AI 中文总结
该研究通过双模粘合阿贝尔范畴，证明粘合半稳定对象存在相位间隙时的支撑性质，结合Collins--Polishchuk相位界，还在dg-逗号范畴构造稳定性条件并分析其在半正交分解突变下的行为。
AI 中文摘要
我们研究通过沿半正交分解粘合得到的稳定性条件的支撑性质。我们的主要工具是一种通过双模沿阿贝尔范畴的新粘合构造：在自然正合性假设下，粘合范畴是阿贝尔的，且它恢复了通过粘合t-结构得到的心脏。这给出了粘合心脏中对象的具体描述，我们利用该描述证明了在粘合半稳定对象存在相位间隙时的支撑性质。特别地，我们在假设Collins--Polishchuk相位界的情况下证明了支撑性质。作为应用，我们在dg-逗号范畴上构造稳定性条件，包括与Alexeev和Kuznetsov意义下的增广曲线相关的例子，并分析该构造在半正交分解突变下的行为。
英文摘要
We study the support property for stability conditions obtained by gluing along semiorthogonal decompositions. Our main tool is a new gluing construction for abelian categories along a bimodule: under natural exactness assumptions the glued category is abelian, and it recovers the heart obtained by gluing $t$-structures. This gives a concrete description of objects in the glued heart which we use to prove the support property given a phase gap for glued semistable objects. In particular, we prove the support property assuming Collins--Polishchuk phase bounds. As applications, we construct stability conditions on dg-comma categories, including examples related to augmented curves in the sense of Alexeev and Kuznetsov, and analyze the behavior of the construction under mutations of semiorthogonal decompositions.
