Nilpotent groups and universal coverings of smooth projective varieties

Characterizing the universal coverings of smooth projective varieties is an old and hard question. Central to the subject is a conjecture of Shafarevich according to which the universal cover X̃ of a smooth projective variety is holomorphically convex, meaning that for every infinite sequence of points without limit points in X̃ there exists a holomorphic function unbounded on this sequence. In this paper we try to study the universal covering of a smooth projective variety X whose fundamental group π1(X) admits an infinite image homomorphism ρ : π1(X) −→ L into a complex linear algebraic group L. We will say that a nonramified Galois covering X ′ → X corresponds to a representation ρ : π1(X) → L if its group of deck transformations is im(ρ). Definition 1.1 We call a representation ρ : π1(X) → L linear, reductive, solvable or nilpotent if the Zariski closure of its image is a linear, reductive, solvable or nilpotent algebraic subgroup in L. We call the corresponding covering linear, reductive, solvable or nilpotent respectively. The natural homomorphism π1(X,x) → π̂uni(X,x) to Malcev’s pro-unipotent completion will be called the Malcev representation and the corresponding covering the Malcev covering. ∗The author was partially supported by A.P. Sloan Dissertational Fellowship