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

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