On the removable singularities for meromorphic mappings

Abstract. If E is a nonempty closed subset of the locally finite Hausdorff (2n-2)measure on an n-dimensional complex manifold Ω and all 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 [7]: Let E be a closed subset of the complex projective space P n (n ≥ 2) such that the Hausdorff (2n − 2)-measure of E (with respect to the Fubini-Study metric) is less then that of any complex hyperplane of P n. If it is true that E is then a removable singularity for meromorphic functions? Using natural projections of P n onto hyperplanes G.Lupacciolu [5] has shown the removability of E under additional conditions on the sizes of E and a maximal ball in the complement. (The projection of P n onto a hyperplane does not decrease Hausdorff measures, as it take place in the Euclidean space.) Using an Oka–Nishino theorem [6] on pseudoconcave sets we prove here the following. Theorem. Let E be a closed subset of the locally finite Hausdorff (2n−2)-measure on an n-dimensional complex manifold Ω and let f be a meromorphic mapping of Ω \\ E into a complex manifold X. If X has the meromorphic extension property and E does not contain any (n − 1)-dimensional closed analytic subset of Ω then f is continued to a meromorphic mapping of Ω into X. Here we say that X has the meromorphic extension property, if any meromorphic map ϕ: T → X of the ”squeezed polydisc” T = (z, w) ∈ C n−1 z × Cw: |z | < r, |w | < 1or|z | < 1, 1 − r < |w | < 1, 0 < r < 1, n ≥ 2, extends to a meromorphic map ˜ϕ: U → X of the unit

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