HYPERBOLIC KÄHLER MANIFOLDS AND PROPER HOLOMORPHIC MAPPINGS TO RIEMANN SURFACES

TERRENCE NAPIER AND MOHAN RAMACHANDRAN

0. Introduction

In [Gv2], Gromov proved that a connected noncompact complex manifold which is a Galois covering of a compact Kähler manifold and which has infinitely many ends admits a proper holomorphic mapping to a Riemann surface. In fact, by [NR1], any complete Kähler manifold which has at least three ends and which has bounded geometry or is weakly 1-complete admits such a mapping to a Riemann surface. This is also the case for a bounded geometry or weakly 1-complete manifold which has two ends and admits a nonconstant holomorphic function.

The main goal of this paper is the following:

Theorem 0.1. Let M be a connected noncompact complete Kähler manifold which does not have exactly two ends and which admits a positive Green’s function that vanishes at infinity. Then M admits a proper holomorphic mapping to a Riemann surface if and only if H_{c}^{1}(M, O) ≠ 0.

Depending on the context, by an end of a connected manifold X we will mean either a connected component of X \ K with noncompact closure, where K is some compact subset of X, or an element of

lim_{←} π_{0}(X \ K),

where the limit is taken as K ranges over the compact subsets of X. In the latter context, we will denote the number of ends by e(X).

A noncompact complex manifold M for which H_{c}^{1}(M, O) = 0 is said to have the Bochner-Hartogs property (see Hartogs [Ha], Bochner [Bo], and Harvey and Lawson [HL]). Equivalently, for every C^{∞} compactly supported form α of type (0, 1) with ¯∂α = 0 on M, there

1991 Mathematics Subject Classification. 32C17.

Key words and phrases. Bochner-Hartogs property, pluriharmonic, q-complete.

Research partially supported by NSF grants DMS9411154 and DMS9971462 and by a Reidler Foundation grant.

Research partially supported by NSF grant DMS9626169.

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is a C^{∞} compactly supported function β on M such that ¯∂β = α. If M has the Bochner-Hartogs property, then every holomorphic function on a neighborhood of infinity with no relatively compact connected components extends to a holomorphic function on M. For, cutting off away from infinity, one gets a C^{∞} function λ on M. Taking α = ¯∂λ and forming β as above, the function λ − β will be the desired extension. In particular, e(M) = 1. Thus, in a sense, the space H_{c}^{1}(M, O) is a function theoretic approximation of the set of (topological) ends of M. An open Riemann surface X (as well as any complex manifold which admits a proper holomorphic mapping onto X) cannot have the Bochner-Hartogs property, because X admits nonconstant meromorphic functions with finitely many poles. Examples of manifolds of dimension n which have the Bochner-Hartogs property include strongly (n−1)-complete complex manifolds (Andreotti and Vesentini [AnV]) and strongly hyper-(n − 1)-convex Kähler manifolds (Grauert and Riemenschneider [GR]).

We will also obtain the following as a consequence of Theorem 0.1:

Theorem 0.2. Let M be a compact Kähler manifold and let π : M̃ → M be a connected infinite Galois covering manifold which does not have exactly two ends. Then M̃ admits a proper holomorphic mapping to a Riemann surface if and only if H_{c}^{1}(M̃, O) ≠ 0.

For a connected infinite Galois covering M̃ of a compact Kähler manifold M, e(M̃) = 1, 2, or ∞ and e(M̃) = 2 if and only if the covering group Γ = π_{1}(M)/π_{1}(M̃) contains an infinite cyclic subgroup of finite index (see Cohen [Co] or Scott and Wall [ScW]). Cousin’s [Cs] example of a ℤ-covering of an Abelian variety which has no nonconstant holomorphic functions shows that Theorem 0.2 fails when e(M̃) = 2. By a result of Arapura, Bressler, and the second author [ArBR], the universal covering of a compact Kähler manifold has at most one end. In fact, as shown in [NR1], any complete noncompact connected Kähler manifold M which satisfies H^{1}(M, ℝ) = 0 and which has bounded geometry or is weakly 1-complete has exactly one end.

