Weak hyperbolicity and many-dimensional analogues of Picard's theorem
The author studies holomorphic mappings between complex manifolds omitting analytic subsets of the target manifold. The main result of the present paper is the following theorem: Let M be an n-dimensional compact complex manifold, \(D\subset M\) a closed analytic hypersurface with normal crossings, and N a complex manifold of dimension \(n-1.\) If \(c_ 1(D)+c_ 1(K_ M)\geq 0,\) then every holomorphic mapping f: \(N\times {\mathbb{C}}\to M\) without essential singularities at infinity, i.e. which extends to a meromorphic mapping f: \(N\times {\bar {\mathbb{C}}}\to M\), is everywhere degenerate. The proof involves some results concerning weakly hyperbolic manifolds - a notion introduced and studied by the author [Sib. Math. J. 29, No.1, 161-163 (1988); translation from Sib. Math. Zh. 29, No.1(167), 209-212 (1988; Zbl 0638.32023)]. The author's theorem generalizes an analogous result in the Griffiths- Carlson-King theory known under the stronger condition \(c_ 1(D)+c_ 1(K_ M)>0\) [\textit{Ph. Griffiths} and \textit{J. King}, Acta Math. 130, 145- 220 (1973; Zbl 0258.32009)].
scientific article; zbMATH DE number 4204792