On the removable singularities for meromorphic mappings

Publicacions Matem`atiques, Vol 40 (1996), 229–232. ON THE REMOVABLE SINGULARITIES FOR MEROMORPHIC MAPPINGS E. M. Chirka Abstract If Eis a closed subset of locally finite Hausdorff (2n−2)-measure on an n-dimensional complex manifold Ω and all the points of Eare nonremovable for a meromorphic mapping of Ω \Einto a compact K¨ahler manifold, then Eis 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 Ebe a closed subset of the complex projective space Pn,n≥2, such that the Hausdorff (2n−2)-measure of Ewith respect to the Fubini-Study metric is less than that of a complex hyperplane in Pn. Is it then true that Eis a set of removable singularities for meromorphic functions? Using natural projections of Pnonto hyperplanes G. Lupacciolu [Lu] has shown the removability of Eunder 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 complex manifolds M,X. A meromorphic map in the sense of Remmert [R] (or strongly meromorphic map) is a holomorphic map f:M\A→Xdefined in the complement of an analytic subset A⊂Mand such that the closure of its graph is an analytic subset of M×Xwith proper projection π in M(the last property is satisfied automatically if the target manifold Xis compact). It follows easily from the definition that the set Supported by Russian Fund of Fundamental Researches, project No 96-01-00530. 230 E. M. Chirka Af={z∈M: dim π−1(z)>0}is an analytic subset of (complex) codimension at least two and fextends holomorphically to M\Af. A meromorphic map in the sense of Stoll [St] (or weakly meromorphic map) is a holomorphic map f:M\A→X, where Ais an analytic subset of codimension larger than or equal to M, such that, for any analytic disk ϕ:∆→Mwith ϕ(∆) ⊂ A, the restriction f◦ϕ:∆\ϕ−1(A)→X extends holomorphically to the disc ∆. We proceed now to state our main result. We point out that there is no difference between strongly and weakly meromorphic maps as far as its statement and proof are concerned, and thus we use the term “meromorphic” to mean always “strongly meromorphic” or always “weakly meromorphic”. Theorem. Let Ebe a closed subset of locally finite Hausdorff (2n−2)-measure on an n-dimensional complex manifold Ωand let f be a meromorphic map from Ω\Einto a complex manifold X.IfX has the meromorphic extension property and Edoes not contain any (n−1)-dimensional closed analytic subset of Ω, then fextends to a meromorphic mapping from Ωinto X. Here we say that Xhas the meromorphic extension property, if every meromorphic mapping ϕ:T→Xon the “squeezed polydisc” T={(z,w)∈Cn−1 z×Cw:|z|<r,|w|<1or|z|<1,1−r<|w|<1}, 0<r<1, n≥2, extends to a meromorphic mapping ˜ϕ:U→Xon the unit polydisc U:|z|<1, |w|<1. By a result of S. M. Ivashkovich [I] every compact K¨ahler manifold Xhas the meromorphic extension property, so we have a lot of nice examples of such X. The case of meromorphic functions (X=P1) is almost trivial in the consideration: every meromorphic function in a squeezed polydisc is represented as a ratio of two holomorphic functions (see [KS]) and thus it extends meromorphically into U. So the answer on the question of Stout is affirmative because the set Ein the question can not contain any complex analytic subset of Pn. (By Chow Theorem such a set is algebraic, hence, its Hausdorff (2n−2)-measure is not less than the measure of a complex hyperplane (see, e.g., [C]).) Note also that the meromorphic extension claimed in the Theorem is not holomorphic in general for holomorphic mappings f:Ω\E→X, even for X=P1(one can take a meromorphic map f:Ω→Xwith nonempty set Afof indeterminacy of fand put E=Af). Removable