In [Ra], the second author proved Theorem 0.2 for M̃ the universal covering, or a Galois covering with infinite covering group of more than quadratic growth, assuming that M̃ admits a nonconstant holomorphic function (one goal of this paper is to remove the rather strong hypothesis that O(M̃) ≠ ℂ). In [NR2], the conclusion of Theorem 0.1 was shown to hold for M a connected noncompact weakly 1-complete Kähler manifold with exactly one end. This too may be considered to be a consequence of Theorem 0.1, because one may apply (the proof of) Theorem 0.1 to the sublevels of the exhaustion function. Finally, it should be remarked that Theorem 0.2 may be relevant to the conjecture of Shafarevich

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that the universal covering of a smooth projective variety is holomorphically convex. For any holomorphically convex complex manifold either has the Bochner-Hartogs property or admits a proper holomorphic mapping to a Riemann surface.

The paper is organized as follows. Some known facts required for the proof of Theorem 0.1 are collected in Sect. 1. With the exception of a required technical lemma which is proved in Sect. 3, the proof Theorem 0.1 is contained in Sect. 2. Theorem 0.2 is considered in Sect. 4. Moreover, by general principles, the mapping on the covering space M̃ in Theorem 0.2 determines a surjective holomorphic mapping of some finite covering of the base manifold onto a compact Riemann surface. This observation and an observation for the case in which e(M̃) = 2 are also considered in Sect. 4. In particular, we get the following:

Corollary 0.3. Let M be a connected compact Kähler manifold. Assume that there exists a connected infinite Galois covering manifold π : M̃ → M which satisfies one of the following:

(i) e(M̃) = 1 and H_{c}^{1}(M, O) ≠ 0, (ii) e(M̃) = 2 and π_{1}(M̃) is not finitely generated, or (iii) e(M̃) = ∞. Then some finite covering of M admits a surjective holomorphic mapping onto a Riemann surface.

Acknowledgement. We would like to thank Jiaping Wang for Lemma 1.3.

1. Preliminary facts

The main known facts required for the proof of Theorem 0.1 are collected in this section. We begin with some terminology and facts from potential theory and pluripotential theory. The Levi form of a real-valued function φ of class C^{2} on a complex manifold M is the Hermitian form

L(φ) = ∑_{i,j} \frac{∂^{2}φ}{∂z_{i}∂z̄_{j}} dz_{i}dz̄_{j}.

The function φ is called plurisubharmonic (strictly plurisubharmonic, pluriharmonic) if L(φ) ≥ 0 (respectively, L(φ) > 0, L(φ) = 0). A complex-valued function is called pluriharmonic if its real and imaginary parts are pluriharmonic. The complex manifold M is called weakly 1-complete if there exists a continuous plurisubharmonic exhaustion function φ : M → ℝ (i.e. φ is locally the limit of a nonincreasing sequence of C^{∞} plurisubharmonic functions and the sublevel { x ∈ M | φ(x) < a } is relatively compact in M for each a ∈ ℝ).

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A connected noncompact Riemannian manifold N which admits a positive symmetric Green’s function G(x, y) is said to be hyperbolic; otherwise, N is called parabolic. Equivalently, N is hyperbolic if, given a relatively compact C^{∞} domain Ω for which no connected component of N \ Ω is compact, there is a connected component E of N \ \overline{Ω} and a (unique) C^{∞} function u_{E} : \overline{E} → [0, 1) such that u_{E} is harmonic on E, u_{E} = 0 on ∂E, and lim sup_{x→∞} u_{E}(x) = 1. We will also say that the end E is hyperbolic. An end E which is not hyperbolic is called parabolic and we set u_{E} = 0. We call the function u : N \Ω → [0, 1) defined by u|\overline{E} = u_{E}, for each connected component E of N \ \overline{Ω}, the harmonic measure of the ideal boundary of N with respect to N \ \overline{Ω}. We normalize the Green’s function G so that, for each point x_{0} ∈ N,

Δ_{distr.}G(·, x_{0}) = −δ_{x_{0}};

where δ_{x_{0}} is the Dirac function at x_{0} and Δ = −(d^{∗}d + dd^{∗}) is the Laplacian. We will use the same notation for the corresponding integral operator G given by

(Gα)(x) = ∫_{N} G(x, y)α(y) dV (y) ∀ x ∈ N

for each suitable function α on N.

