，它是与曲线上的向量丛相关的射影丛的完全交。特别地，当基流形的维度大于2时，模空间不一定具有辛结构。我们将这一结果应用于全纯Casson不变量的计算。最后，我们推广了已建立在Calabi-Yau流形上的Brill-Noether理论。更具体地说，我们将Brill-Noether对偶推广到任意非奇异射影流形上，并进一步给出了Brill-Noether轨迹是模空间的开子集的一个判据。借助于这些结果，我们可以澄清模空间的双调几何。作为应用，我们证明了具有椭圆纤维的Calabi-Yau三重流形上稳定层的模与基曲面0维格式的Hilbert格式上的投影丛是双调的，这表明了Gromov-Witten不变量与全纯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稳定层。较少

项目成果

A construction of stable vector bundles on Calabi-Yau manifolds

Calabi-Yau流形上稳定向量丛的构造

• 发表时间: 2004
• 期刊: Journal of Geometry and Physics 49
• 作者: T.Nakashima;T.Nakashima;M.A.Guest;T.Nakashima
• 通讯作者: T.Nakashima

T.Nakashima: "Moduli spaces of stable bundles on K3 fibered Calabi-Yau threefolds"Communications in Contemporary Mathematics. 5. 119-126 (2003)

T.Nakashima：“K3 纤维 Calabi-Yau 三重上稳定丛的模空间”当代数学通讯。

Codes on Grassmann bundles and related varieties

格拉斯曼束和相关品种的代码

• 发表时间: 2005
• 期刊: Journal of Pure and Applied Algebra 199
• 作者: T.Nakashima;T.Nakashima
• 通讯作者: T.Nakashima

A construction of stable vectorbundles on Calabi-Yau manifolds

Calabi-Yau流形上稳定向量丛的构造

• 发表时间: 2004
• 期刊: Journal of Geometry and Physics 49
• 作者: T.Nakashima;T.Nakashima;M.A.Guest;T.Nakashima;T.Nakashima;T.Nakashima;T.Nakashima
• 通讯作者: T.Nakashima

Complexes of exact hermitian cubes and the Zagier Conjecture

精确厄密立方体的复形和扎吉尔猜想

• 发表时间: 2004
• 期刊: Mathematische Annalen 328
• 作者: T.Nakashima;T.Nakashima;M.A.Guest;T.Nakashima;T.Nakashima;T.Nakashima;T.Nakashima;T.Nakashima;Y.Takeda
• 通讯作者: Y.Takeda