Geometry and Topology of Holomorphic Symplectic Varieties

代数几何是一门研究代数变量的学科，代数变量是由多项式方程的解所描述的空间。这门学科始于古希腊人关于椭圆、抛物线和双曲线的工作，现在它已经发展成为一门现代学科，与物理学和计算机科学有着许多丰富的联系。这项由国家科学基金会资助的研究将集中在一类重要的代数变异上，称为全纯辛变异，它处于几何、表示理论和数学物理的十字路口。更好地理解这些变化将带来对代数几何及其相关领域的深刻见解。在这个项目中，PI将建立紧致和非紧致全纯辛变之间的联系，重点关注P=W猜想，这是一个连接Hitchin的可积系统和Hodge理论的中心猜想。紧致全纯辛变异体是K3曲面的高维推广，已经被广泛研究了几十年。系统地比较紧致与非紧致几何，并从紧致全纯辛变中引入新技术，将为P=W猜想及相关问题带来新的视角和丰硕的成果。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Algebraic geometry is a subject studying algebraic varieties, which are spaces described by solutions of polynomial equations. This subject started with the work of the ancient Greeks about ellipses, parabolas, and hyperbolas, and now it has grown to be a modern subject sharing many rich connections with physics and computer science. The research supported by this NSF award will focus on an important class of algebraic varieties, called the holomorphic symplectic varieties, that lie at the crossroads of geometry, representation theory, and mathematical physics. A better understanding of these varieties will bring deep insights in both algebraic geometry and its related fields.In this project, the PI will build connections between compact and noncompact holomorphic symplectic varieties with a focus on the P=W conjecture, which is a central conjecture connecting Hitchin's integrable systems and Hodge theory. Compact holomorphic symplectic varieties, which are higher dimensional generalizations of K3 surfaces, have been studied intensively for decades. A systematic way to compare compact and noncompact geometries, and to introduce new techniques to Hitchin systems from compact holomorphic symplectic varieties will bring fresh perspectives and fruitful outcomes to the P=W conjecture and related questions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.