1 Introduction, background materials and statement of the main theorem

Let $\unicode[STIX]{x1D6E4}\,\subset \,\text{Aut}(\mathbb{B}^{n})$ be a torsion-free lattice and denote by $X:=\mathbb{B}^{n}/\unicode[STIX]{x1D6E4}$ the quotient manifold. When $\unicode[STIX]{x1D6E4}\,\subset \,\text{Aut}(\mathbb{B}^{n})$ is an arithmetic lattice, by Satake [Reference SatakeSat60] and Baily and Borel [Reference Baily and BorelBB66] $X$ admits a minimal compactification $\overline{X}_{\text{min}}$ by adjoining a finite number of normal isolated singularities. In general $X=\mathbb{B}^{n}/\unicode[STIX]{x1D6E4}$ is of finite volume with respect to the canonical Kähler–Einstein metric, and by Siu and Yau [Reference Siu, Yau and YauSY82], $X$ admits a compactification $X\,\subset \,\overline{X}_{\text{min}}$ by adding a finite number of normal isolated singularities, where $\overline{X}_{\text{min}}$ is exactly the minimal compactification of Satake, Baily and Borel in the arithmetic case. The methods of Siu and Yau [Reference Siu, Yau and YauSY82] are transcendental, and they apply to complete Kähler manifolds of finite volume and of pinched negative sectional curvature. When $X=\mathbb{B}^{n}/\unicode[STIX]{x1D6E4}$ , Mok [Reference MokMok12a] proved that $\overline{X}_{\text{min}}$ is projective. For finite-volume complex ball quotients, by [Reference SatakeSat60], [Reference Baily and BorelBB66], [Reference Siu, Yau and YauSY82] and [Reference MokMok12a] we have the following.

Here and henceforth, without loss of generality we choose the canonical Kähler–Einstein metric so that $(\mathbb{B}^{n},ds_{\mathbb{B}^{n}}^{2})$ and hence $(X,ds_{X}^{2})$ are of constant holomorphic sectional curvature $-2$ . From now on $X=\mathbb{B}^{n}/\unicode[STIX]{x1D6E4}$ will be equipped with the quasi-projective structure inherited from the projective variety $\overline{X}_{\text{min}}$ . To emphasize the dependence on the lattice $\unicode[STIX]{x1D6E4}$ we will now write $X_{\unicode[STIX]{x1D6E4}}$ for $X$ , and $\overline{X_{\unicode[STIX]{x1D6E4}}}$ for $\overline{X}_{\text{min}}$ . In [Reference Mok and EbenfeltMok10] we consider Zariski closures of totally geodesic complex submanifolds on $X_{\unicode[STIX]{x1D6E4}}=\mathbb{B}^{n}/\unicode[STIX]{x1D6E4}$ , and we have proved the following main result.

Denoting by $\widetilde{Z}$ an irreducible component of $\unicode[STIX]{x1D70B}^{-1}(Z)\,\subset \,\mathbb{B}^{n}$ , in the above $Z\,\subset \,X$ is said to be a totally geodesic subset if and only if $\widetilde{Z}\,\subset \,\mathbb{B}^{n}$ is a totally geodesic (complex) submanifold with respect to the canonical Kähler–Einstein metric.

In the current article we prove the Ax–Lindemann–Weierstrass theorem for not necessarily arithmetic finite-volume quotients of $\mathbb{B}^{n}$ , $n\geqslant 2$ . A subvariety $S\,\subset \,\mathbb{B}^{n}$ is said to be an irreducible algebraic subset if and only if it is an irreducible component of the intersection $V\cap \mathbb{B}^{n}$ for some (irreducible) projective subvariety $V\,\subset \,\mathbb{P}^{n}$ , and an algebraic subset $S\,\subset \,\mathbb{B}^{n}$ is the union of a finite number of irreducible algebraic subsets. Note that a totally geodesic complex submanifold of $\mathbb{B}^{n}$ is precisely a nonempty intersection of the form $V\cap \mathbb{B}^{n}$ , where $V\,\subset \,\mathbb{P}^{n}$ is a projective linear subspace of $\mathbb{P}^{n}$ . In this article we prove the following main result generalizing Theorem 1.2.

For the proof of Main Theorem we will consider the universal family $\unicode[STIX]{x1D70C}:\mathscr{U}\rightarrow {\mathcal{K}}$ , $\unicode[STIX]{x1D707}:\mathscr{U}\rightarrow \mathbb{P}^{n}$ over some irreducible component ${\mathcal{K}}$ of the Chow space of $\mathbb{P}^{n}$ and obtain by restriction to $\mathbb{B}^{n}\,\subset \,\mathbb{P}^{n}$ and by descent to $X_{\unicode[STIX]{x1D6E4}}$ a locally homogeneous holomorphic fiber bundle $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6E4}}:\mathscr{U}_{\unicode[STIX]{x1D6E4}}\rightarrow X_{\unicode[STIX]{x1D6E4}}$ equipped with a tautological foliation. The fibered space $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6E4}}:\mathscr{U}_{\unicode[STIX]{x1D6E4}}\rightarrow X_{\unicode[STIX]{x1D6E4}}$ will be embedded as a locally homogeneous fiber subbundle of projective varieties in a certain locally homogeneous projective bundle $\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6E4}}:\mathscr{P}_{\unicode[STIX]{x1D6E4}}\rightarrow X_{\unicode[STIX]{x1D6E4}}$ , which implies in the compact case that $\mathscr{U}_{\unicode[STIX]{x1D6E4}}$ is projective. For the finite-volume and noncompact case we will need to embed the total space $\mathscr{P}_{\unicode[STIX]{x1D6E4}}$ onto a quasi-projective variety. For this purpose we will need the following compactification result of Mok and Zhong [Reference Mok and ZhongMZ89] from Kähler geometry.

For the study of $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6E4}}:\mathscr{U}_{\unicode[STIX]{x1D6E4}}\rightarrow X_{\unicode[STIX]{x1D6E4}}$ we will need the following basic result on Chow spaces of $\mathbb{P}^{n}$ from Chow and van der Waerden [Reference Chow and van der WaerdenCvdW37]. More details will be given in § 2.

Here an element $[W]\in {\mathcal{Q}}$ denotes an $r$ -cycle on $\mathbb{P}^{n}$ , which is neither necessarily irreducible nor reduced. As an $r$ -cycle $[W]=k_{1}[W_{1}]+\cdots +k_{s}[W_{s}]$ , where $s\geqslant 1$ and $W_{i},1\leqslant i\leqslant s$ , are reduced and irreducible $r$ -dimensional subvarieties of $\mathbb{P}^{n}$ .

In the proof of Main Theorem we will study pre-images of Zariski closures of the image of an algebraic subset on $\mathbb{B}^{n}$ under the universal covering map, for which we will need the following result of Klembeck [Reference KlembeckKle78] on the asymptotic behavior of complete Kähler metrics on strictly pseudoconvex domains.

