Some degeneracy theorems for entire functions with values in an algebraic variety

Abstract:

In the first part of this paper we prove the following extension theorem. Let $P_q^ \ast$ be a $q$-dimensional punctured polycylinder, i.e. a product of disks and punctured disks. Let ${W_n}$ be a compact complex manifold such that the bundle of holomorphic $q$-forms is positive in the sense of Grauert. Let $f:P_q^ \ast \to {W_n}$ be a holomorphic map whose Jacobian determinant does not vanish identically. Then $f$ extends as a rational map to the full polycylinder ${P_q}$. In the second half of the paper we prove the following generalization of the little Picard theorem to several complex variables: Let $V \subset {P_n}$ be a hypersurface of degree $d \geqq n + 3$ whose singularities are locally normal crossings. Then any holomorphic map $f:{C^n} \to {P_n} - V$ has identically vanishing Jacobian determinant.