Abstract
We prove that a meromorphic mapping, which sends a peace of a real analytic strictly pseudoconvex hypersurface in \({\mathbb C}^2\) to a compact subset of \({\mathbb C}^N\) which does not contain germs of non-constant complex curves is continuous from the concave side of the hypersurface. This implies the analytic continuability along CR-paths of germs of holomorphic mappings from real analytic hypersurfaces with non-vanishing Levi form to the locally spherical ones in all dimensions.
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Acknowledgements
I am grateful to Rasul Shafikov and Alexander Sukhov for the stimulating discussions on the subject of this paper.
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Appendix: Result of Shafikov–Verma
Appendix: Result of Shafikov–Verma
In this appendix we explain that the paper [13] contains the following statement.
Theorem 5.1
Let M be a smooth real analytic minimal hypersurface in \({\mathbb C}^n\) and let \(M'\) be a smooth compact real algebraic hypersurface in \({\mathbb C}^{n'}, 1 < n \leqslant n'\). Then every germ \(f:(M,x) \rightarrow (M',x')\) of a holomorphic map from M to \(M'\) extends as a meromorphic correspondence F along any CR-path on M and \(F|_M[M]\subset M'\). If, moreover, \(M'\) is strictly pseudoconvex, then F is a meromorphic map.
Recall that a real hypersurface M is called minimal if it does not contain a non-constant germ of a complex hypersurface. The statement of this theorem is implicit in [13] and is hidden inside of the proof of a stronger statement about holomorphicity of F. Unfortunately the proof of the holomorphicity of extension F in [13] contains a gap. To make this point clear we give an outline of the proof of Theorem 5.1 referring step by step to [13].
Proof
The proof of the theorem breaks into several steps. Recall that a real submanifold \(\Sigma \) of \({\mathbb C}^n\) is called generic if its tangent space at any point contains a complex subspace of minimal possible dimension. If \(\mathsf{dim}\,_{{\mathbb R}}\Sigma = 2n-2\) this means simply that \(T_b\Sigma \) is not a complex subspace of \({\mathbb C}^n\) for all \(b\in \Sigma \). Genericity is obviously an open condition.
Step 1 The proof of Theorem 5.1 can be reduced to the following statement: Let\(\Omega \)be a domain inMsuch thatfis meromorphic on\(\Omega \)and let\(b\in \partial \Omega \)be a boundary point such that\(\Sigma :=\partial \Omega \)is a generic submanifold in aneighborhood ofp. Thenfmeromorphically extends to a neighborhood ofb.
The proof is given in the Sect. 4.1 of [13] and relies on the construction of special ellipsoids from [9]. It is true that in Sect. 4.1 of [13] f is already supposed to be holomorphic on \(\Omega \), but the proof goes through for any analytic objects, e.g., for meromorphic correspondences.
Let \(Q_b\) be the Segre variety of M through b. By Proposition 5.1 from [12] there exists a dense open subset \(\omega \) of \(Q_b\) such that for every \(a\in \omega \) the intersection \(Q_a\cap \Omega \) is non-empty. Moreover, since \(I_f\) is of codimension \(\geqslant 2\) for a generic a this intersection will be not contained in \(I_f\). That is, we can find \(\xi \in \Omega \cap Q_a\) such that f is holomorphic in a neighborhood \(V_{\xi }\) of \(\xi \). Let V be a neighborhood of \(Q_{\xi }\).
Step 2For an appropriate choice of\(V_{\xi }\)andVthe set
is analytic in\(V\times {\mathbb C}^{n'}\), extends to an analytic set in\(V\times {\mathbb P}^{n'}\), and this extension contains the graph offover\(V_{\xi }\).
Indeed, if \(Q'\) is the defining polynomial of \(M'\) the condition \(f(Q_z\cap U_{\xi })\subset Q'_{z'}\) can be expressed as
Here \('t = (t_1,\ldots ,t_{n-1})\) is a coordinate on the tangent plane \(T_pM\) and \(t_n=h('t, {\bar{z}})\) is the equation of the Segre variety \(Q_z\). After conjugation it is clear that this equation is holomorphic in \((z,z')\), see [12, 13] for more details. Moreover, since (5.2) is polynomial in \(z'\) our set A closes to an analytic set in \(V\times {\mathbb P}^{n'}\). This closure will be still denoted as A. The fact that A contains the graph of f over \(V_{\xi }\) immediately follows from the invariance of Segre varieties: \(f(Q_z)\subset Q'_{f(z)}\) whenever everything is well defined (i.e.sufficiently localized).
Remark that dimension of A might be bigger than n simply because the set of \(z'\) such that \(Q'_{z'}\supset f(Q_z)\) can have a positive dimension for a fixed z. In order to reduce the size of A proceed as follows. Consider the natural projections \(\pi :A\rightarrow V\) and \(\pi ':A\rightarrow {\mathbb P}^{n'}\). Remark that \(\pi \) is proper simply because \({\mathbb P}^{n'}\) is compact. For an appropriate neighborhoods \(V^*\) of \(Q_a\) and \(V_a\) of a set
Step 3The set\(A^*\)is analytic in\(V^*\times {\mathbb P}^{n'}\)of dimension n and contains the graph offover (a shrinked, if necessary)\(V_{\xi }\).
The rough reason for \(A^*\) to be of dimension n is that both \(\pi ^{-1}(Q_z\cap V_a)\) and \(\pi '^{-1}(Q'_{z'})\) are hypersurfaces in A and therefore the set of \(z'\in {\mathbb P}^{n'}\) such that \(\pi ^{-1}(Q_z\cap V_a)\subset \pi '^{-1}(Q'_{z'})\) for a given \(z\in V^*\) is generically finite. We refer to §3.2 of [13] for more details.
As the graph of the correspondence F which extends f we take the irreducible component of \(A^*\) which contains the germ of the graph of f over \(\xi \). At points \(z\in M\setminus I_F\) all values of F are contained in \(M'\) by unique continuation property of analytic functions and by the connectivity of the graph of F.
Step 4 If \(M'\) is strictly pseudoconvex, then F is generically singlevalued, i.e. is a meromorphic map.
Suppose \(z'\in F(z)\) for some \(z\in M\setminus (I_F\cup R_F)\), where \(R_F\) is the divisor of ramification of F. By the invariance of Segre varieties \(F(Q_z)\subset Q_{z'}'\). This means that for all branches \(F_1,\ldots ,F_d\) of F near z one has \(F_j(Q_z)\subset Q_{z'}'\). Let \(z'=F_1(z)\) for simplicity and suppose that there is \(w'=F_2(z)\) different from \(z'\). Then \(F_1(Q_z)\subset Q_{w'}'\) as well. But due to the strict pseudoconvexity of \(M'\) we have that \(Q_{z'}'\cap M' = \{z'\}\) and \(Q_{w'}'\cap M'=\{w'\}\); i.e., the germs of \(Q_{z'}'\) and \(Q_{w'}'\) are disjoint. This is a contradiction, i.e.\(z'=w'\) and F is singlevalued on \(M\setminus (I_F\cup R_F)\). This implies that \(R_F=\varnothing \) and F is singlevalued on \(M\setminus I_F\).\(\square \)
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Ivashkovich, S. One-Side Continuity of Meromorphic Mappings Between Real Analytic Hypersurfaces. J Geom Anal 30, 2758–2775 (2020). https://doi.org/10.1007/s12220-018-0032-4
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DOI: https://doi.org/10.1007/s12220-018-0032-4