2 Construction of locally homogeneous projective fiber subbundles of projective bundles

Consider the standard inclusions $\mathbb{B}^{n}\Subset \mathbb{C}^{n}\,\subset \,\mathbb{P}^{n}$ . Let $W_{0}\,\subset \,\mathbb{P}^{n}$ be an irreducible subvariety and $S$ be an irreducible component of $W_{0}\cap \mathbb{B}^{n}$ . Let ${\mathcal{K}}$ be an irreducible component of the Chow space of $\mathbb{P}^{n}$ to which the reduced cycle $W_{0}$ belongs, written $[W_{0}]\in {\mathcal{K}}$ . The algebraic group $\text{Aut}(\mathbb{P}^{n})=\mathbb{P}\text{GL}(n+1,\mathbb{C})$ acts on ${\mathcal{K}}$ naturally. Let $\unicode[STIX]{x1D70C}:\mathscr{U}\rightarrow {\mathcal{K}}$ be the universal family associated to ${\mathcal{K}}$ , i.e., $\mathscr{U}=\{([W],x)\in {\mathcal{K}}\times \mathbb{P}^{n}:x\in W\}\,\subset \,{\mathcal{K}}\times \mathbb{P}^{n}$ , and $\unicode[STIX]{x1D70C}:\mathscr{U}\rightarrow {\mathcal{K}}$ is induced from the canonical projection from ${\mathcal{K}}\times \mathbb{P}^{n}$ onto the first factor. For $[W]\in {\mathcal{K}}$ we have $[W]=k_{1}[W_{1}]+\cdots +k_{s}[W_{s}]$ where $s=s(W)\geqslant 1$ and $W_{i},1\leqslant i\leqslant s$ are the reductions of the irreducible components of the pure $r$ -dimensional complex space $W\,\subset \,\mathbb{P}^{n}$ . From the definition of ${\mathcal{K}}$ a general member $[W]\in {\mathcal{K}}$ is irreducible and reduced, i.e., $s=1$ and $k_{1}=1$ . The canonical projection from ${\mathcal{K}}\times \mathbb{P}^{n}$ to $\mathbb{P}^{n}$ restricted to $\mathscr{U}$ gives $\unicode[STIX]{x1D707}:\mathscr{U}\rightarrow \mathbb{P}^{n}$ , called the evaluation map of the universal family $\unicode[STIX]{x1D70C}:\mathscr{U}\rightarrow {\mathcal{K}}$ , which realizes $\mathscr{U}$ as the total space of a homogeneous holomorphic fiber bundle (with respect to Aut $(\mathbb{P}^{n})$ ) with fibers isomorphic to $\mathscr{U}_{0}:=\unicode[STIX]{x1D707}^{-1}(0)$ . Hence, the map $\unicode[STIX]{x1D70C}:\mathscr{U}\rightarrow {\mathcal{K}}$ is a holomorphic fibration with equidimensional fibers $\unicode[STIX]{x1D70C}^{-1}(\unicode[STIX]{x1D705})$ , $\unicode[STIX]{x1D705}=[W]$ . Each fiber $\unicode[STIX]{x1D70C}^{-1}(\unicode[STIX]{x1D705})$ projects via the evaluation map $\unicode[STIX]{x1D707}:\mathscr{U}\rightarrow \mathbb{P}^{n}$ onto the support Supp $(W)=W_{1}\,\cup \,\cdots \,\cup \,W_{s}\,\subset \,\mathbb{P}^{n}$ of the cycles $W\,\subset \,\mathbb{P}^{n}$ belonging to ${\mathcal{K}}$ . (We will henceforth make no notational distinction between $W$ and its support.) The holomorphic fibration $\unicode[STIX]{x1D70C}:\mathscr{U}\rightarrow {\mathcal{K}}$ defines naturally a foliation $\mathscr{F}$ on $\mathscr{U}$ such that, at a smooth point $u\in \mathscr{U}$ where $\unicode[STIX]{x1D70C}$ is a submersion, $\unicode[STIX]{x1D70C}(u)=:[W]\in {\mathcal{K}}$ , the germ of leaf of $\mathscr{F}$ passing through $u$ is just the germ of $\widehat{W}:=\unicode[STIX]{x1D70C}^{-1}(\unicode[STIX]{x1D70C}(u))$ at $u$ , $\widehat{W}$ being the tautological lifting of $W$ to $\mathscr{U}$ . We have $W=\unicode[STIX]{x1D707}(\widehat{W})=\unicode[STIX]{x1D707}(\unicode[STIX]{x1D70C}^{-1}(\unicode[STIX]{x1D70C}(u)))$ .

Write $\mathscr{U}^{\prime }:=\mathscr{U}|_{\mathbb{B}^{n}}$ . The algebraic group $\text{Aut}(\mathbb{P}^{n})$ acts on $\mathscr{U}^{\prime }$ , hence the holomorphic fiber bundle $\unicode[STIX]{x1D707}^{\prime }:\mathscr{U}^{\prime }\rightarrow \mathbb{B}^{n}$ is homogeneous under Aut $(\mathbb{B}^{n})$ , and it descends under the action of a torsion-free discrete subgroup $\unicode[STIX]{x1D6E4}\,\subset \,\text{Aut}(\mathbb{B}^{n})$ to $X_{\unicode[STIX]{x1D6E4}}$ to give a complex space $\mathscr{U}_{\unicode[STIX]{x1D6E4}}:=\mathscr{U}^{\prime }/\unicode[STIX]{x1D6E4}$ equipped with the evaluation map $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6E4}}:\mathscr{U}_{\unicode[STIX]{x1D6E4}}\rightarrow X_{\unicode[STIX]{x1D6E4}}$ , realizing the latter as a locally homogeneous holomorphic fiber bundle with fibers isomorphic to $\mathscr{U}_{0}$ . The fiber bundle $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6E4}}:\mathscr{U}_{\unicode[STIX]{x1D6E4}}\rightarrow X_{\unicode[STIX]{x1D6E4}}$ will be our major object of study. In the case where $X_{\unicode[STIX]{x1D6E4}}$ is compact, we will show that $\mathscr{U}_{\unicode[STIX]{x1D6E4}}$ is a projective variety by embedding $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6E4}}:\mathscr{U}_{\unicode[STIX]{x1D6E4}}\rightarrow X_{\unicode[STIX]{x1D6E4}}$ into some projective bundle $\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6E4}}:\mathscr{P}_{\unicode[STIX]{x1D6E4}}\rightarrow X_{\unicode[STIX]{x1D6E4}}$ . (Projective bundles are understood to be holomorphic.) By the Kodaira embedding theorem, we have the following well-known result on projective bundles.

We proceed to construct a projective bundle $\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6E4}}:\mathscr{P}_{\unicode[STIX]{x1D6E4}}\rightarrow X_{\unicode[STIX]{x1D6E4}}$ which admits $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6E4}}:\mathscr{U}_{\unicode[STIX]{x1D6E4}}\rightarrow X_{\unicode[STIX]{x1D6E4}}$ naturally as an embedded complex subspace. In what follows we denote by $G:=\mathbb{P}\text{GL}(n+1,\mathbb{C})$ the automorphism group of $\mathbb{P}^{n}$ , by $G_{0}:=\mathbb{P}U(n,1)\,\subset \,G$ the automorphism group of $\mathbb{B}^{n}\Subset \mathbb{C}^{n}\,\subset \,\mathbb{P}^{n}$ , and by $\unicode[STIX]{x1D6E4}\,\subset \,G_{0}$ a torsion-free lattice. We write $P\,\subset \,G$ for the parabolic subgroup at $0\in \mathbb{P}^{n}$ , and $K\,\subset \,G_{0}$ for the isotropy subgroup of $(\mathbb{B}^{n},ds_{\mathbb{B}^{n}}^{2})$ at $0\in \mathbb{B}^{n}$ . Lie algebras of real or complex Lie groups will be denoted by corresponding fraktur characters, so that $\mathfrak{g}$ stands for the Lie algebra of the complex Lie group $G$ , and $\mathfrak{g}_{0}$ for the Lie algebra of the real Lie group $G_{0}$ , etc. On the complex unit ball $\mathbb{B}^{n}\Subset \mathbb{C}^{n}$ , $K=U(n)$ acts as a compact group of linear transformations on $\mathbb{C}^{n}$ extending to projective linear transformations on $\mathbb{P}^{n}$ , and we denote by $K^{\mathbb{C}}$ its complexification, i.e., $K^{\mathbb{C}}=\text{GL}(n,\mathbb{C})$ , preserving $\mathbb{C}^{n}\,\subset \,\mathbb{P}^{n}$ and acting as a group of automorphisms of $\mathbb{P}^{n}$ which restricts to $\mathbb{C}^{n}$ to give the usual action of $\text{GL}(n,\mathbb{C})$ on $\mathbb{C}^{n}\,\subset \,\mathbb{P}^{n}$ . The complex Lie algebra $\mathfrak{g}$ , considered as the Lie algebra of holomorphic vector fields on $\mathbb{P}^{n}$ , admits the Harish-Chandra decomposition $\mathfrak{g}=\mathfrak{m}^{+}\oplus \,\mathfrak{k}^{\mathbb{C}}\oplus \,\mathfrak{m}^{-}$ , where $\mathfrak{k}^{\mathbb{C}}=\mathfrak{k}\,\otimes _{\mathbb{R}}\mathbb{C}$ is the complexification of $\mathfrak{k}$ , equivalently the Lie algebra of $K^{\mathbb{C}}$ , $\mathfrak{m}^{-}\,\subset \,\mathfrak{g}$ is the Abelian Lie subalgebra consisting of holomorphic vector fields on $\mathbb{P}^{n}$ vanishing to the order ${\geqslant}2$ at $0\in \mathbb{P}^{n}$ , and $\mathfrak{m}^{+}\,\subset \,\mathfrak{g}$ is the Abelian Lie subalgebra whose elements restrict to constant vector fields on $\mathbb{C}^{n}\,\subset \,\mathbb{P}^{n}$ . The vector subspace $\mathfrak{p}:=\mathfrak{k}^{\mathbb{C}}\oplus \mathfrak{m}^{-}\,\subset \,\mathfrak{g}$ is the parabolic subalgebra at $0\in \mathbb{P}^{n}$ . Writing $M^{+}=\exp (\mathfrak{m}^{+})$ and $M^{-}=\exp (\mathfrak{m}^{-})$ , the mapping $\unicode[STIX]{x1D706}:M^{+}\times K^{\mathbb{C}}\times M^{-}\rightarrow G$ defined by $\unicode[STIX]{x1D706}(m^{+},k,m^{-})=m^{+}km^{-}\in G$ maps $M^{+}\times K^{\mathbb{C}}\times M^{-}$ biholomorphically onto a dense open subset of $G$ . We note that $K^{\mathbb{C}}M^{-}=P$ is a Levi decomposition of $P\,\subset \,G$ , with $K^{\mathbb{C}}\,\subset \,P$ being a Levi factor. By means of $\unicode[STIX]{x1D706}$ , and writing $e\in G$ for the identity element, the orbit of $0=eP\in G/P$ under $M^{+}$ is identified with $M^{+}\cong \mathbb{C}^{n}$ , from which the Harish-Chandra realization $\mathbb{B}^{n}\Subset \mathbb{C}^{n}$ arises as the obvious projection of $\unicode[STIX]{x1D706}^{-1}(G_{0})$ into $M^{+}\cong \mathbb{C}^{n}$ . The description above applies in general to bounded symmetric domains, cf. Wolf [Reference WolfWol72] and Mok [Reference MokMok89].

