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

基本信息

批准号：
1664296
负责人：
Izzet Coskun
金额：
$ 27.54万
依托单位：
University of Illinois at Chicago
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-08-01 至 2021-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1664296&HistoricalAwards=false
关键词：
FRG Collaborative Research Moduli Spaces

项目摘要

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.

从信用卡交易到向日葵的增长，生活中的许多过程都是由多项式方程式建模的。代数几何研究这种系统的解决方案。这些系统的主要特征是它们在家庭中通过改变多项式的系数而变化。家庭中的某些平衡更容易解决，并且可以从更简单的系统的解决方案中推导出更复杂的系统的属性。研究人员研究了由多项式方程定义的某些空间的几何形状，这些空间在数学和物理学中无处不在，称为载体束的模量空间。他们通过使用最近的称为Bridgeland稳定性的突破将这些空间的几何不变性与更简单的空间联系起来来计算这些空间的几何不变。研究人员还致力于培训下一代美国科学家和研究人员。在这个项目中，他们将培训本科，研究生和博士后研究人员使用新技术Bridgeland稳定性。专注的研究小组Grant将支持这些年轻研究人员与几位高级研究人员一起访问和合作，并参加有关该主题的会议和讲习班。调查人员还将组织两个大型会议和四个研讨会，以帮助吸引年轻的人才进入该地区。矢量束的模具空间是代数几何形状的基本对象，并应用于通勤代数，代表理论，组合术和数学物理学。在过去的五年中，布里奇兰的稳定条件彻底改变了对表面上矢量束的模量空间的理解。他们允许在这些模量空间上计算足够和有效的除数锥，并导致解决了长期存在的问题，例如在K3类型的某些Hyperkähler歧管上存在Lagrangian振动，以及在平面上普通吊杆的较高等级插值问题。将这些新技术应用于表面和三倍的矢量束模量空间的几何形状中的中心问题。该专注的研究小组项目集中在三条询问方面：（1）证明了使用Bridgeland稳定性证明结果消失的结果，因此在表面和三倍的表面上构建了Ulrich捆绑包，并在表面上的矢量挤出物的模量空间上构建了有效的Brill-Neether分隔线。（2）确定特殊捆绑包（例如Lazarsfeld-Mukai捆绑包或表面上的无效捆绑包和三倍）是Bridgeland稳定的。将稳定性应用于Syzygies和Koszul共同体的经典问题。（3）研究Bridgeland稳定物体模量空间的哲学，通过墙壁交叉。调查人员计划通过研究参与该项目来培训十名大学生，十名研究生和七个博士后协会。