This
book presents recent advances on Kobayashi hyperbolicity in complex geometry,
especially in connection with projective hypersurfaces. This is a very active
field, not least because of the fascinating relations with complex algebraic
and arithmetic geometry. Foundational works of Serge Lang and Paul A. Vojta,
among others, resulted in precise conjectures regarding the interplay of these
research fields (e.g. existence of Zariski dense entire curves should
correspond to the (potential) density of rational points).







Perhaps
one of the conjectures which generated most activity in Kobayashi hyperbolicity
theory is the one formed by Kobayashi himself in 1970 which predicts that a
very general projective hypersurface of degree large enough does not contain
any (non-constant) entire curves. Since the seminal work of Green and Griffiths
in 1979, later refined by J.-P. Demailly, J. Noguchi, Y.-T. Siu and others, it
became clear that a possible general strategy to attack this problem was to
look at particular algebraic differential equations (jet differentials) that
every entire curve must satisfy. This has led to some several spectacular
results. Describing the state of the art around this conjecture is the main
goal of this work.