Let $E$ be a finite-dimensional complex vector space, and $\unicode[STIX]{x1D6F7}:P\rightarrow \mathbb{P}\text{GL}(E)$ be a projective linear representation. Introduce an equivalence relation ${\sim}$ on $G\times \mathbb{P}(E)$ by declaring $(g,e)\sim (g^{\prime },e^{\prime })$ ; $g,g^{\prime }\in G$ , $e,e^{\prime }\in E$ ; if and only if there exists $p\in P$ such that $g^{\prime }=gp^{-1}$ and $e^{\prime }=pe$ , where $pe$ means $\unicode[STIX]{x1D6F7}(p)e$ , and define $G\times _{P}\mathbb{P}(E)=(G\times \mathbb{P}(E))/\sim$ . Denote by $[g,e]$ the equivalence class of $(g,e)$ with respect to ${\sim}$ . The natural map $\unicode[STIX]{x1D70F}:G\times _{P}\mathbb{P}(E)\rightarrow G/P$ defined by $\unicode[STIX]{x1D70F}([g,e])=gP$ realizes $G\times _{P}\mathbb{P}(E)$ as the total space of a projective bundle $\unicode[STIX]{x1D71B}:\mathscr{P}\rightarrow G/P=\mathbb{P}^{n}$ . Left multiplication on $G$ induces a holomorphic action of $G$ on $\mathscr{P}$ compatible with the natural transitive holomorphic action of $G$ on $G/P=\mathbb{P}^{n}$ . By a homogeneous projective bundle on $\mathbb{P}^{n}$ we will always mean $\unicode[STIX]{x1D71B}:\mathscr{P}\rightarrow \mathbb{P}^{n}$ arising this way. We have $K=P\cap G_{0}$ . Defining $G_{0}\times _{K}\mathbb{P}(E)$ analogously we have on $G_{0}\times _{K}\mathbb{P}(E)$ the structure of a smooth projective bundle $\unicode[STIX]{x1D71B}^{\prime }:\mathscr{P}^{\prime }\rightarrow G_{0}/K=\mathbb{B}^{n}$ . The latter is a priori only a smooth projective bundle since $K$ is a real Lie group, but embedding $G_{0}\times _{K}\mathbb{P}(E)$ canonically into $G\times _{P}\mathbb{P}(E)$ one identifies $G_{0}\times _{K}\mathbb{P}(E)$ as an open subset of $G\times _{P}\mathbb{P}(E)$ , and hence $\mathscr{P}^{\prime }$ as the restriction of $\mathscr{P}$ over $\mathbb{B}^{n}\,\subset \,\mathbb{P}^{n}$ , so that $\mathscr{P}^{\prime }=\mathscr{P}|_{\mathbb{B}^{n}}$ inherits the structure of a (holomorphic) projective bundle over $\mathbb{B}^{n}$ . From now on we will identify $\mathscr{P}^{\prime }$ with $\mathscr{P}|_{\mathbb{B}^{n}}$ . The fiber of $\mathscr{P}$ over a point $x\in \mathbb{P}^{n}$ will be denoted by $\mathscr{P}_{x}$ .

Let $\unicode[STIX]{x1D708}:{\mathcal{E}}\rightarrow M$ be a locally trivial holomorphic fiber bundle over a complex manifold $M$ . By a locally trivial holomorphic fiber subbundle ${\mathcal{E}}^{\prime }\,\subset \,{\mathcal{E}}$ we mean a subvariety ${\mathcal{E}}^{\prime }\,\subset \,{\mathcal{E}}$ such that, writing ${\mathcal{E}}_{0}^{\prime }:={\mathcal{E}}_{0}\cap {\mathcal{E}}^{\prime }$ , and shrinking the neighborhoods $U$ if necessary, the trivializations $\unicode[STIX]{x1D711}:\unicode[STIX]{x1D708}^{-1}(U)\overset{\cong }{\longrightarrow }{\mathcal{E}}_{0}\times U$ can be chosen such that $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D708}^{-1}(U)\cap {\mathcal{E}}^{\prime })={\mathcal{E}}_{0}^{\prime }\times U\,\subset \,{\mathcal{E}}_{0}\times U$ , endowed with the canonical projection map $\unicode[STIX]{x1D708}^{\prime }:{\mathcal{E}}^{\prime }\rightarrow M$ given by $\unicode[STIX]{x1D708}^{\prime }=\unicode[STIX]{x1D708}|_{{\mathcal{E}}^{\prime }}$ . The adjective ‘locally trivial’ will be understood and omitted in what follows. For $M=X_{\unicode[STIX]{x1D6E4}}$ a holomorphic fiber (sub)bundle is said to be locally homogeneous if it descends from a homogeneous holomorphic fiber (sub)bundle on $\mathbb{B}^{n}$ under the action of $G_{0}$ .

In order to embed $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6E4}}:\mathscr{U}_{\unicode[STIX]{x1D6E4}}\rightarrow X_{\unicode[STIX]{x1D6E4}}$ into a projective bundle we will embed $\unicode[STIX]{x1D707}:\mathscr{U}\rightarrow \mathbb{P}^{n}$ into a projective bundle $\unicode[STIX]{x1D71B}:\mathscr{P}\rightarrow \mathbb{P}^{n}$ in such a way that $G$ acts canonically on $\mathscr{P}$ . For this we need some basic facts about Chow $(\mathbb{P}^{n})$ . This is provided by the Chow coordinates (Chow and van der Waerden [Reference Chow and van der WaerdenCvdW37]). Fix an integer $r$ , $0<r<n$ , and an integer $d>0$ . For an irreducible and reduced $r$ -cycle $W$ of degree $d$ consider the set $\mathscr{F}$ of $(r+1)$ -tuples $(\unicode[STIX]{x1D6F1}_{1},\ldots ,\unicode[STIX]{x1D6F1}_{r+1})$ , $1\leqslant i\leqslant r+1$ , of hyperplanes $\unicode[STIX]{x1D6F1}_{i}\,\subset \,\mathbb{P}^{n}$ , such that $\unicode[STIX]{x1D6F1}_{1}\cap \cdots \cap \unicode[STIX]{x1D6F1}_{r+1}\cap W\neq \emptyset$ . Writing $\mathbb{P}^{n}=\mathbb{P}(H)$ , $H\cong \mathbb{C}^{n+1}$ , hyperplanes $\unicode[STIX]{x1D6F1}\,\subset \,\mathbb{P}(H)$ are parametrized by the dual projective space $\check{\mathbb{P}}^{n}=\mathbb{P}(H^{\ast })$ . The set $\mathscr{F}\,\subset \,\check{\mathbb{P}}^{n}\times \cdots \times \check{\mathbb{P}}^{n}$ is then a hypersurface and it is the zero set of some $\unicode[STIX]{x1D70E}=\unicode[STIX]{x1D70E}_{W}\in \unicode[STIX]{x1D6E4}(\check{\mathbb{P}}^{n}\times \cdots \times \check{\mathbb{P}}^{n},\unicode[STIX]{x1D716}_{1}^{\ast }{\mathcal{O}}(d)\otimes \cdots \otimes \unicode[STIX]{x1D716}_{r+1}^{\ast }{\mathcal{O}}(d))$ , $\check{\mathbb{P}}^{n}=\mathbb{P}(H^{\ast })$ , where $\unicode[STIX]{x1D716}_{k}:\check{\mathbb{P}}^{n}\times \cdots \times \check{\mathbb{P}}^{n}\rightarrow \check{\mathbb{P}}^{n}$ is the canonical projection onto the $k$ th factor. Writing ${\mathcal{O}}(d,\ldots ,d)$ for $\unicode[STIX]{x1D716}_{1}^{\ast }{\mathcal{O}}(d)\otimes \cdots \otimes \unicode[STIX]{x1D716}_{r+1}^{\ast }{\mathcal{O}}(d)$ , $\unicode[STIX]{x1D70E}$ corresponds to a plurihomogeneous polynomial of multi-degree $(d,\ldots ,d)$ , which is called the Chow form of $[W]$ , uniquely determined up to a nonzero multiplicative scalar. In the general case of $[W]=k_{1}[W_{1}]+\cdots +k_{s}[W_{s}]$ one defines $\unicode[STIX]{x1D70E}_{W}=\unicode[STIX]{x1D70E}_{W_{1}}^{k_{1}}\cdots \unicode[STIX]{x1D70E}_{W_{s}}^{k_{s}}$ , which is again of multi-degree $(d,\ldots ,d)$ . Writing ${\mathcal{Q}}$ for the set of all $r$ -cycles of degree $d$ in $\mathbb{P}^{n}$ , the mapping $\unicode[STIX]{x1D6F9}$ associating $[W]\in {\mathcal{Q}}$ to $[\unicode[STIX]{x1D70E}_{W}]\in \mathbb{P}(J)$ , $J=\unicode[STIX]{x1D6E4}(\check{\mathbb{P}}^{n}\times \cdots \times \check{\mathbb{P}}^{n},{\mathcal{O}}(d,\ldots ,d))^{\ast }$ , is injective, and we have defined the structure of a projective variety on ${\mathcal{Q}}$ by identifying it with $\unicode[STIX]{x1D6F9}({\mathcal{Q}})\,\subset \,\mathbb{P}(J)$ , noting that $G$ acts canonically on $\mathbb{P}(J)$ . Restricting to an irreducible component ${\mathcal{K}}$ of ${\mathcal{Q}}$ we summarize the relevant statements in the following lemma using the notation in the above.