If α is a C^{∞} compactly supported function, then β ≡ −Gα is a C^{∞} bounded function with finite energy (i.e. ∫_{N} |∇β|^{2} dV < ∞) and Δβ = α. Moreover, β(x_{ν}) → 0 if {x_{ν}} is a sequence in N with x_{ν} → ∞ and G(·, x_{ν}) → 0 (equivalently, u(x_{ν}) → 1). Such a sequence {x_{ν}} always exists (for N hyperbolic) and will be called a regular sequence.

As in [Gv3], [ArBR], [Ra], [NR1], and [NR2], a holomorphic mapping to ℙ^{1} will be obtained as a quotient of holomorphic 1-forms (see Lemma 1.4). The main tool for producing such holomorphic 1-forms from a ¯∂-closed compactly supported form will be the following (as in [Ra] and [NR2]):

Lemma 1.1. Let (M, g) be a connected noncompact complete hyperbolic Kähler manifold and let α be a C^{∞} compactly supported form of type (0, 1) on M such that ¯∂α = 0. Then there exist an L^{2} harmonic form γ of type (0, 1) which is closed and coclosed and a C^{∞} bounded function β : M → ℂ with finite energy such that γ = α − ¯∂β and β(x_{ν}) → 0 for every regular sequence {x_{ν}} in M.

Remark. In particular, γ̄ is a holomorphic 1-form on M and β is pluriharmonic on the complement of the support of α.

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Proof of Lemma 1.1. Let G denote the Green’s function on M as well as the associated integral operator. Since ¯∂^{∗}α is a C^{∞} function with compact support on M, the function

β ≡ 2G(¯∂^{∗}α)

is a C^{∞} bounded function with finite energy which vanishes at infinity along any regular sequence and which satisfies ¯∂^{∗} ¯∂β = −\frac{1}{2}Δβ = ¯∂^{∗}α. The form

γ ≡ α − ¯∂β

is therefore L^{2} and harmonic, and the Gaffney theorem [Ga] implies that γ is closed and coclosed.

The next lemma is a special case of [NR1, Theorem 2.6] and is contained implicitly in the work of Sario, Nakai, and their collaborators (see also Li [L] and Li and Tam [LT]). It will be applied in the proof of Theorem 0.1 in the same way it was applied in [NR2] (i.e. to the case in which the function β in Lemma 1.1 is real-valued).

Lemma 1.2. Let (M, g) be a connected noncompact complete hyperbolic Kähler manifold with Green’s function G, let K be a compact subset of M, and let E be a connected component of M \K with noncompact closure. Assume that G vanishes at infinity along the end E and that M \ E contains a hyperbolic end (in particular, e(M) > 1). Then there exists a real-valued pluriharmonic function ρ on M such that ρ has finite energy, 0 < ρ < 1 on M, lim_{x→∞} ρ|\overline{E}(x) = 1, and lim_{ν→∞} ρ(x_{ν}) = 0 for any regular sequence \{x_{ν}\} in M \ E.

Sketch of the proof. Clearly, we may assume that the end E is smooth at its compact boundary ∂E. By choosing a C^{∞} nondecreasing convex function χ : (0, 1) → (0, 1) which vanishes near 0, which is linear near 1, and which approaches 1 at 1, and by forming the composition with the harmonic measure of the ideal boundary of M with respect to E and extending by 0, one gets a C^{∞} subharmonic function η : M → [0, 1) which vanishes on M \ E, which is harmonic on the end E outside a relatively compact neighborhood of ∂E, and which approaches 1 at infinity along E. The function ρ = η + GΔη is harmonic with finite energy, and is therefore pluriharmonic by the Gaffney theorem [Ga]. Moreover, ρ has the required limits at infinity because GΔη approaches zero along any regular sequence. We also have ρ ≥ η ≥ 0 and, by forming an exhaustion of M by C^{∞} relatively compact domains and writing G as the limit of the corresponding sequence of Green’s function, one sees that ρ ≤ 1. Therefore, by the maximum principle, we have 0 < ρ < 1.

