稳定性条件、扭转理论和倾斜
Stability conditions, torsion theories and tilting
摘要：
三角化范畴上的稳定性条件空间自然地被划分为具有给定中心$A$的稳定性条件子集$U(A)$。如果$A$有有限长度，而$n$是简单对象，则$U(A)$有简单几何，只依赖于$n$。此外，Bridgeland已经表明，如果$B$是通过简单的倾斜从$A$获得的，即，通过倾斜于由一个简单对象产生的挠理论，则$U(A)$和$U(B)$的闭包的交具有余维数1。假设$A$，以及任何由一个（左或右）倾斜于简单对象的有限序列从它获得的心，具有有限的长度和200多个不可分解的对象。然后我们证明了$U(A)$和$U(B)$的闭包相交当且仅当$A$和$B$通过倾斜相关，并且相交的维数可以由挠理论确定。在这种情况下，子集$U(B)$的并集（其中$B$是通过简单倾斜的有限序列从$A$获得的）形成稳定性条件空间的分量。我们说明了这一点，通过计算（的一个组成部分）的空间的稳定性条件的可构造的导出范畴的复射影线分层的一个点和它的补充。
英文摘要：
The space of stability conditions on a triangulated category is naturally partitioned into subsets $U(A)$ of stability conditions with a given heart $A$. If $A$ has finite length and $n$ simple objects then $U(A)$ has a simple geometry, depending only on $n$. Furthermore, Bridgeland has shown that if $B$ is obtained from $A$ by a simple tilt, i.e.\ by tilting at a torsion theory generated by one simple object, then the intersection of the closures of $U(A)$ and $U(B)$ has codimension one. Suppose that $A$, and any heart obtained from it by a finite sequence of (left or right) tilts at simple objects, has finite length and finitely many indecomposable objects. Then we show that the closures of $U(A)$ and $U(B)$ intersect if and only if $A$ and $B$ are related by a tilt, and that the dimension of the intersection can be determined from the torsion theory. In this situation the union of subsets $U(B)$, where $B$ is obtained from $A$ by a finite sequence of simple tilts, forms a component of the space of stability conditions. We illustrate this by computing (a component of) the space of stability conditions on the constructible derived category of the complex projective line stratified by a point and its complement.
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