We proceed to embed $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6E4}}:\mathscr{U}_{\unicode[STIX]{x1D6E4}}\rightarrow X_{\unicode[STIX]{x1D6E4}}$ into a projective bundle. We have the following.

In the case where $\unicode[STIX]{x1D6E4}\,\subset \,G_{0}$ is a torsion-free nonuniform lattice, we will study $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6E4}}:\mathscr{U}_{\unicode[STIX]{x1D6E4}}\rightarrow X_{\unicode[STIX]{x1D6E4}}$ by constructing Hermitian metrics with positive curvature (1,1)-form on a homogeneous holomorphic line bundle on $\mathscr{P}_{\unicode[STIX]{x1D6E4}}$ and prove quasi-projectivity by means of techniques of compactification of complete Kähler manifolds of finite volume.

As a preparation we consider the general situation of a representation $\unicode[STIX]{x1D6F7}:P\rightarrow \mathbb{P}\text{GL}(E)$ and the associated homogeneous projective bundle $\unicode[STIX]{x1D71B}:\mathscr{P}\rightarrow \mathbb{P}^{n}$ .

Writing $E:\cong \mathbb{C}^{m+1}$ we have $\mathbb{P}\text{GL}(E)\cong \mathbb{P}\text{GL}(m+1,\mathbb{C})\cong \text{SL}(m+1,\mathbb{C})/Z$ , where $Z=\{\unicode[STIX]{x1D706}I_{m+1}:\unicode[STIX]{x1D706}^{m+1}=1\}.$ Recall that $K\cong U(n)$ is the isotropy subgroup of $(\mathbb{B}^{n},ds_{\mathbb{B}^{n}}^{2})$ at $0\in \mathbb{B}^{n}$ , $K=P\,\cap \,G_{0}$ . Denoting by $\unicode[STIX]{x1D6FC}:\text{SL}(m+1,\mathbb{C})\rightarrow \text{SL}(m+1,\mathbb{C})/Z\cong \mathbb{P}\text{GL}(E)$ the quotient map, let $Q\,\subset \,\text{SL}(m+1,\mathbb{C})$ be the subgroup $\unicode[STIX]{x1D6FC}^{-1}(\unicode[STIX]{x1D6F7}(K))$ . Now let $\unicode[STIX]{x1D701}$ be a Hermitian Euclidean metric on the complex vector space $E$ which is invariant under the compact subgroup $Q\,\subset \,\text{GL}(E)$ . The metric $\unicode[STIX]{x1D701}$ induces on $\mathbb{P}(E)$ a Fubini–Study metric $g_{c}$ with Kähler form $\unicode[STIX]{x1D714}_{c}$ such that, with respect to the canonical projection $\unicode[STIX]{x1D6FD}:E-\{0\}\rightarrow \mathbb{P}(E)$ , $\unicode[STIX]{x1D6FD}^{\ast }(\unicode[STIX]{x1D714}_{c})=\sqrt{-1}\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}\log \Vert w\Vert _{\unicode[STIX]{x1D701}}^{2}$ , where $w=(w_{1},\ldots ,w_{m+1})$ are Euclidean coordinates on $E$ and $\Vert w\Vert _{\unicode[STIX]{x1D701}}$ denotes the norm of $w$ measured in terms of $\unicode[STIX]{x1D701}$ .

Restricting $\unicode[STIX]{x1D6F7}$ to $G_{0}\,\cap \,P=K$ , from $\unicode[STIX]{x1D6F7}|_{K}:K\rightarrow \mathbb{P}\text{GL}(E)$ we obtain $\unicode[STIX]{x1D71B}^{\prime }:\mathscr{P}^{\prime }=G_{0}\times _{K}\mathbb{P}(E)\rightarrow G/K=\mathbb{B}^{n}$ as a homogeneous projective bundle over $\mathbb{B}^{n}$ such that $\mathscr{P}^{\prime }$ is identified with $\mathscr{P}|_{\mathbb{B}^{n}}$ . Since $g_{c}$ is invariant under $Q$ , identifying $\mathbb{P}(E)$ with the fiber $\mathscr{P}_{0}=\unicode[STIX]{x1D71B}^{-1}(0)$ we have correspondingly a Fubini–Study metric $g_{0}$ on $\mathscr{P}_{0}$ . By the $Q$ -invariance of $g_{0}$ , the latter metric can be transported by means of $G_{0}$ -action to fibers $\mathscr{P}_{x}$ over $x\in \mathbb{B}^{n}$ , yielding thereby a $G_{0}$ -invariant Hermitian metric $g$ on the relative tangent bundle $T_{\unicode[STIX]{x1D71B}^{\prime }}$ . Denoting by $L$ the determinant line bundle det $(T_{\unicode[STIX]{x1D71B}^{\prime }})$ and writing $h$ for det $(g)$ , we have a Hermitian holomorphic line bundle $(L,h)$ on $\mathscr{P}^{\prime }$ which is invariant under the $G_{0}$ -action described. The curvature form $\unicode[STIX]{x1D6E9}(L,h)$ , as a closed (1,1)-form on $\mathscr{P}^{\prime }$ , is $G_{0}$ -invariant and positive when restricted to each of the fibers $\mathscr{P}_{x},\,x\in \mathbb{B}^{n}$ . Let $(A,t)$ be a positive homogeneous Hermitian holomorphic line bundle on $\mathbb{B}^{n}$ . By the $G_{0}$ -invariance of $\unicode[STIX]{x1D703}$ and the compactness of the fibers $\mathscr{P}_{x}\cong \mathbb{P}(E)$ , for a sufficiently large integer $k$ , $\unicode[STIX]{x1D703}:=\unicode[STIX]{x1D6E9}(L\otimes ({\unicode[STIX]{x1D71B}^{\prime }}^{\ast }A)^{k},h\otimes ({\unicode[STIX]{x1D71B}^{\prime }}^{\ast }(t))^{k})>0$ . Since $\unicode[STIX]{x1D703}$ is $G_{0}$ -invariant, for a torsion-free lattice $\unicode[STIX]{x1D6E4}\,\subset \,G_{0}$ , $\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6E4}}:\mathscr{P}_{\unicode[STIX]{x1D6E4}}=\mathscr{P}/\unicode[STIX]{x1D6E4}\rightarrow X_{\unicode[STIX]{x1D6E4}}=\mathbb{B}^{n}/\unicode[STIX]{x1D6E4}$ , the Hermitian holomorphic line bundle $(\unicode[STIX]{x1D6EC},s):=(L\otimes ({\unicode[STIX]{x1D71B}^{\prime }}^{\ast }A)^{k},h\otimes ({\unicode[STIX]{x1D71B}^{\prime }}^{\ast }(t))^{k})$ descends to a positive line bundle on $\mathscr{P}_{\unicode[STIX]{x1D6E4}}$ .

We summarize the discussion above to the following result which will be used in the case of finite-volume noncompact quotient manifolds.

We prove now an embedding theorem for $\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6E4}}:\mathscr{P}_{\unicode[STIX]{x1D6E4}}\rightarrow X_{\unicode[STIX]{x1D6E4}}$ .

Proof. By Proposition 2.3, $\mathscr{P}^{\prime }=\mathscr{P}|_{\mathbb{B}^{n}}$ is equipped with a homogeneous Hermitian holomorphic line bundle $(\unicode[STIX]{x1D6EC},s)$ whose curvature form $\unicode[STIX]{x1D703}$ is positive definite. We can now equip $\mathscr{P}^{\prime }$ with the Kähler form $\unicode[STIX]{x1D703}$ , which descends to a Kähler form $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D6E4}}$ on $\mathscr{P}_{\unicode[STIX]{x1D6E4}}$ . From the definition we have

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703} & := & \displaystyle \unicode[STIX]{x1D6E9}(L\otimes ({\unicode[STIX]{x1D71B}^{\prime }}^{\ast }A)^{k},h\otimes ({\unicode[STIX]{x1D71B}^{\prime }}^{\ast }(t))^{k})=\unicode[STIX]{x1D6E9}(L,h)+k{\unicode[STIX]{x1D71B}^{\prime }}^{\ast }(\unicode[STIX]{x1D6E9}(K_{\mathbb{B}^{n}},t))\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D6E9}(L,h)+k(n+1){\unicode[STIX]{x1D71B}^{\prime }}^{\ast }(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D6E4}})>0\,.\nonumber\end{eqnarray}$$

Recall that $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D6E4}}$ is the Kähler form of the complete Kähler metric on $X_{\unicode[STIX]{x1D6E4}}$ with constant holomorphic sectional curvature $-2$ . Replacing $k$ by $k+1$ we may assume furthermore that $\unicode[STIX]{x1D703}>(n+1){\unicode[STIX]{x1D71B}^{\prime }}^{\ast }(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D6E4}})>{\unicode[STIX]{x1D71B}^{\prime }}^{\ast }(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D6E4}})$ . It follows that the length of any smooth curve $\unicode[STIX]{x1D6FE}:[0,1]\rightarrow \mathscr{P}_{\unicode[STIX]{x1D6E4}}$ with respect to $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D6E4}}$ dominates the length of the smooth curve $\unicode[STIX]{x1D6FE}\,\circ \,\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6E4}}:[0,1]\rightarrow X_{\unicode[STIX]{x1D6E4}}$ with respect to $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D6E4}}$ . Since $(X_{\unicode[STIX]{x1D6E4}},\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D6E4}})$ is a complete Kähler manifold, we conclude that $(\mathscr{P}_{\unicode[STIX]{x1D6E4}},\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D6E4}})$ is also a complete Kähler manifold.