The following useful observation is due to J. Wang (it will rule out the case γ = 0 when Lemma 1.1 is applied in the proof of Theorem 0.1):

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Lemma 1.3 (Wang). Let M be a connected noncompact Kähler manifold. Suppose that, for some compact subset K of M, there exists a holomorphic function f on the complement M \ K which vanishes at infinity but is not identically zero on any open subset. Then M is parabolic.

Proof. The function φ = − log |f|^{2} is superharmonic on M \ K and φ(x) → ∞ as x → ∞ in M. In particular, we may assume that φ is positive. Fix a C^{∞} relatively compact domain Ω in M such that K ⊂ Ω and such that each connected component of M \ \overline{Ω} has noncompact closure. Let u be the harmonic measure of the ideal boundary of M with respect to M \ \overline{Ω}. Given ϵ > 0, we have ϵφ > 0 = u on ∂Ω and ϵφ > 1 > u on the complement of a sufficiently large compact subset of M. Therefore 0 ≤ u < ϵφ on M \ Ω for every ϵ > 0 and hence u ≡ 0.

The next lemma will be used for the production of a holomorphic mapping to ℙ^{1} (as in [Gv3], [ArBR], [NR1], and [NR2]).

Lemma 1.4. If θ_{1} and θ_{2} are two linearly independent closed holomorphic 1-forms satisfying θ_{1} ∧ θ_{2} ≡ 0 on a connected complex manifold M, then the meromorphic function f ≡ θ_{1}/θ_{2} has no points of indeterminacy in M. In other words, f is a holomorphic mapping of M into ℙ^{1}.

Remarks. 1. One can prove this by considering holomorphic equivalence relations (for an elementary proof, see [NR2, Lemma 2.2]).

2. Let S_{i} ≡ { x ∈ M | (θ_{i})_{x} = 0 } for i = 1, 2. Then f is locally constant on the analytic set S_{1} ∪ S_{2} (see [NR2]).

3. If f is nonconstant and A = S_{1} ∩ S_{2}, then the levels of f|_{M\A} are precisely the (smooth) leaves of the holomorphic foliation determined by θ_{1} and θ_{2} on M \ A. For, on a neighborhood U of a point at which θ_{2} ≠ 0, we have local coordinates z = (z_{1}, …, z_{n}) in which θ_{2} = dz_{1} and f = f(z_{1}). Hence the levels of f|_{U} are smooth coordinate slices which are leaves of the foliation in U. A similar argument applies when θ_{1} ≠ 0 and the claim follows. In particular, if L is a level of f which does not meet A (for example, if L lies over a point not in the countable set f(A)), then L is precisely a leaf of the foliation.

4. If f is nowhere locally constant and, for i = 1, 2, θ_{i} = ∂ρ_{i}, where ρ_{i} is a real-valued pluriharmonic function on M, then ρ_{i} is constant on each level L of f. For ρ_{i} is clearly locally constant on the zero set of θ_{i} (since dρ_{i} = 0 at each point in this set), and, outside this set, ρ_{i} is constant on the leaves of the foliation described above and these leaves locally agree with levels of f.

After Lemma 1.4 gives a mapping of an open set to ℙ^{1}, the following lemma, which is essentially contained in [NR1], will yield a proper mapping to a Riemann surface:

Lemma 1.5. Let (M, g) be a connected noncompact complete hyperbolic Kähler manifold. Assume that the Green’s function vanishes at infinity and that there exists an open subset U ⊂ M and a holomorphic mapping f : U → ℙ^{1} with at least one (nonempty) compact level. Then there exists a proper holomorphic mapping of M onto a Riemann surface.

Sketch of the proof. By hypothesis, the set V of points which lie in a compact level of f is nonempty. This set is open for point-set topological reasons. Applying Stein factorization [St], one gets a Riemann surface X and a proper surjective holomorphic mapping Φ : V → X whose fibers are precisely the compact levels of f.

