The survey is on hyperbolic analysis with the emphasis on its algebraic geometry aspects. It can serve as an excellent introduction to the subject not pretending completeness but reflecting the main ideas including those which appeared at the last decade (after Kobayashi’s survey, 1976). It contains also a number of old and new problems concerning both the hyperbolic analysis itself and its applications to Diophantine equations; among them the author’s (1974) multidimensional versions of Mordell’s conjecture. The central one asserts that a hyperbolic projective variety is Mordellic (i.e. the set of its rational points is finite). The only known case is the Faltings’ theorem in dimension one.
Starting with the notion of the Kobayashi hyperbolicity §§1 and 2 contain the full proofs of certain hyperbolicity criteria - Brody’s one for compact varieties (absence of entire curves) and M. Green’s one for subvarieties of a complex torus (absence of translated subtori). There are given also some variations on these themes.
During the whole article the author adheres to an idea of ”pseudofication”, i.e. a system of relative analogues of the notions and facts under consideration (usually modulo some ”exceptional” sets). The analytic exceptional set Exc(X) of a variety X is defined to be the Zariski closure of the union of all entire curves in X, and an algebraic exceptional set to be the union of all compact rational and elliptic curves in X. One of the important open questions is whether (the algebraic characterization of hyperbolicity).
Sections 3 and 4 are devoted to the Chern-Ricci forms, Ahlfors’ lemma and its multidimensional version working then in the differential geometric conditions for hyperbolicity and measure hyperbolicity.
The problem in §5 are the canonical and ”pseudocanonical” varieties (i.e. respectively the varieties with ample canonical bundle and the varieties of general type). It is established their measure hyperbolicity and discussed the validity of the converse. The Diophantine conjectures 5.7 and 5.8 say: the set of rational points of a pseudocanonical variety X is not Zariski dense; moreover and is Mordellic.
The short §6 is mainly concerned with minimal models. - In §7 the hyperbolicity of a manifold with ample cotangent bundle is proved by the construction of a negatively curved length function (”Finsler metric”). - The Green-Griffiths’ construction of nonpositively curved length functions on jet bundles is given in §8 where also the Green- Griffith’s exceptional set containing Exc(X) is defined.
In the appendix there are some historical comments on the functional analogues of Mordell’s conjecture and its multidimensional versions.