We also equip $(\mathscr{P}_{\unicode[STIX]{x1D6E4}},\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D6E4}})$ with the Hermitian–Einstein positive line bundle $(\unicode[STIX]{x1D6EC}_{\unicode[STIX]{x1D6E4}},s_{\unicode[STIX]{x1D6E4}})$ . Since the Kähler form $\unicode[STIX]{x1D703}$ on $\mathscr{P}^{\prime }$ is $G_{0}$ -invariant, for any open subset ${\mathcal{O}}\Subset \mathbb{B}^{n}$ and any $\unicode[STIX]{x1D711}\in G_{0}$ , Volume $(\unicode[STIX]{x1D71B}^{-1}({\mathcal{O}}),\unicode[STIX]{x1D703})=\text{Volume}(\unicode[STIX]{x1D71B}^{-1}(\unicode[STIX]{x1D711}({\mathcal{O}})),\unicode[STIX]{x1D703})<\infty$ . Recall that $\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6E4}}:\mathscr{P}_{\unicode[STIX]{x1D6E4}}\rightarrow X_{\unicode[STIX]{x1D6E4}}$ is of fiber dimension $s$ . Since both (1,1)-forms $\unicode[STIX]{x1D703}$ and $\unicode[STIX]{x1D71B}^{\ast }(\unicode[STIX]{x1D714})$ are invariant under $G_{0}$ , both $\unicode[STIX]{x1D703}^{s+n}$ and $\unicode[STIX]{x1D703}^{s}\wedge (\unicode[STIX]{x1D71B}^{\ast }(\unicode[STIX]{x1D714}))^{n}$ are $G_{0}$ -invariant volume forms on $\mathscr{P}^{\prime }$ , and the function $h:\mathscr{P}^{\prime }\rightarrow \mathbb{R}$ defined by $\unicode[STIX]{x1D703}^{s+n}(p)=h(p)\unicode[STIX]{x1D703}^{s}\wedge (\unicode[STIX]{x1D71B}^{\ast }(\unicode[STIX]{x1D714}))^{n}$ is a $G_{0}$ -invariant function on $\mathscr{P}^{\prime }$ . Since any point $p\in \mathscr{P}^{\prime }$ is equivalent under the action of $G_{0}$ to some point on $\mathscr{P}_{0}$ , which is compact, we observe that there exists a constant $C_{0}>0$ such that $0<h(p)\leqslant C_{0}$ for any $p\in \mathscr{P}^{\prime }$ . For any open subset ${\mathcal{O}}\,\subset \,X_{\unicode[STIX]{x1D6E4}}$ , by Fubini’s theorem we have

$$\begin{eqnarray}\displaystyle \int _{\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6E4}}^{-1}({\mathcal{O}})}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D6E4}}^{s}\wedge (\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6E4}}^{\ast }(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D6E4}}))^{n} & = & \displaystyle \int _{\mathscr{P}_{0}}\unicode[STIX]{x1D703}^{s}\times \int _{{\mathcal{O}}}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D6E4}}^{n}\,;\nonumber\\ \displaystyle \text{Volume}(\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6E4}}^{-1}({\mathcal{O}}),\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D6E4}}) & = & \displaystyle {\displaystyle \frac{1}{(s+n)!}}\displaystyle \int _{\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6E4}}^{-1}({\mathcal{O}})}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D6E4}}^{s+n}\leqslant {\displaystyle \frac{C_{0}}{(s+n)!}}\int _{\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6E4}}^{-1}({\mathcal{O}})}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D6E4}}^{s}\wedge (\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6E4}}^{\ast }(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D6E4}}))^{n}\nonumber\\ \displaystyle & = & \displaystyle {\displaystyle \frac{n!}{(s+n)!}}\cdot C_{0}\displaystyle \int _{\mathscr{P}_{0}}\unicode[STIX]{x1D703}^{s}\times \displaystyle \int _{{\mathcal{O}}}{\displaystyle \frac{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D6E4}}^{n}}{n!}}=C\cdot \text{Volume}({\mathcal{O}},\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D6E4}}),\nonumber\end{eqnarray}$$

where $C=n!/(s+n)!\cdot C_{0}\displaystyle \int _{\mathscr{P}_{0}}\unicode[STIX]{x1D703}^{s}$ . In particular,

$$\begin{eqnarray}\text{Volume}(\mathscr{P}_{\unicode[STIX]{x1D6E4}},\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D6E4}})\leqslant C\cdot \text{Volume}(X_{\unicode[STIX]{x1D6E4}},\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D6E4}})<\infty .\end{eqnarray}$$

From the $G_{0}$ -invariance of $(\mathscr{P}^{\prime },\unicode[STIX]{x1D703})$ and the compactness of the fibers of $\unicode[STIX]{x1D71B}^{\prime }:\mathscr{P}^{\prime }\rightarrow \mathbb{B}^{n}$ , it follows that $(\mathscr{P}^{\prime },\unicode[STIX]{x1D703})$ is of bounded sectional curvature. Since $X_{\unicode[STIX]{x1D6E4}}$ is a quasi-projective manifold (as in Theorem 1.1), it is of finite topological type. Hence, $\mathscr{P}_{\unicode[STIX]{x1D6E4}}$ is also of finite topological type. As a consequence, one can apply the embedding result on complete Kähler manifolds of finite volume given by Theorem 1.3 here (from Mok and Zhong [Reference Mok and ZhongMZ89]) to complete the proof of the proposition.◻

Without loss of generality we may assume $V$ to be normal, and we also write $\mathscr{P}_{\unicode[STIX]{x1D6E4}}\,\subset \,\overline{\mathscr{P}_{\unicode[STIX]{x1D6E4}}}:=V$ for the compactification of $\mathscr{P}_{\unicode[STIX]{x1D6E4}}$ as a normal projective variety. Since $\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6E4}}:\mathscr{P}_{\unicode[STIX]{x1D6E4}}\rightarrow X_{\unicode[STIX]{x1D6E4}}$ is a proper holomorphic map, by the Riemann extension theorem and the normality of $\overline{\mathscr{P}_{\unicode[STIX]{x1D6E4}}}$ , it extends holomorphically to $\overline{\mathscr{P}_{\unicode[STIX]{x1D6E4}}}$ , to be denoted as $\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6E4}}^{\sharp }:\overline{\mathscr{P}_{\unicode[STIX]{x1D6E4}}}\rightarrow \overline{X_{\unicode[STIX]{x1D6E4}}}$ .

We now return to our study of the locally homogeneous holomorphic fiber subbundle $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6E4}}:\mathscr{U}_{\unicode[STIX]{x1D6E4}}\rightarrow X_{\unicode[STIX]{x1D6E4}}$ of $\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6E4}}:\mathscr{P}_{\unicode[STIX]{x1D6E4}}\rightarrow X_{\unicode[STIX]{x1D6E4}}$ . We have the following result on $\mathscr{U}_{\unicode[STIX]{x1D6E4}}\subset \mathscr{P}_{\unicode[STIX]{x1D6E4}}$ .

3 Uniruling on Zariski closures of images of algebraic subsets under the uniformization map

Let $\unicode[STIX]{x1D6E4}\,\subset \,G_{0}$ be a torsion-free lattice, $X_{\unicode[STIX]{x1D6E4}}:=\mathbb{B}^{n}/\unicode[STIX]{x1D6E4}$ , $n\geqslant 2$ , $\overline{X_{\unicode[STIX]{x1D6E4}}}$ be its minimal compactification, and $\unicode[STIX]{x1D70B}:\mathbb{B}^{n}\rightarrow X_{\unicode[STIX]{x1D6E4}}$ be the universal covering map. We consider an irreducible algebraic subset $S\,\subset \,\mathbb{B}^{n}$ of positive dimension and define $Z\,\subset \,X_{\unicode[STIX]{x1D6E4}}$ to be the Zariski closure of $\unicode[STIX]{x1D70B}(S)$ in $X_{\unicode[STIX]{x1D6E4}}$ . To characterize $Z$ we will resort to studying meromorphic foliations on holomorphic fiber bundles over $X_{\unicode[STIX]{x1D6E4}}$ and on compactifications of total spaces of such fiber bundles. We start with some generalities about complex spaces (cf. Grauert, Peternell and Remmert [Reference Grauert, Peternell and RemmertGPR94, p. 100ff]) and about meromorphic foliations on them.

