FRG: Collaborative Research: Moduli Spaces, Birational Geometry, and Stability Conditions

FRG：协作研究：模空间、双有理几何和稳定性条件

• 批准号: 1664215
• 负责人: Alina Marian
• 金额: $ 15.58万
• 依托单位: Northeastern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-08-01 至 2023-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1664215&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Moduli; Spaces

Many processes in life, ranging from credit card transactions to the growth of a sunflower, are modeled by systems of polynomial equations. Algebraic geometry studies solutions of such systems. A major feature of these systems is that they vary in families by varying the coefficients of the polynomials. Some equations in the family are easier to solve, and properties of more complicated systems can be deduced from the solutions of the simpler systems. The investigators study the geometry of certain spaces defined by polynomial equations that are ubiquitous in mathematics and physics, called moduli spaces of vector bundles. They compute geometric invariants of these spaces by relating them to simpler spaces using a recent breakthrough called Bridgeland stability. The investigators are also dedicated to training the next generation of U.S. scientists and researchers. In this project, they will train undergraduate, graduate, and postdoctoral researchers to use the new technique of Bridgeland stability. The Focused Research Group grant will support these young researchers to visit and collaborate with several senior researchers and to attend conferences and workshops on the topic. The investigators will also organize two large conferences and four workshops to help attract young talent to the area.Moduli spaces of vector bundles are fundamental objects in algebraic geometry, with applications to commutative algebra, representation theory, combinatorics, and mathematical physics. In the last five years, Bridgeland stability conditions have revolutionized the understanding of moduli spaces of vector bundles on surfaces. They have allowed the computation of the ample and effective cones of divisors on these moduli spaces and led to the solution of longstanding problems such as the existence of Lagrangian fibrations on certain hyperkähler manifolds of K3 type and the higher rank interpolation problem for general sheaves on the plane. It is timely to apply these new techniques to central problems in the geometry of moduli spaces of vector bundles on surfaces and threefolds. This Focused Research Group project centers on three lines of inquiry:(1) Prove cohomology vanishing results using Bridgeland stability and consequently construct Ulrich bundles on surfaces and threefolds and effective Brill-Noether divisors on moduli spaces of vector bundles on surfaces. Give applications to Le Potier's Strange Duality Conjecture.(2) Determine when special bundles, such as Lazarsfeld-Mukai bundles or null-correlation bundles on surfaces and threefolds, are Bridgeland stable. Apply the stability to classical problems on syzygies and Koszul cohomology.(3) Study the birational geometry of moduli spaces of Bridgeland stable objects via wall-crossing. The investigators plan to train ten undergraduates, ten graduate students, and seven postdoctoral associates through research involvement in the project.

生活中的许多过程，从信用卡交易到向日葵的生长，都是通过多项式方程组来建模的，这些系统的一个主要特征是它们通过改变系数而在族中变化。多项式族中的一些方程更容易求解，并且可以从更简单系统的解中推导出更复杂系统的性质。研究人员研究由普遍存在的多项式方程定义的某些空间的几何形状。他们利用最近的一项名为布里奇兰稳定性的突破来计算这些空间的几何不变量，称为数学和物理学的模空间。在这个项目中，他们将培训本科生、研究生和博士后研究人员使用布里奇兰稳定性新技术，重点研究小组拨款将支持这些年轻研究人员访问几位高级研究人员并与之合作，并参加有关该主题的会议和研讨会。调查人员还将组织两大型会议和四个研讨会，以帮助吸引年轻人才来到该领域。矢量丛的模空间是代数几何中的基本对象，应用于交换代数、表示论、组合学和数学物理。在过去的五年里，布里奇兰稳定性条件。彻底改变了对曲面上向量丛模空间的理解，它们允许计算这些模空间上充足且有效的除数锥，并解决了长期存在的问题，例如某些 K3 型超卡勒流形上的拉格朗日振动以及平面上一般滑轮的高阶插值问题 现在正是将这些新技术应用于曲面和三重向量束模空间几何的中心问题的时候。小组项目集中于三个研究方向：（1）使用布里奇兰稳定性证明上同调消失结果，从而在曲面和三重上构造乌尔里希丛，并有效的布里尔-诺特曲面上向量丛模空间上的除数。 (2) 确定曲面和三重上的特殊丛（例如 Lazarsfeld-Mukai 丛或零相关丛）何时是布里奇兰稳定的。 syzygies和Koszul上同调的经典问题。(3)通过穿墙研究Bridgeland稳定物体模空间的双有理几何。计划通过该项目的研究参与培养十名本科生、十名研究生和七名博士后。