The hyperbolic Ax-Lindemann-Weierstraß conjecture

Abstract

1.1. Bi-algebraic geometry and the Ax-Lindemann-Weierstras property. — Let X and S be complex algebraic varieties and suppose π : Xan −→ San is a complex analytic, nonalgebraic, morphism between the associated complex analytic spaces. In this situation the image π(Y) of a generic algebraic subvariety Y ⊂ X is usually highly transcendental and the pairs (Y ⊂ X,V ⊂ S) of irreducible algebraic subvarieties such that π(Y) = V are rare and of particular geometric significance. We will say that an irreducible subvariety Y ⊂ X (resp. V ⊂ S) is bi-algebraic if π(Y) is an algebraic subvariety of S (resp. any analytic irreducible component of π−1(V) is an irreducible algebraic subvariety of X). Notice that V ⊂ S is bi-algebraic if and only if any analytic irreducible component of π−1(V) is bi-algebraic.