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.

研究人员和他的合作者已经在射影空间中的卡斯德尔诺沃子簇理论和几何肖特基问题之间建立了联系，即根据极化的几何性质在所有主要极化的阿贝尔簇中识别雅可比行列式。这尤其包括与著名的韦尔特斯三割线猜想的关系。他们现在正在利用这些联系以及同调 $M$ 正则性方法直接逼近三割猜想。他们还有兴趣开发一种“更高的卡斯特努沃-肖特基理论”，该理论基于射影空间中的子簇与阿贝尔簇的子簇之间更深层次的相似性。在这方面，他们感兴趣的要点之一是德巴尔猜想，该猜想是代表最小类的主要极化的阿贝尔簇的子变体。三割猜想或奇怪的对偶猜想是各个方向上最突出的猜想之一（粗略地可以称为theta函数的阿贝尔和非阿贝尔理论）。正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加深该领域的知识，将具有广泛的应用，并将与不同背景的人们进行互动。该提案中的许多问题都是相邻领域的研究人员感兴趣的，例如复分析（theta函数）、复分析 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。