Positivity and Convexity in Algebraic Geometry

代数几何中的正性和凸性

• 批准号: RGPIN-2015-04776
• 负责人: Smith, Gregory
• 金额: $ 1.24万
• 依托单位: Queen's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675846
• 关键词: Positivity; Convexity; Algebraic; Geometry

Although the origins of positivity and convexity are found in the natural total ordering on the real numbers, these basic structures emerge in several important and distinct ways within contemporary algebraic geometry. For instance, the theory of normal toric varieties over an algebraically closed field builds a robust dictionary between projective varieties and rational convex polytopes. In contrast, when working over a real closed field, the sections of a line bundle, whose evaluation at any real point is positive, form a convex cone that is rarely polyhedral. As a third example, Boij-Söderberg theory characterizes all of the cohomology groups for vector bundles on projective space via convex geometry. The broad aim of this research program is to understand the deep and subtle relations between these various incarnations of positivity and convexity. Because these fundamental problems connect many different areas of mathematics including algebraic geometry, commutative algebra, optimization, combinatorics, and convex geometry, advances will likely impact and influence a large community.****The long-term goals are to discover new frameworks for positivity inside algebraic geometry and to refine our understanding of specific convex cones appearing in algebraic geometry. The research will produce new mathematical results and new open-source software tools. In the short-term, we will concentrate on the following three problems: ***(a) Create a comprehensive dictionary between projectivized torus-equivariant vector bundles over a complete toric variety and appropriate collections of convex polytopes. ***(b) Given any nonnegative form f of degree 2d on a real projective subvariety, develop effective bounds on the integers e for which there exists a sum of squares g of forms degree e such that the product fg is a sum of squares of forms of degree 2(d+e). ***(c) Invent new homological mechanisms for representing coherent sheaves on toric varieties as short complexes of arithmetically-free vector bundles (also known as direct sums of line bundles).***The graduate students, postdoctoral fellows, and undergraduate students who contribute to this research program, will not only obtain valuable technical and scientific skills, but they will also become competent communicators.  With their training, they will be well-positioned for a variety of careers in the mathematical sciences. **

尽管正性和凸性的起源是在实数的自然全序中找到的，但这些基本结构在当代代数几何中以几种重要且独特的方式出现。 例如，代数闭域上的正态环面簇理论在射影簇和有理凸多面体之间建立了一个强大的字典。 相反，当在实闭场上工作时，线束的截面在任何实点处的评估都是正的，形成很少是多面体的凸锥体。 作为第三个例子，Boij-Söderberg 理论通过凸几何来表征射影空间上向量丛的所有上同调群。 该研究项目的总体目标是了解积极性和凸性的不同体现之间的深刻而微妙的关系。 因为这些基本问题连接着许多不同的数学领域，包括代数几何、交换代数、优化、组合学和凸几何，所以进步可能会影响一个大的社区。****长期目标是发现代数几何内部积极性的新框架，并完善我们对代数几何中出现的特定凸锥的理解。 该研究将产生新的数学结果和新的开源软件工具。 短期内，我们将集中精力解决以下三个问题： ***(a) 在完整的环面簇上的投影环面等变向量丛和适当的凸多面体集合之间创建一个综合字典。 ***(b) 给定实射影子簇上任何 2d 次非负形式 f，在整数 e 上建立有效界，其中存在 e 次形式的平方和 g，使得乘积 fg 是 2(d+e) 次形式的平方和。 ***(c) 发明新的同调机制，将环面簇上的连贯滑轮表示为无算术向量丛的短复合体（也称为线丛直和）。***为该研究项目做出贡献的研究生、博士后研究员和本科生不仅将获得宝贵的技术和科学技能，而且还将成为有能力的传播者。  通过培训，他们将为数学科学领域的各种职业做好准备。 **