Let $(Y,{\mathcal{O}}_{Y})$ be a reduced irreducible complex space, assumed to be embedded as a subvariety of a complex manifold $M$ . Let $\unicode[STIX]{x1D6FA}_{M}={\mathcal{O}}(T_{M}^{\ast })$ be the cotangent sheaf on $M$ , and $\mathscr{I}_{Y}\,\subset \,{\mathcal{O}}_{M}$ be the ideal sheaf of $Y\,\subset \,M$ . Define now $\mathscr{S}\,\subset \,\unicode[STIX]{x1D6FA}_{M}$ to be the subsheaf spanned by $\mathscr{I}_{Y}\unicode[STIX]{x1D6FA}_{M}$ and $\{df:f\in \mathscr{I}_{Y}\}$ . Then, $\mathscr{S}\,\subset \,\unicode[STIX]{x1D6FA}_{M}$ is a coherent subsheaf and $\unicode[STIX]{x1D6FA}_{Y}:=\unicode[STIX]{x1D6FA}_{M}/\mathscr{S}$ is called the cotangent sheaf of $Y$ . The tangent sheaf ${\mathcal{T}}_{Y}$ is by definition the coherent sheaf $\mathscr{H}om_{{\mathcal{O}}_{Y}}(\unicode[STIX]{x1D6FA}_{Y},{\mathcal{O}}_{Y})$ , which is naturally identified with a coherent subsheaf of ${\mathcal{T}}_{M}|_{Y}$ . The tangent sheaf ${\mathcal{T}}_{Y}$ on $Y$ thus defined is unique up to isomorphisms independent of the embedding $Y\,\subset \,M$ . When the assumption $Y\,\subset \,M$ is dropped, the tangent sheaf ${\mathcal{T}}_{Y}$ is defined using an atlas $\mathscr{U}=\{{U_{\unicode[STIX]{x1D6FC}}\}}_{\unicode[STIX]{x1D6FC}\in A}$ on $Y$ consisting of subvarieties of coordinate open sets by gluing together the tangent sheaves of $U_{\unicode[STIX]{x1D6FC}}$ .

A meromorphic foliation $\mathscr{F}$ on a reduced irreducible complex space $Y$ is by definition given by a coherent subsheaf ${\mathcal{T}}_{\mathscr{F}}$ of the tangent sheaf ${\mathcal{T}}_{Y}$ such that, outside some subvariety $A\,\subsetneq \,Y$ , $A\supset \text{Sing}(Y)$ , ${\mathcal{T}}_{\mathscr{F}}|_{Y-A}$ is a locally free subsheaf and, writing ${\mathcal{T}}_{\mathscr{F}}|_{Y-A}:={\mathcal{O}}(F)$ for a holomorphic distribution $F\,\subset \,T_{Y-A}$ , $F$ satisfies the involutive property $[F,F]=F$ . The tangent subsheaf of the meromorphic foliation ${\mathcal{T}}_{\mathscr{F}}\,\subset \,{\mathcal{T}}_{Y}$ is also uniquely determined subject to the conditions that ${\mathcal{T}}_{\mathscr{F}}$ agrees with ${\mathcal{O}}(F)$ over $Y-A$ and that ${\mathcal{T}}_{Y}/{\mathcal{T}}_{\mathscr{F}}$ is torsion-free, in which case ${\mathcal{T}}_{\mathscr{F}}\,\subset \,{\mathcal{T}}_{Y}$ is said to be saturated. In what follows, ${\mathcal{T}}_{\mathscr{F}}\,\subset \,{\mathcal{T}}_{Y}$ is always assumed saturated. The singular locus $\text{Sing}(\mathscr{F})\,\subset \,Y$ is the union of Sing $(Y)$ and the locus on Reg $(Y)$ over which $\mathscr{F}$ fails to be a locally free subsheaf. The subset $\text{Sing}(\mathscr{F})\subsetneq Y$ is a subvariety, and we write $\text{Reg}(\mathscr{F}):=Y-\text{Sing}(\mathscr{F})$ . Given an irreducible complex-analytic subvariety $Z\,\subset \,Y$ such that $Z\,\cap \,\text{Reg}(\mathscr{F})\neq \emptyset$ , we say that $Z$ is saturated with respect to $\mathscr{F}$ to mean that for any point $z_{0}\in \text{Reg}(Z)\,\cap \,\text{Reg}(\mathscr{F})\neq \emptyset$ , the leaf ${\mathcal{L}}(z_{0})$ of $\mathscr{F}$ passing through $z_{0}$ must necessarily lie on $Z$ . When $Y$ is projective, we have the following result concerning Zariski closures of leaves of $\mathscr{F}$ .

Let $M$ be a complex manifold, $\unicode[STIX]{x1D70B}:\widetilde{M}\rightarrow M$ be a covering map, and $Y\,\subset \,M$ be an irreducible subvariety. Let ${\mathcal{H}}$ be a reduced irreducible complex space and ${\mathcal{R}}\,\subset \,{\mathcal{H}}\times \widetilde{M}$ be a subvariety such that the canonical projection $\unicode[STIX]{x1D70E}:{\mathcal{R}}\rightarrow {\mathcal{H}}$ is surjective and the fibers $\unicode[STIX]{x1D70E}^{-1}(t)=:\{t\}\times R_{t}$ are equidimensional. Denote by $\unicode[STIX]{x1D708}:{\mathcal{H}}\times \widetilde{M}\rightarrow \widetilde{M}$ the canonical projection. We have the following notion of uniruling of subvarieties $Y\,\subset \,M$ . In what follows openness and denseness of subsets are defined in terms of the complex topology.

We will apply the notion of uniruling to the case where $M=\mathbb{B}^{n}/\unicode[STIX]{x1D6E4}=X_{\unicode[STIX]{x1D6E4}}$ is a complex ball quotient by a torsion-free lattice $\unicode[STIX]{x1D6E4}\,\subset \,\text{Aut}(\mathbb{B}^{n})$ equipped with the quasi-projective structure given by $X_{\unicode[STIX]{x1D6E4}}\,\subset \,\overline{X_{\unicode[STIX]{x1D6E4}}}$ , ${\mathcal{H}}$ is some $G$ -invariant irreducible subvariety of Chow $(\mathbb{P}^{n})$ , and where $Y\,\subset \,X_{\unicode[STIX]{x1D6E4}}$ is an algebraic subvariety. We start with some preparation relating meromorphic foliations to meromorphic sections of Grassmann bundles.

Let $M$ be a complex manifold, $Y\,\subset \,M$ be an irreducible subvariety, and ${\mathcal{T}}_{Y}\,\subset \,{\mathcal{T}}_{M}|_{Y}$ be the tangent sheaf of $Y$ . Let ${\mathcal{T}}_{\mathscr{F}}\,\subset \,{\mathcal{T}}_{Y}$ be the tangent sheaf of a meromorphic foliation $\mathscr{F}$ of leaf dimension $r$ , $0<r<\dim (M)$ , defined on $Y$ . Write $B:=\text{Sing}(\mathscr{F})$ , and write $F\,\subset \,T_{Y-B}$ for the involutive holomorphic distribution such that $T_{\mathscr{F}}={\mathcal{O}}(F)$ . Define the holomorphic section $\unicode[STIX]{x1D711}:Y-B\rightarrow \text{Gr}(r,T_{M})$ by setting $\unicode[STIX]{x1D711}(y)=[F_{y}]\in \text{Gr}(r,T_{Y,y})\,\subset \,\text{Gr}(r,T_{M,y})$ . Since $\mathscr{F}$ is locally finitely generated, at every point $y_{0}\in Y$ , there exist $r$ holomorphic sections $\unicode[STIX]{x1D712}_{1},\ldots ,\unicode[STIX]{x1D712}_{r}\in \unicode[STIX]{x1D6E4}(U,{\mathcal{T}}|_{{\mathcal{F}}})$ defined on some neighborhood $U$ of $y_{0}$ , such that $\unicode[STIX]{x1D712}_{1}(y^{\prime }),\ldots ,\unicode[STIX]{x1D712}_{r}(y^{\prime })$ span ${\mathcal{T}}_{\mathscr{F},y^{\prime }}$ at a general point $y^{\prime }\in U-B$ . Then, by using Plücker coordinates, the formula $\unicode[STIX]{x1D6F7}(y)=[\unicode[STIX]{x1D712}_{1}(y)\wedge \cdots \wedge \unicode[STIX]{x1D712}_{r}(y)]\in \text{Gr}(r,T_{M,y})\,\subset \,\mathbb{P}(\unicode[STIX]{x1D6EC}^{r}T_{M,y})$ determines a unique meromorphic section over $U$ of $\text{Gr}(r,T_{M})$ extending $\unicode[STIX]{x1D711}|_{U-B}$ . From uniqueness it follows that $\unicode[STIX]{x1D711}$ extends meromorphically to $\unicode[STIX]{x1D711}^{\flat }:Y{\dashrightarrow}\text{Gr}(r,T_{M})$ . We call $\unicode[STIX]{x1D711}^{\flat }$ the meromorphic section of $\text{Gr}(r,T_{M})$ associated to $(Y,\mathscr{F})$ . We have the following lemma on meromorphic extensions of foliations.

