One-Side Continuity of Meromorphic Mappings Between Real Analytic Hypersurfaces

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.

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

$$\begin{aligned} A:=\{(z,z')\in V\times {\mathbb C}^{n'}: f(Q_z\cap V_{\xi })\subset Q'_{z'}\} \end{aligned}$$

(5.1)

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

$$\begin{aligned} Q'(f('t, h('t, {\bar{z}}), {\bar{z}}') =0. \end{aligned}$$

(5.2)

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

$$\begin{aligned} A^* :=\{(z,z')\in V^*\times {\mathbb P}^{n'}: \pi ^{-1}(Q_z\cap V_a)\subset \pi '^{-1}(Q'_{z'})\}. \end{aligned}$$

(5.3)

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 $M\setminus I_F$.$\square$