We now apply arguments from the proof of [NR1, Theorem 4.6]. By replacing X by a small coordinate neighborhood of some regular value of Φ, we may assume that X is a region in the plane and Φ is a proper holomorphic submersion with connected fibers. Let D_{1} and D_{2} be two disks which are relatively compact in X and which have disjoint closures. Then the domain N ≡ M \ Φ^{−1}(\overline{D}_{1} ∪ \overline{D}_{2}) has at least three ends. Moreover, N admits a complete Kähler metric for which there is a positive Green’s function which vanishes at infinity in M and at ∂N = Φ^{−1}(∂D_{1} ∪ ∂D_{2}). Therefore, by Lemma 1.2, there exists a pluriharmonic function ρ : N → (0, 1) which approaches 1 at infinity in M and 0 at ∂N. Thus − log ρ−log(1−ρ) is a C^{∞} plurisubharmonic exhaustion function on N. The main result of [NR1] now implies that there exists a proper holomorphic mapping Ψ of N onto a Riemann surface Y; again, with connected fibers. The maps Φ and Ψ determine a proper holomorphic mapping of M onto the Riemann surface Z obtained by identifying each point x in X \ (\overline{D}_{1} ∪ \overline{D}_{2}) with the point Ψ(Φ^{−1}(x)) in Y.

Remarks. 1. The results of [NR1] together with the above arguments imply that the lemma also holds if, in place of the existence of a positive Green’s function which vanishes at infinity, one assumes that M has bounded geometry or M is weakly 1-complete.

2. The weakly 1-complete case is a special case of a theorem of Nishino [Ns] who proved it without the assumption that M is Kähler.

3. For the proof of the above, one may instead apply the theory of Barlet spaces [Ba], because the graph over some 1-dimensional irreducible component of the Barlet space of compact analytic (dim M − 1)-cycles will map properly and surjectively onto M.

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2. Proof of the main result

The proof of Theorem 0.1 will require the following two lemmas which may be thought of as versions of Gromov’s cup product lemma [Gv3]:

Lemma 2.1. Suppose M is a connected noncompact complex manifold of dimension n, ρ and τ are two real-valued pluriharmonic functions on M, and ρ has a compact fiber F. Then ∂ρ ∧ ∂τ ≡ 0. In fact, if the differentials dρ and dτ are linearly independent for some such pair of functions, then there exists a surjective proper holomorphic mapping of some open subset of M onto a Riemann surface.

Remark. Two real-valued pluriharmonic functions ρ and τ have linearly dependent differentials on a connected complex manifold M (i.e. the functions 1, ρ, and τ are linearly dependent) if and only if dρ ∧ dτ ≡ 0.

Lemma 2.2. Let (M, g) be a connected noncompact complete hyperbolic Kähler manifold of dimension n with exactly one end, let K ⊂ M be a compact subset with connected complement E = M \ K, and let ρ_{1} and ρ_{2} be real-valued pluriharmonic functions on E which vanish at infinity in M. Then ∂ρ_{1} ∧ ∂ρ_{2} ≡ 0.

Remark. Lemma 2.2 also holds if M is weakly 1-complete or M has bounded geometry.

The proof of Lemma 2.1 is given below. The proof of Lemma 2.2 is slightly technical and will be postponed until the next section.

Proof of Lemma 2.1. Since the image of the nowhere dense complex analytic set

{ x ∈ M | (dρ)_{x} = 0 } = { x ∈ M | (∂ρ)_{x} = 0 }

under ρ is countable and since the restriction of ρ to the complement is an open mapping, there exist points arbitrarily close to the compact fiber F which lie over regular values of ρ. Moreover, levels of ρ through such points near F are compact. Thus ρ has a compact level N which lies over a regular value. The restriction of τ to N assumes its maximum at some point x_{0} ∈ N and, since ρ is real-valued, the leaf L of the foliation determined by ∂ρ through x_{0} is contained in N. Therefore τ|_{L} attains its maximum at x_{0} and hence τ ≡ τ(x_{0}) on L. Thus ∂ρ = ∂τ = 0 on T^{1,0}L and hence, since dim L = n − 1, we have ∂ρ ∧ ∂τ = 0 at each point in L.