We return now to the study of Zariski closures of images of algebraic subsets $S\,\subset \,\mathbb{B}^{n}$ , $n\geqslant 2$ , under the universal covering map $\unicode[STIX]{x1D70B}:\mathbb{B}^{n}\rightarrow X_{\unicode[STIX]{x1D6E4}}$ for a torsion-free lattice $\unicode[STIX]{x1D6E4}\,\subset \,G_{0}$ , $X_{\unicode[STIX]{x1D6E4}}:=\mathbb{B}^{n}$ . When $\unicode[STIX]{x1D6E4}\,\subset \,G_{0}$ is not cocompact, we embed $X_{\unicode[STIX]{x1D6E4}}$ into its minimal compactification $\overline{X_{\unicode[STIX]{x1D6E4}}}$ by adding a finite number of normal isolated singularities. Recall that $\overline{X_{\unicode[STIX]{x1D6E4}}}$ is projective and $X_{\unicode[STIX]{x1D6E4}}\,\subset \,\overline{X_{\unicode[STIX]{x1D6E4}}}$ inherits a canonical quasi-projective structure.

Recall that the irreducible algebraic subset $S\,\subset \,\mathbb{B}^{n}$ in the statement of Main Theorem is an irreducible component of $W_{0}\,\cap \,\mathbb{B}^{n}$ for some irreducible projective subvariety $W_{0}\,\subset \,\mathbb{P}^{n}$ . Consider the universal family $\unicode[STIX]{x1D70C}:\mathscr{U}\rightarrow {\mathcal{K}}$ , where ${\mathcal{K}}$ is an irreducible component of the Chow space of $\mathbb{P}^{n}$ containing $[W_{0}]$ . By means of the stratification given by singular loci we are going to define a $G$ -invariant irreducible subvariety ${\mathcal{H}}\,\subset \,{\mathcal{K}}$ , $[W_{0}]\in {\mathcal{H}}$ , such that $[W_{0}]$ is a smooth point of ${\mathcal{H}}$ , and consider the restriction over ${\mathcal{H}}$ of the universal family $\unicode[STIX]{x1D70C}:\mathscr{U}\rightarrow {\mathcal{K}}$ , as follows. Write ${\mathcal{K}}_{1}:={\mathcal{K}}$ , and consider the Zariski open subset ${\mathcal{K}}_{1}^{0}\,\subset \,{\mathcal{K}}_{1}$ consisting of smooth points $[F]\in {\mathcal{K}}$ such that $F\,\subset \,\mathbb{P}^{n}$ is irreducible and reduced. In case $[W_{0}]\in {\mathcal{K}}_{1}^{0}$ we define ${\mathcal{H}}={\mathcal{K}}_{1}={\mathcal{K}}$ . Otherwise $[W_{0}]\in {\mathcal{K}}_{1}-{\mathcal{K}}_{1}^{0}$ . Now let ${\mathcal{K}}_{2}\subsetneq {\mathcal{K}}_{1}$ be an irreducible component of ${\mathcal{K}}_{1}-{\mathcal{K}}_{1}^{0}$ containing the point $[W_{0}]$ . Since $G=\mathbb{P}\text{GL}(n+1,\mathbb{C})$ is connected, the irreducible component ${\mathcal{K}}_{2}$ of the $G$ -invariant subvariety ${\mathcal{K}}_{1}-{\mathcal{K}}_{1}^{0}$ is also $G$ -invariant. Define now ${\mathcal{K}}_{2}^{0}\,\subset \,{\mathcal{K}}_{2}$ to consist of all $[F]\in {\mathcal{K}}_{2}$ such that $F$ is irreducible and reduced. Iterating the process, we obtain a finite sequence of $G$ -invariant irreducible subvarieties ${\mathcal{K}}_{m}\subsetneq \cdots \subsetneq {\mathcal{K}}_{1}={\mathcal{K}}\,\subset \,\text{Chow}(\mathbb{P}^{n})$ such that $[W_{0}]$ is a smooth point on ${\mathcal{K}}_{m}$ . We define now ${\mathcal{H}}={\mathcal{K}}_{m}$ and $\mathscr{V}:=\unicode[STIX]{x1D70C}^{-1}({\mathcal{K}}_{m})$ . Clearly ${\mathcal{H}}$ is also $G$ -invariant. We denote by $\unicode[STIX]{x1D70E}:\mathscr{V}\rightarrow {\mathcal{H}}$ the restriction of the universal family $\unicode[STIX]{x1D70C}:\mathscr{U}\rightarrow {\mathcal{K}}$ over ${\mathcal{H}}\,\subset \,{\mathcal{K}}$ .

A torsion-free lattice $\unicode[STIX]{x1D6E4}\,\subset \,G_{0}$ acts properly discontinuously on $\mathscr{V}^{\prime }=\mathscr{V}|_{\mathbb{B}^{n}}$ without fixed points, and we have a quotient space $\mathscr{V}_{\unicode[STIX]{x1D6E4}}:=\mathscr{V}^{\prime }/\unicode[STIX]{x1D6E4}$ equipped with a map $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D6E4}}:\mathscr{V}_{\unicode[STIX]{x1D6E4}}\rightarrow X_{\unicode[STIX]{x1D6E4}}=\mathbb{B}^{n}/\unicode[STIX]{x1D6E4}$ , $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D6E4}}=\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6E4}}|_{\mathscr{V}_{\unicode[STIX]{x1D6E4}}}$ , realizing $\mathscr{V}_{\unicode[STIX]{x1D6E4}}$ as a locally homogeneous holomorphic fiber subbundle of $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6E4}}:\mathscr{U}_{\unicode[STIX]{x1D6E4}}\rightarrow X_{\unicode[STIX]{x1D6E4}}$ . The foliation $\mathscr{F}$ on $\mathscr{U}$ restricts to $\mathscr{V}=\unicode[STIX]{x1D70C}^{-1}({\mathcal{H}})$ , and we will use the same notation $\mathscr{F}$ for the restriction to $\mathscr{V}$ . The foliation $\mathscr{F}$ descends to a foliation $\mathscr{F}_{\unicode[STIX]{x1D6E4}}$ on $\mathscr{V}_{\unicode[STIX]{x1D6E4}}$ .

In the case of torsion-free nonuniform lattices $\unicode[STIX]{x1D6E4}\,\subset \,G_{0}$ we will be dealing with quasi-projective varieties $X_{\unicode[STIX]{x1D6E4}}$ and we have to consider their minimal compactifications $X_{\unicode[STIX]{x1D6E4}}\,\subset \,\overline{X_{\unicode[STIX]{x1D6E4}}}$ together with projective compactifications of various holomorphic fiber bundles over $X_{\unicode[STIX]{x1D6E4}}$ . Recall that $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6E4}}:\mathscr{U}_{\unicode[STIX]{x1D6E4}}\rightarrow X_{\unicode[STIX]{x1D6E4}}$ is embedded as a locally homogeneous holomorphic fiber subbundle into $\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6E4}}:\mathscr{P}_{\unicode[STIX]{x1D6E4}}\rightarrow X_{\unicode[STIX]{x1D6E4}}$ , $\mathscr{P}_{\unicode[STIX]{x1D6E4}}$ is quasi-projective by Proposition 2.4 and $\mathscr{U}_{\unicode[STIX]{x1D6E4}}\,\subset \,\mathscr{P}_{\unicode[STIX]{x1D6E4}}$ is quasi-projective by Proposition 2.5. In fact, identifying $\mathscr{U}_{\unicode[STIX]{x1D6E4}}$ as a subset of $\mathscr{P}_{\unicode[STIX]{x1D6E4}}$ , compactifying $\mathscr{P}_{\unicode[STIX]{x1D6E4}}$ to a projective variety $\overline{\mathscr{P}_{\unicode[STIX]{x1D6E4}}}\,\subset \,\mathbb{P}^{\ell }$ and extending the canonical projection $\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6E4}}:\mathscr{P}_{\unicode[STIX]{x1D6E4}}\rightarrow X_{\unicode[STIX]{x1D6E4}}$ to a holomorphic map $\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6E4}}^{\sharp }:\overline{\mathscr{P}_{\unicode[STIX]{x1D6E4}}}\rightarrow \overline{X_{\unicode[STIX]{x1D6E4}}}$ , the topological closure $\overline{\mathscr{U}_{\unicode[STIX]{x1D6E4}}}\,\subset \,\overline{\mathscr{P}_{\unicode[STIX]{x1D6E4}}}\,\subset \,\mathbb{P}^{\ell }$ is a projective subvariety. The same proof as in Proposition 2.5 shows that for the holomorphic fiber bundle $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D6E4}}:\mathscr{V}_{\unicode[STIX]{x1D6E4}}\rightarrow X_{\unicode[STIX]{x1D6E4}}$ , the topological closure $\overline{\mathscr{V}_{\unicode[STIX]{x1D6E4}}}\,\subset \,\overline{\mathscr{U}_{\unicode[STIX]{x1D6E4}}}\,\subset \,\overline{\mathscr{P}_{\unicode[STIX]{x1D6E4}}}\,\subset \,\mathbb{P}^{\ell }$ is a projective subvariety, and we have $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D6E4}}^{\sharp }:\overline{\mathscr{V}_{\unicode[STIX]{x1D6E4}}}\rightarrow \overline{X_{\unicode[STIX]{x1D6E4}}}$ , $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D6E4}}^{\sharp }:=\unicode[STIX]{x1D71B}_{\unicode[STIX]{x1D6E4}}^{\sharp }|_{\overline{\mathscr{V}_{\unicode[STIX]{x1D6E4}}}}$ . We need now to extend $\mathscr{F}_{\unicode[STIX]{x1D6E4}}$ on $\mathscr{V}_{\unicode[STIX]{x1D6E4}}$ to a meromorphic foliation $\mathscr{F}_{\unicode[STIX]{x1D6E4}}^{\sharp }$ on $\overline{\mathscr{V}_{\unicode[STIX]{x1D6E4}}}$ . Denoting the normalization of a reduced irreducible complex space $Y$ by $Y^{\mathfrak{n}}$ , $\mathscr{F}_{\unicode[STIX]{x1D6E4}}$ induces a meromorphic foliation $\mathscr{F}_{\unicode[STIX]{x1D6E4}}^{\mathfrak{n}}$ on $\mathscr{V}_{\unicode[STIX]{x1D6E4}}^{\mathfrak{n}}$ , and the aforementioned extension problem on $\mathscr{F}_{\unicode[STIX]{x1D6E4}}$ is equivalently the problem of extending $\mathscr{F}_{\unicode[STIX]{x1D6E4}}^{\mathfrak{n}}$ meromorphically from $\mathscr{V}_{\unicode[STIX]{x1D6E4}}^{\mathfrak{n}}$ to $\overline{\mathscr{V}_{\unicode[STIX]{x1D6E4}}}^{\mathfrak{n}}$ , hence we may apply Lemma 3.1. For the purpose of proving the meromorphic extension of $\mathscr{F}_{\unicode[STIX]{x1D6E4}}^{\mathfrak{n}}$ to $\overline{\mathscr{V}_{\unicode[STIX]{x1D6E4}}}^{\mathfrak{n}}$ we will need the following Hartogs-type extension theorem for meromorphic functions in Mok and Zhang [Reference Mok and ZhangMZ19, Lemma 7.4].

