ABELIAN VARIETIES, MODULI OF VECTOR BUNDLES AND MODULI OF COURVES

阿贝尔簇、向量丛模和曲线模

• 批准号: 0200150
• 负责人: Mihnea Popa
• 金额: $ 11.06万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-08-01 至 2005-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0200150&HistoricalAwards=false
• 关键词: ABELIAN; VARIETIES; MODULI; VECTOR; BUNDLES

In work with G. Pareschi, Popa has introduced a notion of Mukai regularity for coherent sheaves on abelian varieties, defined using the Fourier-Mukai transform, and developed its theory. Popa and Pareschi are now interested in using the concept of Mukai regularity in the study of irregular varieties, especially to give effective results on adjoint linear series and pluiricanonical maps. Also, Popa and Pareschi intend to use invariants arising naturally from this theory in order to differentiate between Jacobians and other abelian varieties, and thus approach the Schottky problem in a new way. Using some of their results that carry through in the more general context of an arbitrary Fourier transform, Popa and Pareschi intend to obtain constraints on the existence of equivalences of derived categories, which is a main concern in this line of approach to mirror symmetry. In other work, Popa has established effective results for linear series on moduli spaces of vector bundles on curves. Popa is interested in approaching the Strange Duality by refining his methods based on the Fourier-Mukai transform of Verlinde bundles on Jacobians and combining it with representation theory techniques. Popa also intends to approach some optimal conjectures on effective base point freeness on these moduli spaces for general curves, using degeneration to stable curves. In work with G. Farkas, Popa has obtained Brill-Noether type non-existence results for rank 2 vector bundles on general curves, via generalized limit linear series and stability of pairs on reducible curves. Popa and Farkas intend to further refine these techniques in order to prove a conjecture of Bertram-Feinberger-Mukai on Brill-Noether theory for rank 2 vector bundles with canonical determinant, and to find non-existence results in higher ranks. This will allow them to define new divisors in moduli spaces of stable curves for appropriate genera. The ultimate goal of this project is to compute the class of these divisors and determine whether they provide counterexamples to the Harris-Morrison Slope Conjecture as expected. Algebraic curves and abelian varieties are ubiquitous objects in mathematics. Apart from their realizations in algebraic geometry, they appear in complex analysis (as Riemann surfaces or complex tori) or algebra (as field extensions or group schemes), and play a fundamental role in recent progress in mathematical physics. One of the most important problems in algebraic geometry is to classify algebraic varieties. For one-dimensional varieties (that is, for algebraic curves) this problem is approached by understanding the geometry of a space M(g) parametrizing all curves of given genus. In the case of abelian varieties, one way to approach this is to understand the totality of a specific kind of algebraic objects (called coherent sheaves) which can be associated to them. The investigator is pursuing these directions in research related to this proposal.

在与G。Pareschi，Popa为阿贝尔簇上的相干层引入了Mukai正则性的概念，使用Fourier-Mukai变换定义，并发展了其理论。 Popa和Pareschi现在有兴趣使用Mukai正则性的概念来研究不规则簇，特别是在伴随线性序列和pluiricanonical映射上给出有效的结果。 此外，波帕和Pareschi打算使用自然产生的不变量从这个理论，以区分雅可比和其他阿贝尔品种，从而接近肖特基问题的一种新的方式。 波帕和帕雷斯基利用他们在任意傅立叶变换的更一般的上下文中所得到的一些结果，试图获得对导出范畴的等价性存在的限制，这是镜像对称性研究中的一个主要问题。 在其他工作中，波帕建立了有效的结果线性系列模空间的向量丛曲线。 波帕有兴趣接近奇怪的对偶性，通过改进他的方法的基础上傅立叶-Mukai变换的Verlinde束的雅可比矩阵，并结合它与表示论技术。 Popa还打算利用退化到稳定曲线的方法，在这些模空间上对一般曲线的有效基点自由度进行一些优化。 在与G。Farkas，Popa利用广义极限线性级数和可约曲线上对的稳定性，得到了一般曲线上秩为2的向量丛的Brill-Noether型不存在性结果。 Popa和Farkas打算进一步改进这些技巧，以证明Bertram-Feinberger-Mukai关于Brill-Noether理论的一个猜想，其中秩为2的向量丛具有标准行列式，并在更高的秩中找到不存在的结果。 这将使他们能够定义新的因子在模空间的稳定曲线适当的一般。 这个项目的最终目标是计算这些因子的类，并确定它们是否像预期的那样为Harris-Morrison斜率猜想提供反例。代数曲线和阿贝尔簇是数学中普遍存在的对象。除了它们在代数几何中的实现，它们还出现在复分析（如黎曼曲面或复环面）或代数（如场扩张或群方案）中，并在数学物理的最新进展中发挥着重要作用。代数簇的分类是代数几何中的一个重要问题。对于一维簇（即代数曲线），这个问题是通过理解空间M（g）的几何来解决的，参数化给定亏格的所有曲线。在阿贝尔簇的情况下，一种接近这一点的方法是理解可以与它们相关联的特定类型的代数对象（称为凝聚层）的总体。调查员正在与本提案相关的研究中遵循这些方向。