The Kobayashi canonical-ampleness conjecture asserts that the canonical bundle 
is ample for every compact connected Kähler manifold
of positive complex dimension that is Brody hyperbolic.
Thus the hypothesis excludes every nonconstant holomorphic
map 
, and the conclusion says that a positive tensor power
of 
defines a holomorphic embedding into projective
space.
Projectivity follows from canonical ampleness. For a compact Riemann surface of genus 
, the canonical bundle
has degree 
and is ample. The higher-dimensional assertion connects
the absence of entire curves with positivity of top-degree
holomorphic forms.