We deduce from Lemma 3.2 the following extension theorem for meromorphic foliations. We will apply the setting of Lemma 3.1 to the case where $\unicode[STIX]{x1D6FA}=B$ and where $B$ and ${\mathcal{X}}$ are normal.

We conclude this section with the following result on $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D6E4}}:\mathscr{V}_{\unicode[STIX]{x1D6E4}}\rightarrow X_{\unicode[STIX]{x1D6E4}}$ and on the compactification $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D6E4}}^{\sharp }:\overline{\mathscr{V}_{\unicode[STIX]{x1D6E4}}}\rightarrow \overline{X_{\unicode[STIX]{x1D6E4}}}$ for torsion-free lattices $\unicode[STIX]{x1D6E4}\,\subset \,G_{0}$ .

The uniruling of $Z$ by subvarieties belonging to ${\mathcal{H}}$ is the key statement in Proposition 3.2 which will allow us to establish Main Theorem in § 4 by analytic methods.

4 Proof of Main Theorem and generalizations

Recall that $\mathscr{S}\,\subset \,\mathscr{V}$ is the tautological lifting of $S$ to the total space $\mathscr{V}$ of the $G$ -invariant subfamily $\unicode[STIX]{x1D70E}:\mathscr{V}\rightarrow {\mathcal{H}}$ of the universal family $\unicode[STIX]{x1D70C}:\mathscr{U}\rightarrow {\mathcal{K}}$ , and that $\mathscr{Z}:=\overline{\widetilde{\unicode[STIX]{x1D70B}}(\mathscr{S})}^{\mathscr{Z}ar}\,\subset \,\mathscr{V}_{\unicode[STIX]{x1D6E4}}$ . By Proposition 3.1, $\mathscr{Z}$ is saturated with respect to the tautological foliation $\mathscr{F}$ on $\mathscr{V}_{\unicode[STIX]{x1D6E4}}$ . By Proposition 3.2, $Z=\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D6E4}}(\mathscr{Z})$ is uniruled by subvarieties belonging to ${\mathcal{H}}$ in the sense of Definition 3.1. Consider the germ at some point $b\in \unicode[STIX]{x2202}\widetilde{Z}$ of a complex submanifold $\unicode[STIX]{x1D6F4}$ which is the union of nonempty open subsets of a certain family of subvarieties $W_{\unicode[STIX]{x1D702}}\,\subset \,\mathbb{P}^{n}$ belonging to ${\mathcal{H}}$ and which exits the boundary of $\mathbb{B}^{n}$ near $b\in \unicode[STIX]{x2202}\widetilde{Z}\,\subset \,\unicode[STIX]{x2202}\mathbb{B}^{n}$ . We are going to prove that $Z\,\subset \,X_{\unicode[STIX]{x1D6E4}}$ is a totally geodesic subset by exploiting the asymptotic total geodesy of $(\unicode[STIX]{x1D6F4}\,\cap \,\widetilde{Z},g|_{\unicode[STIX]{x1D6F4}\,\cap \,\widetilde{Z}})$ at $b\in \unicode[STIX]{x2202}\mathbb{B}^{n}\,\cap \,\unicode[STIX]{x1D6F4}$ using asymptotic curvature properties resulting from Klembeck [Reference KlembeckKle78] (Theorem 1.5 here).

By Proposition 3.2, $\mathscr{Z}$ is uniruled by subvarieties belonging to ${\mathcal{H}}$ . In the notation there consider now the mapping $\unicode[STIX]{x1D6FD}:=\unicode[STIX]{x1D708}|_{\unicode[STIX]{x1D70E}^{-1}({\mathcal{S}})}:\unicode[STIX]{x1D70E}^{-1}({\mathcal{S}})\rightarrow \mathbb{P}^{n}$ . Note that dim $(\unicode[STIX]{x1D70E}^{-1}({\mathcal{S}}))=(d-r)+r=d$ and that the image $\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D70E}^{-1}({\mathcal{S}}))$ contains the open set $\widetilde{{\mathcal{O}}}\,\subset \,\widetilde{Z}$ . Since by construction $\unicode[STIX]{x1D6FD}$ is a local biholomorphism into $\widetilde{Z}$ at $\widetilde{v}\in \widetilde{Z}$ , $\unicode[STIX]{x1D6FD}$ must necessarily be an immersion into $\mathbb{P}^{n}$ at a general smooth point of $\unicode[STIX]{x1D70E}^{-1}({\mathcal{S}})$ .

Observe that singularities of $\unicode[STIX]{x1D70E}^{-1}({\mathcal{S}})$ are of complex codimension ${\geqslant}1$ while $\unicode[STIX]{x1D6FD}^{-1}(\unicode[STIX]{x2202}\mathbb{B}^{n})=\unicode[STIX]{x1D708}^{-1}(\unicode[STIX]{x2202}\mathbb{B}^{n})\,\cap \,\unicode[STIX]{x1D70E}^{-1}({\mathcal{S}})$ is a real hypersurface in $\unicode[STIX]{x1D70E}^{-1}({\mathcal{S}})$ . It follows that, for a general point $\widehat{b}\in \unicode[STIX]{x1D6FD}^{-1}(\unicode[STIX]{x2202}\mathbb{B}^{n})$ , $\widehat{b}$ is a smooth point of $\unicode[STIX]{x1D70E}^{-1}({\mathcal{S}})$ and $\unicode[STIX]{x1D6FD}$ is an immersion at $\widehat{b}$ . Thus, for some open neighborhood ${\mathcal{W}}$ of $\widehat{b}$ in $\unicode[STIX]{x1D70E}^{-1}({\mathcal{S}})$ , $\unicode[STIX]{x1D6FD}|_{{\mathcal{W}}}:{\mathcal{W}}\overset{\cong }{\longrightarrow }\unicode[STIX]{x1D6F4}\,\subset \,\mathbb{C}^{n}$ maps ${\mathcal{W}}$ biholomorphically onto some complex submanifold $\unicode[STIX]{x1D6F4}\,\subset \,U$ of some open set $U\,\subset \,\mathbb{C}^{n}$ , $b\in U$ , such that $\unicode[STIX]{x1D6F4}\,\cap \,\mathbb{B}^{n}$ is a nonempty open subset of $\widetilde{Z}$ . One may say that $\widetilde{Z}$ is analytically continued by grafting $\unicode[STIX]{x1D6F4}$ to $\widetilde{Z}$ at $\widehat{b}$ . For convenience we may assume that both $\unicode[STIX]{x1D6F4}$ and $\unicode[STIX]{x1D6F4}\,\cap \,\mathbb{B}^{n}\,\subset \,\unicode[STIX]{x1D6F4}$ are connected, and that $\unicode[STIX]{x1D6F4}\,\cap \,\mathbb{B}^{n}=\widetilde{Z}\,\cap \,U$ .