Griffiths positivity

Remarks

Intrinsically, the condition says that $h(i\Theta(\xi, \bar\xi) v, v) > 0$ for every nonzero tangent vector $\xi \in T_x X$ and every nonzero $v \in E_x$. A holomorphic vector bundle is called Griffiths positive if it admits a metric with this property. For $r = 1$ the condition is exactly that of a positive line bundle.

Example: 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. Then $c_{jk\lambda\mu} = a_{jk} \delta_{\lambda\mu}$, and the sum equals $\left(\sum a_{jk} \xi_j \bar \xi_k\right) \abs{v}^2 > 0$.

Non-example: $E = L \oplus \mathcal{O}_X$ with the sum of the metric of $L$ and the trivial metric. The curvature vanishes on the second summand, so the sum is $0$ for $v = (0, 1)$, and the condition fails.

Quotient bundles of Griffiths positive bundles are Griffiths positive, and Griffiths positive bundles on compact manifolds are ample.