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  "body": "\n## Definition\n\nLet $\\mathcal{D}$ be an essentially small [[triangulated category|T2V7GJJ2EN]] with shift functor $[1]$, and let $K(\\mathcal{D})$ be its Grothendieck group.\nA *stability condition* on $\\mathcal{D}$ is a pair $(Z, \\mathcal{P})$ with the following data.\nThe central charge $Z \\colon K(\\mathcal{D}) \\to \\C$ is a [[group homomorphism|0HV9Y7WG5R]].\nThe slicing $\\mathcal{P}$ assigns to each $\\phi \\in \\R$ a full additive subcategory $\\mathcal{P}(\\phi)$ of $\\mathcal{D}$.\nThese data must satisfy:\n\n1. If $E$ is a nonzero object of $\\mathcal{P}(\\phi)$, then $Z(E) = m(E) e^{i \\pi \\phi}$ for some real number $m(E) \u003e 0$.\n2. $\\mathcal{P}(\\phi + 1) = \\mathcal{P}(\\phi)[1]$ for all $\\phi \\in \\R$.\n3. If $\\phi_1 \u003e \\phi_2$ and $A_j$ is an object of $\\mathcal{P}(\\phi_j)$ for $j = 1, 2$, then $\\Hom(A_1, A_2) = 0$.\n4. Each nonzero object $E$ of $\\mathcal{D}$ has a Harder–Narasimhan filtration. This means real numbers $\\phi_1 \u003e \\phi_2 \u003e \\cdots \u003e \\phi_n$ and morphisms $0 = E_0 \\to E_1 \\to \\cdots \\to E_n = E$. Each step sits in a [[distinguished triangle|ADKK56VZHW]] $E_{j-1} \\to E_j \\to A_j \\to E_{j-1}[1]$ with $A_j$ a nonzero object of $\\mathcal{P}(\\phi_j)$.\n\nHere $Z(E)$ means $Z$ applied to the class of $E$ in $K(\\mathcal{D})$.\nThe nonzero objects of $\\mathcal{P}(\\phi)$ are the semistable objects of phase $\\phi$.\n\n## Remarks\n\nExample: let $k$ be a field and let $\\mathcal{D}$ be the bounded derived category of finite-dimensional vector spaces over $k$.\nEach object is a finite direct sum of shifts $k[n]$, and $\\Hom(k[a], k[b]) = 0$ for $a \\ne b$.\nPut $Z(E) = -\\chi(E)$, where $\\chi(E) = \\sum_i (-1)^i \\dim H^i(E)$.\nLet $\\mathcal{P}(\\phi)$ be the finite direct sums of copies of $k[\\phi - 1]$ when $\\phi$ is an integer, and $0$ otherwise.\nThen $Z(k[n]) = (-1)^{n+1} = e^{i \\pi (n+1)}$, so clause 1 holds, and clause 2 holds since $k[n][1] = k[n+1]$.\nClause 3 holds because $\\Hom(k[a], k[b]) = 0$ for $a \u003e b$.\nClause 4 holds by splitting an object into its shifted summands, taken in order of decreasing phase.\n\nNon-example: keep the same $Z$ but move every slice down by one, so that $k$ lies in $\\mathcal{P}(0)$.\nThen $Z(k) = -1$ while $e^{i \\pi \\cdot 0} = 1$, so clause 1 fails.\n\nMany authors add a finiteness hypothesis, such as local finiteness or the support property, so that the set of stability conditions becomes a complex manifold.\nEquivalently, 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.\n",
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