Thisbook presents recent advances on Kobayashi hyperbolicity in complex geometry especially in
connection with projective hypersurfaces. This is a very activefield not least because of the
fascinating relations with complex algebraicand arithmetic geometry. Foundational works of
Serge Lang and Paul A. Vojta among others resulted in precise conjectures regarding the
interplay of theseresearch fields (e.g. existence of Zariski dense entire curves
shouldcorrespond to the (potential) density of rational points).Perhapsone of the conjectures
which generated most activity in Kobayashi hyperbolicitytheory is the one formed by Kobayashi
himself in 1970 which predicts that avery general projective hypersurface of degree large
enough does not containany (non-constant) entire curves. Since the seminal work of Green and
Griffithsin 1979 later refined by J.-P. Demailly J. Noguchi Y.-T. Siu and others itbecame
clear that a possible general strategy to attack this problem was tolook at particular
algebraic differential equations (jet differentials) thatevery entire curve must satisfy. This
has led to some several spectacularresults. Describing the state of the art around this
conjecture is the maingoal of this work.
