Hyperbolicity is the study of the geometry of holomorphic entire curves f:C→X, with values in a given complex manifold X. In this introductary first talk, we will give some definitions and provide historical examples motivating the study of the hyperbolicity of complements Pn∖Xd of projective hypersurfaces Xd having sufficiently high degree d≫n.
Then, we will introduce the formalism of jets, that can be viewed as a coordinate free description of the differential equations that entire curves may satisfy, and explain a successful general strategy due to Bloch, Demailly, Siu, that relies in an essential way on the relation between entire curves and jet differentials vanishing on an ample divisor.