ON THE REMOVABLE SINGULARITIES FOR MEROMORPHIC MAPPINGS

If E is a closed subset of locally finite Hausdorff (2n − 2)-measure on an n-dimensional complex manifold Ω and all the points of E are nonremovable for a meromorphic mapping of Ω \ E into a compact Kähler manifold, then E is a pure (n − 1)-dimensional complex analytic subset of Ω. 1. This paper was inspired by the following question of E. L. Stout [S]: Let E be a closed subset of the complex projective space Pn, n ≥ 2, such that the Hausdorff (2n−2)-measure of E with respect to the Fubini-Study metric is less than that of a complex hyperplane in Pn. Is it then true that E is a set of removable singularities for meromorphic functions? Using natural projections of Pn onto hyperplanes G. Lupacciolu [Lu] has shown the removability of E under additional conditions on E. A complete affirmative solution of the problem was given by M. Lawrence [La]. In this paper we consider an extension of the problem for meromorphic mappings on an arbitrary complex manifold and solve it (for special target manifolds) using the Oka-Nishino Theorem [N] on pseudoconcave sets. There are two general notions of meromorphic mapping between com-plex manifolds M, X. A meromorphic map in the sense of Remmert [R] (or strongly mero-morphic map) is a holomorphic map f: M \ A → X defined in the complement of an analytic subset A ⊂ M and such that the closure of its graph is an analytic subset of M × X with proper projection π in M (the last property is satisfied automatically if the target mani-fold X is compact). It follows easily from the definition that the se