Nakano positivity

Summary

Nakano positivity is a notion of positive curvature for a holomorphic Hermitian vector bundle, stronger than Griffiths positivity. It views the curvature of the Chern connection as a Hermitian form on the tensor product of the tangent space and the fibre. Then it asks that this form be positive definite. It 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 $X$. For line bundles it agrees with Griffiths positivity.