Birational Geometry and Bridgeland Stability Conditions

派生范畴上的稳定性条件的概念产生于研究弦理论中狄里克莱膜的高能物理文献。除了最初的动机之外，稳定性条件理论还有更深一步的发展，与计数不变量、表示理论、同调镜像对称和经典代数几何有关。这个项目位于这些领域的结合点；主题是应用布里奇兰稳定性条件技术来解决由二次几何和数学物理引起的问题。稳定性条件以内在的方式标识派生类别中的一类对象，称为稳定对象。了解某些物体是否稳定可以提供有关代数簇几何的重要信息。稳定性条件取决于某些参数的选择。了解稳定的物体如何通过改变这些参数而变化是该理论许多应用的关键，也是该项目的技术核心。该项目有三个主要目标。首先，研究了Bridgeland稳定性条件的存在性问题。研究人员和合作者通过某些猜想不等式，提出了构造三重Bridgeland稳定性条件的一般框架。新的观点是观察特定乘积中的纤颤，并在纤颤的基础上从某一除数类的正性推导出这样的不等式。第二个目标是研究非交换K3曲面上的稳定性条件，并利用Bridgeland稳定对象的模空间来重新解释和推广几何结构，例如三次四折曲面。第三个目标是研究这种模空间是否总体上是“行为良好”的射影簇，并在K3曲面的情况下研究它们的局部结构。应用包括研究关于二元不可约全纯辛簇的派生等价的一个猜想。

The notion of stability conditions on derived categories emerged from high energy physics literature in the study of Dirichlet branes in string theory. Besides its original motivation, the theory of stability conditions has a much further reach, with connections to counting invariants, representation theory, homological mirror symmetry, and classical algebraic geometry. This project lies at the nexus of these areas; the theme is to apply Bridgeland stability condition techniques to solve questions arising from birational geometry and mathematical physics. A stability condition identifies a class of objects in the derived category, called stable objects, in an intrinsic way. Understanding if certain objects are stable or not gives important information on the geometry of algebraic varieties. A stability condition depends on the choice of certain parameters. Understanding how stable objects change by varying these parameters is the key to many applications of the theory and is the technical core of this project.There are three main goals in the project. The first is to investigate the existence problem for Bridgeland stability conditions. A general framework for constructing Bridgeland stability conditions on threefolds was introduced by the investigator and collaborators, via certain conjectural inequalities. The new viewpoint is to look at fibrations in particular products, and to deduce such inequalities from the positivity of a certain divisor class on the base of the fibration. The second goal is to study stability conditions on non-commutative K3 surfaces and to use Bridgeland's theory to reinterpret and generalize geometric constructions, for example for cubic fourfolds, by using moduli spaces of Bridgeland stable objects. The third goal is to investigate whether such moduli spaces are "well-behaved" projective varieties in general and to study their local structure in the case of K3 surfaces. Applications include investigation of a conjecture on derived equivalences of birational irreducible holomorphic symplectic varieties.