Kodaira embedding theorem can be regarded as a vast generalizaton of the projectivity criterion for complex tori: indeed, the Riemann conditions essentially say that the line bundle defined by the polarisation is positive.
If $k$ is a complete field endowed with non-Archimedean absolute value, then Riemann conditions for analytic tori $\mathbb{G}_m^n(k)/\mathbb{Z}^n$ can also be given, polarisation becomes a map $\varphi: \mathbb{Z}^n \to \mathrm{Hom}(\mathbb{G}_m^n, \mathbb{G}_m)$ such that $\varphi(\lambda)(\lambda')=\varphi(\lambda')(\lambda)$ for any $\lambda, \lambda' \in \mathbb{Z}^n$, and the positivity condition now reads as that the form $\sigma(\lambda, \lambda')=-\log |\varphi(\lambda)(\lambda')|$ is positive definite (all this is nicely exposed in Fresnel and van der Put's book "Rigid analytic geometry and its applications", Section 6.5).
It is then natural to ask: does there exist a criterion for a line bundle over a rigid analytic space to be ample?
While there is a notion of a metrized line bundle over a rigid analytic space (a metric on a line bunde $L$ is just a map of sheaves $h: L \to C(X^{an},\mathbb{R})$, where $C(X^{an},\mathbb{R})$ is the sheaf of continuous $\mathbb{R}$-valued functions on $X^{an}$, such that for local sections $s$ and local functions $f$, $h(f \cdot s) = |f|\cdot h(s)$), is there a notion of positive curvature for such a metric?
