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.

研究者及其合作者在投射空间中的Castelnuovo次视角理论与几何肖特基问题之间建立了障碍，即基于主要极化的阿贝尔品种中基于极化的几何形式的主要极化的阿布尔品种中的雅各布人的几何问题。这尤其包括与著名的三角星猜想的关系。他们现在使用这些连接直接对三角构想直接进行了猜想，以及ashomological $ m $ regularity方法。他们还有兴趣开发一个“高级库诺沃诺沃 - 索特基理论”，基于投影空间和阿贝尔品种的亚地区之间的更深层次的相似之处。 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).正如文献中许多地方所记录的那样，证明的陈述甚至取得了重大进展，都会产生大量后果。渐近方法的进一步发展有可能提供对每种光滑投射品种的几何形状，代数几何学研究的主要对象的新见解。该项目的所有部分都将进一步进一步了解该领域的知识，并具有广泛的应用程序，并将与此求发的人相互分析，而与研究相互分析的相互作用。几何（代数几何学中的先验方法）和保形场理论（保形块），并将导致与其中一些的互相互动。