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  "body": "\n## Summary\n\nNakano positivity is a notion of positive curvature for a holomorphic Hermitian vector bundle, stronger than [[Griffiths positivity|FVCDGMNSAN]].\nIt views the curvature of the [[Chern connection|7RC9CN2Q49]] as a Hermitian form on the tensor product of the tangent space and the fibre.\nThen it asks that this form be positive definite.\nIt is the hypothesis under which the Bochner–Kodaira–Nakano identity gives vanishing of cohomology, as in Nakano's vanishing theorem $H^q(X, K_X \\otimes E) = 0$ for $q \\ge 1$ on compact Kähler $X$.\nFor line bundles it agrees with Griffiths positivity.\n\n## Explanations\n\n### L1 human\n\nThe curvature of a vector bundle eats a tangent direction and a fibre vector.\nGriffiths positivity only tests \"pure\" pairs, one tangent vector together with one fibre vector.\nNakano positivity tests all combinations, that is, all elements of the tensor product of the two spaces, including sums of such pairs.\nSo it is a stronger requirement, and in return it gives stronger vanishing theorems.\n",
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  "name": "Nakano positivity",
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