Now if the analytic set A = { x ∈ M | (∂ρ ∧ ∂τ)_{x} = 0 } is nowhere dense in M, then L contains some connected open subset V of A_{reg} (again, since dim L = n−1) and, therefore, the pluriharmonic function ρ is constant on the irreducible component A_{1} containing V.

HYPERBOLIC KÄHLER MANIFOLDS 9

Thus A_{1} ⊂ N and hence A_{1} is compact. On the other hand, ρ is constant on every compact irreducible component of A, so we may choose the level N so as to avoid the compact irreducible components of A. Thus we have arrived at a contradiction and we may conclude that ∂ρ ∧ ∂τ ≡ 0 on M as claimed.

Suppose now that dρ and dτ are linearly independent. Then, by Lemma 1.4, the quotient ∂ρ/∂τ determines a holomorphic mapping f : M → ℙ^{1}. If f is nonconstant, then ρ is constant on the level of f through x_{0}. It follows that this level is compact and hence that the set of points which lie in a compact level of f is a nonempty open subset of M which admits a proper holomorphic mapping onto a Riemann surface. If f is constant, then ∂ρ and ∂τ are linearly dependent and therefore, for some nonzero constant a ∈ ℂ, the function h ≡ ρ + aτ is holomorphic and nonconstant on M. Since τ = (Im a)^{−1}(Im h) and ρ = h − aτ are constant on the level of h through x_{0}, we again get the required proper holomorphic mapping to a Riemann surface on some open set.

For the proof of Theorem 0.1, we will consider the set up γ = α − ¯∂β as in Lemma 1.1. The case in which β is real-valued is contained in the next lemma. This lemma was essentially proved in [NR2, pp. 1361–1363], but, for the convenience of the reader, the proof is reproduced here.

Lemma 2.3. Let (M, g) be a connected noncompact hyperbolic complete Kähler manifold. Assume that the Green’s function G vanishes at infinity and that there exist a compact subset K with connected complement E = M \ K, a nonzero closed harmonic form γ of type (0, 1) on M, and a real-valued pluriharmonic function τ on E such that

γ|_{E} = −¯∂τ and τ → 0 at infinity in M.

Then M admits a proper holomorphic mapping to a Riemann surface.

Proof. Fix a point x_{0} ∈ E and let Γ be the image of π_{1}(E, x_{0}) in π_{1}(M, x_{0}). Then Γ is a proper subgroup. For if Γ = π_{1}(M, x_{0}), then, since the C^{∞} closed real 1-form θ = −γ − γ̄ on M is equal to dτ on E, we get a well-defined extension of τ to a C^{∞} function τ_{0} on M by setting

τ_{0}(x) = τ(x_{0}) + ∫_{x_{0}}^{x} θ ∀ x ∈ M.

Moreover, τ_{0} is pluriharmonic because dτ_{0} = θ and hence ∂ ¯∂τ_{0} = −∂γ = 0. On the other hand, τ = τ_{0}|_{E} vanishes at infinity in M, so we must have τ_{0} ≡ 0. Since γ is not everywhere 0, this is impossible. Thus Γ ≠ π_{1}(M, x_{0}).

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Now let π : M̃ → M be a connected covering space with π_{∗}(π_{1}(M̃, x_{1})) = Γ for some point x_{1} ∈ π^{−1}(x_{0}). Clearly, we may assume that E is smooth at its boundary and hence that π maps a neighborhood of the closure \overline{E}_{1} of the connected component E_{1} of Ẽ = π^{−1}(E) containing x_{1} isomorphically onto a neighborhood of \overline{E}. In particular, E_{1} is a hyperbolic end of M̃ with respect to the complete Kähler metric g̃ = π^{∗}g and the Green’s function G̃ on M̃ vanishes at infinity along E_{1}. Since Γ is a proper subgroup, M̃ \ E_{1} is noncompact (i.e. e(M̃) > 1). Moreover, the lifting to M̃ of a negative subharmonic function on M which vanishes at infinity must vanish at infinity along some sequence in M̃\E_{1}. So M̃\E_{1} must contain a hyperbolic end. Therefore, by Lemma 1.2, there exists a pluriharmonic function ρ̃ on M̃ such that 0 < ρ̃ < 1, ρ̃ has finite energy, lim_{x→∞} ρ̃|_{\overline{E}_{1}}(x) = 1, and lim_{ν→∞} ρ̃(x_{ν}) = 0 for any regular sequence {x_{ν}} in M̃ \ E_{1}. Since π maps E_{1} isomorphically onto E, the restriction ρ̃|_{E_{1}} determines a pluriharmonic function ρ on E such that 0 < ρ < 1 and ρ → 1 at infinity.

