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  "body": "\n## Definition\n\nLet $X$ be a [[complex manifold|JSGWAN2RJV]] of dimension $n$ and let $E \\to X$ be a [[holomorphic vector bundle|TKA958Z4KZ]] of rank $r$ with a [[Hermitian metric|Q6X6W2N29G]] $h$.\nLet $\\Theta$ be the curvature of the [[Chern connection|7RC9CN2Q49]] of $(E, h)$.\nNear a point $x$, choose local coordinates $z_1, \\dots, z_n$ and a frame $e_1, \\dots, e_r$ of $E$ that is orthonormal at $x$, and write\n$$i \\Theta(x) = i \\sum_{j,k=1}^{n} \\sum_{\\lambda,\\mu=1}^{r} c_{jk\\lambda\\mu}\\, dz_j \\wedge d\\bar z_k \\otimes e_\\lambda^* \\otimes e_\\mu.$$\nThe bundle $(E, h)$ is *Griffiths positive* if at every point $x \\in X$\n$$\\sum_{j,k,\\lambda,\\mu} c_{jk\\lambda\\mu}\\, \\xi_j \\bar\\xi_k v_\\lambda \\bar v_\\mu \u003e 0$$\nfor all nonzero $\\xi = (\\xi_1, \\dots, \\xi_n) \\in \\C^n$ and all nonzero $v = (v_1, \\dots, v_r) \\in \\C^r$.\n\n## Remarks\n\nIntrinsically, the condition says that $h(i\\Theta(\\xi, \\bar\\xi) v, v) \u003e 0$ for every nonzero tangent vector $\\xi \\in T_x X$ and every nonzero $v \\in E_x$.\nA holomorphic vector bundle is called Griffiths positive if it admits a metric with this property.\nFor $r = 1$ the condition is exactly that of a [[positive line bundle|J45KWFHENP]].\n\nExample: let $L$ be a positive line bundle with metric $h_L$ and curvature $i\\Theta_L = i \\sum a_{jk} dz_j \\wedge d\\bar z_k$, and let $E = L^{\\oplus r}$ with the direct sum metric.\nThen $c_{jk\\lambda\\mu} = a_{jk} \\delta_{\\lambda\\mu}$, and the sum equals $\\left(\\sum a_{jk} \\xi_j \\bar \\xi_k\\right) \\abs{v}^2 \u003e 0$.\n\nNon-example: $E = L \\oplus \\mathcal{O}_X$ with the sum of the metric of $L$ and the trivial metric.\nThe curvature vanishes on the second summand, so the sum is $0$ for $v = (0, 1)$, and the condition fails.\n\nQuotient bundles of Griffiths positive bundles are Griffiths positive, and Griffiths positive bundles on compact manifolds are ample.\n",
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