Abelian Varieties, Asymptotic Invariants in Higher Dimensional Geometry, and Moduli of Vector Bundles

阿贝尔簇、高维几何中的渐近不变量以及向量丛的模

• 批准号: 0500985
• 负责人: Mihnea Popa
• 金额: --
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2005-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500985&HistoricalAwards=false
• 关键词: Abelian; Varieties; Asymptotic; Invariants; Higher

The investigator and his collaborator have established aconnection between the Castelnuovo Theory of subvarieties in projective space, andthe geometric Schottky problem, i.e. that of identifying Jacobians amongall principally polarized abelian varieties based on geometricproperties of the polarization. This includes in particular a relationship with the celebrated Trisecant Conjecture of Welters. They are nowapproaching directly the Trisecant Conjecture, using these connections, as well ashomological $M$-regularity methods. They are also interested in developing a "higherCastelnuovo-Schottky theory", based on deeper similaritiesbetween subvarieties in projective space and those of abelian varieties. In this respect, one of the main points they are interested in is a conjecture of Debarre on which are the subvarieties of principally polarized abelian varieties representing minimal classes.The Trisecant Conjecture, or the Strange Duality Conjecture, are among themost prominent conjectures in the respective directions (which can be called roughly speaking the abelian and non-abelian theory of theta functions). The proved statements, or even significant progress towards them, would have a large number of consequences, as documented in numerous places in the literature. Further development of the asymptotic methods has the potential to provide new insight into the geometry of every smooth projective variety, the main objects of study in algebraic geometry.All parts of the project would further the knowledge in the field,would have a broad range of applications, and willcreate interaction with people of different backgrounds.Many of the problems in this proposal are of interest to researchersin adjacent fields, like complex analysis (theta functions), complexanalytic geometry (transcendental methods in algebraic geometry)and conformal field theory (conformal blocks), and will lead tointeractions with some of them.

研究者和他的合作者已经建立了射影空间中子簇的Castelnuovo理论和几何Schottky问题之间的联系，即根据极化的几何性质识别Jacobian--所有主要是极化的阿贝尔簇。这尤其包括与著名的韦尔特三割猜想的关系。他们现在直接接近三割线猜想，使用这些联系，以及ashomological $M$-正则性的方法。他们也有兴趣发展一个“higherCastelnuovo-Schottky理论”，基于更深层次的相似性之间的子品种在射影空间和阿贝尔品种。在这方面，他们感兴趣的一个要点是一个猜想的Debarre上的子品种主要极化阿贝尔品种代表最小类。该Trisecant猜想，或奇怪的对偶猜想，是其中最突出的aesthetures在各自的方向（这可以被称为粗略地说阿贝尔和非阿贝尔理论的theta函数）。如文献中许多地方所记载的那样，经过证实的陈述，甚至是朝着这些陈述取得的重大进展，都会产生大量的后果。渐近方法的进一步发展有可能为每个光滑射影簇的几何提供新的见解，这是代数几何的主要研究对象。该项目的所有部分都将进一步丰富该领域的知识，将具有广泛的应用范围，并将与不同背景的人进行互动。该提案中的许多问题引起了邻近领域研究人员的兴趣，像复分析（theta函数），复解析几何（代数几何中的超越方法）和共形场论（共形块），并将导致与其中一些的相互作用。