Derived Categories and Mirror Symmetry

镜像对称开始于超共形场论之间的对偶。1991年，马克西姆·孔采维奇（Maxim Kontsevich）将这个概念从物理学中重新解释为一种令人难以置信的深刻和普遍存在的数学对偶性，现在被称为同调镜像对称（HMS）。在他著名的菲尔兹奖的地址，他提出了一种新的几何类型的基础上派生类别，创造了狂热的活动，在数学界导致了显着的协同作用，不同的数学学科：辛几何，代数几何和范畴理论。HMS现在是一个巨大的活跃的数学研究领域的基石，它利用了弦理论中不断进步的最先进的技术。 该提案构成了一项为期5年的计划，旨在研究代数几何中的衍生范畴，更具体地说，它们与弦理论和镜像对称中的构造的关系。 基于我以前的工作，我希望应用范围从霍奇猜想（“千年问题”）的部分解决方案，以深川俣双有理几何和派生类别之间的关系的猜想。 我打算彻底解决后一个问题。 在此过程中，我将得到包括孔采维奇本人在内的国际合作者网络的帮助。 该建议的主要数学目标是： 1)通过分层研究具有群作用的代数簇，目的是使用这些分层来扩展几何不变理论和派生类别之间的关系。 2)统一、发展和扩展镜像对称结构与规范群在设定的衍生类别和应用上述理论对经典代数几何不变量：诺特-莱夫谢茨轨迹，代数循环（和霍奇猜想），格里菲斯群。 第一个目标是在许多方面的基本研究群作用的代数几何。 另一方面，它有巨大的潜力，促进我的领域的具体利益。 事实上，通过目标1，我设想一个新的理解半正交分解和等价的派生类别，我希望重写和统一目前的文献。 这是高度证明了我以前的工作派生类别和几何不变的理论恢复了许多最令人兴奋的定理派生类别和代数几何。 第二个目标有一个外在的重点，对镜像对称和代数几何更大的领域。 我以前的大部分工作都是朝着这个目标发展的。发展规范群的镜像对称结构对现有文献来说是很自然的，并且非常重要，因为它扩展了复曲面镜像对称的高度影响力。 此外，对经典的代数几何不变量的应用是极大的兴趣，以更大的代数几何社区。

Mirror symmetry began as a duality between superconformal field theories. In 1991, Maxim Kontsevich re-interpreted this concept from physics as an incredibly deep and ubiquitous mathematical duality now known as homological mirror symmetry (HMS). In his famous Fields Medal address, he proposed a new type of geometry based on derived categories, creating a frenzy of activity in the mathematical community which lead to a remarkable synergy of diverse mathematical disciplines: symplectic geometry, algebraic geometry, and category theory. HMS is now the cornerstone of an immense field of active mathematical research which utilizes ever-advancing state-of-the-art techniques from string theory. This proposal constitutes a 5-year plan to study derived categories in algebraic geometry and more specifically, their relationship to constructions in string theory and mirror symmetry.  Based on my previous work, I expect applications to range from partial solutions of the Hodge Conjecture (a "Millennium Problem")  to a deep conjecture of Kawamata on the relationship between birational geometry and derived categories.  I intend to fully solve the latter problem.  In doing so, I will be aided by an international network of collaborators which includes Kontsevich himself.   The central mathematical objectives of this proposal are: 1) Study algebraic varieties with group actions via stratifications with the intent to use these stratifications to extend the relationship between geometric invariant theory and derived categories. 2) Unify, develop, and expand mirror symmetry constructions with gauge groups in the setting of derived categories and apply the theory above towards classical algebro-geometric invariants: Noether-Lefschetz loci, algebraic cycles (and the Hodge conjecture), Griffiths groups. The first objective is in many ways a fundamental study of group actions in algebraic geometry.  On the other hand, it has enormous potential for furthering the specific interests of my field.  Indeed, through Objective 1, I envision a new understanding of semi-orthogonal decompositions and equivalences of derived categories which I expect to re-write and unify the current literature.  This is highly evidenced by my previous work on derived categories and geometric invariant theory which recovers many of the most exciting theorems in derived categories and algebraic geometry. The second objective has an outward focus on the greater field of mirror symmetry and algebraic geometry.  Much of my previous work has been building towards this objective. Developing mirror symmetry constructions with gauge groups is natural to the existing literature and is of great importance, as it expands on the highly influential role of toric mirror symmetry.  In addition, applications towards classical algebro-geometric invariants are of great interest to the larger algebraic geometry community.