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  "body": "\n## Summary\n\nA holomorphic line bundle is positive when it has a Hermitian metric whose [[Chern connection|7RC9CN2Q49]] has positive curvature.\nThen $i$ times the curvature is a Kähler form, so a compact complex manifold with a positive line bundle is a [[Kähler manifold|X60JG9AEQJ]].\nBy the [[Kodaira embedding theorem|7G7CSM2NZ9]], on a compact complex manifold the positive line bundles are exactly the [[ample line bundles|RRK8P6TXG2]].\nPositivity is the analytic hypothesis of the Kodaira and Kodaira–Nakano vanishing theorems.\n\n## Explanations\n\n### L1 human\n\nThe curvature of a line bundle with a metric is a $2$-form, and \"positive\" means that this form is like a Kähler form at every point.\nThe model is the line bundle $\\mathcal{O}(1)$ on projective space, whose curvature is the Fubini–Study form.\nPositive bundles have many holomorphic sections in their high powers, enough to embed the manifold in projective space.\nThe trivial bundle on a compact manifold is never positive: its curvature can be made zero, but never strictly positive everywhere.\n",
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