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  "body": "\n## Definition\n\nLet $X$ be a [[complex manifold|JSGWAN2RJV]] and let $L \\to X$ be a [[holomorphic line bundle|QAC7BYK9KX]].\nFor a [[Hermitian metric|Q6X6W2N29G]] $h$ on $L$, let $\\Theta_h$ be the curvature of its [[Chern connection|7RC9CN2Q49]], a $(1,1)$-form on $X$.\nIn local coordinates $z_1, \\dots, z_n$ write $i \\Theta_h = i \\sum_{j,k} a_{jk}\\, dz_j \\wedge d\\bar z_k$; the matrix $(a_{jk})$ is Hermitian.\nThe line bundle $L$ is *positive* if some Hermitian metric $h$ on $L$ has $(a_{jk}(x))$ [[positive definite|0KX66CZMBB]] at every point $x \\in X$.\n\n## Remarks\n\nIf $e$ is a local holomorphic frame of $L$ and $\\abs{e}_h^2 = e^{-\\varphi}$, then $i \\Theta_h = i \\partial \\bar\\partial \\varphi$.\nSo $L$ is positive exactly when some metric has local weights $\\varphi$ that are strictly plurisubharmonic.\nThe form $\\frac{i}{2\\pi} \\Theta_h$ represents the first Chern class $c_1(L)$, so positivity implies that $c_1(L)$ contains a Kähler form.\n\nExample: $\\mathcal{O}(1)$ on $\\mathbb{P}^n$ with the metric dual to the standard metric on $\\mathcal{O}(-1)$.\nOn $\\set{z_0 \\ne 0}$ with affine coordinates $z$, the dual frame $e^*$ has $\\abs{e^*}^2 = (1 + \\abs{z}^2)^{-1}$, so $\\varphi = \\log(1 + \\abs{z}^2)$.\nThen $i \\partial \\bar\\partial \\varphi$ is the Fubini–Study form, which is positive definite at each point.\n\nNon-example: the trivial line bundle on a compact connected complex manifold $X$ of positive dimension.\nAny metric is $\\abs{1}_h^2 = e^{-\\varphi}$ for a global smooth function $\\varphi$.\nAt a maximum point of $\\varphi$ the complex Hessian $(\\partial^2 \\varphi / \\partial z_j \\partial \\bar z_k)$ is negative semidefinite, so it is not positive definite there.\n",
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