singularities 231 2. Here we prove the Theorem. Let Ebe the set of points a∈Esuch that fextends meromorphically into a neighbourhood of a. Then S:= E\Eis closed. As the complement of Eis locally connected in Ω, these local meromorphic extensions of fto points of Eglue together into the unique meromorphic mapping f:Ω\S→X(we keep the notation f). The proof of the Theorem is accomplished now in two steps. First we prove that Sis locally pseudoconcave (Lemma 1), and then we prove that Sis complex analytic (Lemma 2). (A closed subset Σ of a complex manifold is called locally pseudoconcave if for every point a∈Σ there exists a Stein neighborhood Vof asuch that V\Σ is Stein.) It follows then from the condition of the Theorem that Sis empty, and hence, f extends meromorphically into Ω. Lemma 1. Sis locally pseudoconcave in Ω. Proof: Let a∈S, let Vbe a neighborhood of awhich is biholomorphic to a ball in Cnand let ϕ:T→V\Sbe a holomorphic embedding. Then ϕextends to a holomorphic embedding ˜ϕ:U→V(see [DG]) because Vis Stein. As Xhas the meromorphic extension property, the meromorphic map f◦ϕ:T→Xextends to a meromorphic map of U into Xand thus fextends meromorphically into the domain ˜ϕ(U)⊂V. By the definition of S,˜ϕ(U) does not intersect S, and thus we have proved that V\Ssatisfies the condition of “p7-convexity” of Docquier- Grauert [DG]. It follows that V\Sis Stein. Lemma 2. Let Sbe a nonempty locally pseudoconcave subset of finite Hausdorff (2n−2)-measure of an n-dimensional complex manifold Ω. Then Sis a pure (n−1)-dimensional complex analytic subset of Ω. Proof: The statement is local, so we can assume that Ω is a domain in Cn. Let a∈Sand choose coordinates (z,w), z=(z1,... ,z n−1)inCnin such a way that a= 0 and the set S∩{z=0}is finite or countable (this can be done in the obvious way). Then there exists a neighbourhood V=V×Vnof 0 such that the projection of S∩Vinto Vis proper. It follows that the fibres S∩V∩{z=c}are finite for almost every c∈V. By the Oka-Nishino Theorem [N]S∩Vis then a complex analytic set of pure dimension n−1. References [C] E. M. Chirka,“Complex analytic sets,” Nauka, Moscow, 1985. 232 E. M. Chirka English transl., Kluwer Acad. Publishers, Dordrecht-Boston-Lon- don, 1989. [DG] F. Docquier and H. Grauert, Levisches Problem und Rungescher Satz f¨ur Teilgebiete Steinscher M¨annigfaltigkeiten, Math. Ann. 140 (1960), 94–123. [I] S. M. Ivashkovich, The Hartogs-type extension theorem for meromorphic maps into compact K¨ahler manifolds, Invent. Math. 109 (1992), 47–54. [KS] J. Kajiwara and E. Sakai, Generalization of Levi-Oka’s theorem concerning meromorphic functions, Nagoya Math. J. 29 (1967), 75–84. [La] M. Lawrence, Polynomial hulls and geometric function theory of several complex variables, Thesis, University of Washington, Seattle (1991). [Lu] G. Lupacciolu, Meromorphic continuation in projective space, Bollettini U. M. I. (7) 4-A (1990), 391–397. [N] T. Nishino, Sur les ensembles pseudoconcaves, J. Math. Kyoto Univ. 1-2 (1962), 225–245. [R] R. Remmert, Holomorphe und meromorphe Abbildungen komplexer R¨aume, Math. Ann. 133 (1957), 328–370. [S] E. L. Stout, Removable sets for holomorphic functions of several complex variables, Publ. Math. 33 (1989), 345–362. [St] W. Stoll,¨ Uber meromorphe Modifikation, II, Math. Z. 61 (1955), 467–488. Keywords. meromorphic mappings, removable singularities, pseudoconcave sets, analytic varieties. 1991 Mathematics subject classifications: Primary 32D15, 32D20; Secondary 32D10. Steklov Mathematical Institute Vavilov st. 42 Moscow GSP-1 117966 RUSSIA email: chirk[email protected] Rebut el 4 de Desembre de 1995