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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=613157
• 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.

尽管在实际数字上的自然总顺序中发现了积极性和凸度的起源，但这些基本结构以当代代数几何形状以几种重要和不同的方式出现。例如，在代数封闭场上的正常曲曲面理论建立了投射品种和理性凸多属性之间的强大词典。相反，当在实际闭合场上工作时，线束的部分（在任何实际点的评估都是正面的），形成很少是多面体的凸锥。作为第三个例子，Boij -Söderberg理论特征通过凸线上的矢量束的所有共同体组组。该研究计划的广泛目的是了解这些积极和凸性的各种化身之间的深厚和微妙的关系。因为这些基本问题将数学的许多不同领域连接起来，包括代数几何，交换代数，优化，组合和凸几何形状，因此进步可能会影响和影响大型社区。 长期目标是发现代数几何内部积极的新框架，并完善我们对代数几何形状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。在短期内，我们将专注于以下三个问题： （a）在完整的感谢您的多种多样的凸层和适当的凸层杂音集合之间创建一个综合词典。 （b）如果在真实的投射亚变量上有任何非负数f度2d的非负形式，则在整数e上发展有效的界限。 （c）发明新的同源机制，用于代表托里克变化上的相干滑轮，作为算术载体束的短复合物（也称为直接束束总和）。 为这项研究计划做出贡献的研究生，博士后研究员和本科生，不仅将获得有价值的技术和科学技能，而且还将成为合格的沟通者。通过他们的培训，他们将在数学科学中的各种职业中得到充分的态度。