Derived Categories and Mirror Symmetry

镜像对称最初是超共形场论之间的对偶。1991年，马克西姆·康茨维奇（Maxim Kontsevich）从物理学中将这一概念重新解释为一种难以置信的深刻和无处不在的数学对偶，现在被称为同调镜像对称（HMS）。在他著名的菲尔兹奖演讲中，他提出了一种基于派生范畴的新型几何，在数学界引起了狂热的活动，导致了不同数学学科的显著协同：辛几何、代数几何和范畴论。HMS现在是一个巨大的积极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。***这一提议构成了一个五年计划，研究代数几何中的派生范畴，更具体地说，研究它们与弦理论和镜像对称结构的关系。基于我以前的工作，我期望应用范围从霍奇猜想（一个“千年问题”）的部分解到关于两族几何和派生范畴之间关系的Kawamata的深度猜想。我打算彻底解决后一个问题。在这样做的过程中，我将得到包括康采维奇本人在内的国际合作者网络的帮助。***本提案的中心数学目标是：***1)通过分层研究具有群作用的代数变异，目的是利用这些分层来扩展几何不变理论与派生范畴之间的关系。***2)在派生范畴的集合中统一、发展和扩展规范群的镜像对称结构，并将上述理论应用于经典的代数-几何不变量：Noether-Lefschetz轨迹、代数循环（和Hodge猜想）、Griffiths群。第一个目标是在许多方面对代数几何中的群行为进行基础研究。另一方面，它有巨大的潜力来促进我所在领域的特定兴趣。事实上，通过目标1，我设想了对派生类别的半正交分解和等价的新理解，我希望重写和统一当前的文献。我以前在派生范畴和几何不变理论方面的工作充分证明了这一点，它恢复了派生范畴和代数几何中许多最激动人心的定理。***第二个目标是向外聚焦于镜像对称和代数几何的更大领域。我以前的大部分工作都是为了实现这一目标。发展规范群的镜像对称结构对现有文献来说是很自然的，而且非常重要，因为它扩展了环面镜像对称的高度影响作用。此外，经典代数几何不变量的应用也引起了代数几何学界的极大兴趣。*** ***

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. *** *** *** *** **