It has been known for a long time that there is a very strong link between convexity and curvature on one side, and the existence of holomorphic objects on the other side. We will illustrate this philosophy by recent results relating curvature of a complex manifold with the existence of holomorphic sections and holomorphic entire curves. These ideas can be used to study e.g. the Green-Griffiths conjecture in value distribution theory, which asserts that entire curves contained in a projective variety of general type are contained in a proper algebraic subvariety.