Stability condition

Remarks

Example: let $k$ be a field and let $\mathcal{D}$ be the bounded derived category of finite-dimensional vector spaces over $k$. Each object is a finite direct sum of shifts $k[n]$, and $\Hom(k[a], k[b]) = 0$ for $a \ne b$. Put $Z(E) = -\chi(E)$, where $\chi(E) = \sum_i (-1)^i \dim H^i(E)$. Let $\mathcal{P}(\phi)$ be the finite direct sums of copies of $k[\phi - 1]$ when $\phi$ is an integer, and $0$ otherwise. Then $Z(k[n]) = (-1)^{n+1} = e^{i \pi (n+1)}$, so clause 1 holds, and clause 2 holds since $k[n][1] = k[n+1]$. Clause 3 holds because $\Hom(k[a], k[b]) = 0$ for $a > b$. Clause 4 holds by splitting an object into its shifted summands, taken in order of decreasing phase.

Non-example: keep the same $Z$ but move every slice down by one, so that $k$ lies in $\mathcal{P}(0)$. Then $Z(k) = -1$ while $e^{i \pi \cdot 0} = 1$, so clause 1 fails.

Many authors add a finiteness hypothesis, such as local finiteness or the support property, so that the set of stability conditions becomes a complex manifold. Equivalently, a stability condition is a bounded t-structure with heart $\mathcal{A}$ together with a stability function on $\mathcal{A}$ that has the Harder–Narasimhan property.