Derived Categories and Mirror Symmetry

镜像对称性起源于超共形场理论之间的对偶性。1991年，马克西姆·康采维奇将这个概念从物理学重新解释为一种极其深刻和普遍的数学二元性，现在被称为同源镜像对称性(HMS)。在他著名的菲尔兹奖章演讲中，他提出了一种基于派生范畴的新型几何，在数学界掀起了一股狂热的活动，导致了不同数学学科的显著协同：辛几何、代数几何和范畴理论。HMS现在是一个巨大的活跃数学研究领域的基石，该领域利用了弦理论中不断进步的最先进技术。 这项建议构成了一个研究代数几何中派生范畴的五年计划，更具体地说，研究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，我预计应用的范围从Hodge猜想(一个“千禧年问题”)的部分解到川端康成关于二元几何和派生范畴之间关系的深层猜想。我打算全面解决后一个问题。在这样做的过程中，我将得到包括Kontsevich本人在内的国际合作者网络的帮助。-- 这项建议的主要数学目标是： 1)通过分层来研究具有群作用的代数簇，目的是利用这些分层来扩展几何不变量理论与派生范畴之间的关系。 2)在派生范畴中统一、发展和扩展具有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、代数圈(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作用进行基础研究。另一方面，它在促进我所在领域的具体利益方面具有巨大的潜力。事实上，通过目标1，我设想了对派生范畴的半正交分解和等价的新理解，我预计这将重写和统一当前的文献。这在我以前关于派生范畴和几何不变量理论的工作中得到了高度的证明，这些工作恢复了派生范畴和代数几何中许多最令人兴奋的定理。 第二个目标向外聚焦于镜面对称和代数几何这一更大的领域。我以前的大部分工作都是为了这个目标。用规范群发展镜像对称结构对现有文献来说是自然的，也是非常重要的，因为它扩展了环面镜像对称的高度影响的作用。此外，更大的代数几何社区对经典代数几何不变量的应用非常感兴趣。

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.