Holomorphic mappings between spaces of different dimensions. I
The author generalizes some results of equidimensional holomorphic mappings to the nonequidimensional case in which there are generalizations of the Schwarz lemma and a Kodaira's theorem [\textit{K. Kodaira}, J. Differ. Geom. 6, 33-46 (1971; Zbl 0227.32008)]. He also discusses some conjectures asked by M. Green, P. Griffiths and S. Lang. In particular, the following result follows from his theorems: Theorem A. Let \(N\) be a Hermitian manifold of dimension \(n\) with Ricci curvature bounded from above by a negative constant and let \(f:\mathbb{C}^ n \to N\) be a holomorphic map. Then 1) \(\text{rank }f < n\); 2) if the holomorphic tangent bundle \(T_ N\) of \(N\) is generated by its sections, and if \(N\) is compact, then rank \(f < m\). In particular, \(N\) is hyperbolic; 3) if \(N\) is projective algebraic, then \(T_ N\) is not ample. Theorem B. Let \(N\) be a smooth projective algebraic variety of dimension \(n\) and let \(f: \mathbb{C}^ n \to N\) be a holomorphic map of rank \(\min(m,n)\) which is rational. If \(m\geq n\), or if \(m < n\), but \(f\) is algebraically nondegenerate and there exist holomorphic vector fields \(X_ 1,\dots,X_ n\) on \(N\) such that \(X_ 1 \wedge \dots \wedge X_ n \not\equiv 0\), then all the plurigenera \(P_ k\) of \(N\) vanish. Theorem C. Let \(M\) be a smooth affine variety of dimension \(m\) and \(N\) a smooth projective variety of general type of dimension \(n\). Let \(f: M\to N\) be a holomorphic map of rank \(m\). If \(m<n\), and if \(T_ n\) is generated by its sections or there exist holomorphic vector fields \(X_ 1,\dots,X_ n\) on \(N\) such that \(X_ 1 \wedge \dots \wedge X_ n \not\equiv 0\), then either \(f\) is rational, or \(f(M)\) is contained in a proper subvariety of \(N\). In particular, if \(M = \mathbb{C}^ m\), then \(f(\mathbb{C}^ m)\) is contained in a proper subvariety of \(N\). The author also gives some other corollaries and a defect relation.
scientific article; zbMATH DE number 567427
scientific article