Because ρ has a compact fiber, Lemma 2.1 and Lemma 1.5 imply that it suffices to show that the differentials dρ and dτ are linearly independent. If this is not the case, then there exist real constants r and s such that τ = rρ + s on E. The lifting γ̃ = π^{∗}γ of γ is then a closed form of type (0, 1) on M̃ which is equal to the form −¯∂(rρ̃ + s) on the nonempty open set E_{1} and hence on the entire manifold M̃. Therefore, on the nonempty open set Ẽ \ \overline{E}_{1}, we have

−¯∂(τ ∘ π) = γ̃ = −¯∂(rρ̃ + s).

Hence the restriction of the function (τ∘π)−(rρ̃+s) to Ẽ\\overline{E}_{1} is real-valued and holomorphic and is therefore locally constant. Thus if E_{2} is a connected component of Ẽ which is not equal to E_{1}, then, for some real constant s′, we have τ ∘ π = rρ̃ + s′ on E_{2}. Now since π(E_{1}) = π(E_{2}) = E, we may choose a sequence {x_{ν}} in E with x_{ν} → ∞ in M and sequences {y_{ν}} and {z_{ν}} in E_{1} and E_{2}, respectively, such that π(y_{ν}) = π(z_{ν}) = x_{ν} for each ν. The sequences {y_{ν}} and {z_{ν}} are then regular sequences in M̃, because the lifting v of the function −G(x_{0}, ·) to M̃ is a negative subharmonic function and v(y_{ν}), v(z_{ν}) → 0. Therefore ρ̃(y_{ν}) → 1 and ρ̃(z_{ν}) → 0. Since τ vanishes at infinity, we get

0 = lim τ(x_{ν}) = lim(rρ̃(y_{ν}) + s) = r + s and 0 = lim τ(x_{ν}) = lim(rρ̃(z_{ν}) + s′) = s′.

Therefore, for each point x ∈ E and each pair of points y ∈ E_{1}∩π^{−1}(x) and z ∈ E_{2}∩π^{−1}(x), we have ρ̃(y) − 1 = r^{−1}τ(x) = ρ̃(z). But this is impossible because 0 < ρ̃ < 1 on M̃. Thus dρ and dτ are linearly independent and the lemma follows.

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Proof of Theorem 0.1. If e(M) ≥ 3, then we may proceed as in the proof of Lemma 1.5. More precisely, there exists a smooth relatively compact domain Ω in M such that each of the (finitely many) connected components E_{1}, …, E_{m} of M \ \overline{Ω} has noncompact closure and such that m ≥ 3. By Lemma 1.2, there exists a pluriharmonic function ξ : M → (0, 1) which approaches 1 at infinity along E_{1} and 0 at infinity along E_{2}, …, E_{m}. Therefore, the function − log(1−ξ)−log ξ is a C^{∞} plurisubharmonic exhaustion function on M and hence, by the main result of [NR1], M admits a proper holomorphic mapping onto a Riemann surface.

Assume now that e(M) = 1 and H_{c}^{1}(M, O) ≠ 0. Then there is a C^{∞} compactly supported form α of type (0, 1) on M such that ¯∂α = 0 and such that α ∉ ¯∂C_{c}^{∞}(M). We may fix a connected compact set K such that the complement E = M \ K is connected and such that α ≡ 0 on E. Applying Lemma 1.1, we get a (closed and coclosed) L^{2} harmonic form γ of type (0, 1) on M and a C^{∞} bounded function β : M → ℂ with finite energy such that

γ = α − ¯∂β and β → 0 at infinity.

In particular, γ̄ is a holomorphic 1-form on M and β is pluriharmonic on the end E. Observe also that γ is not identically zero on M. For if this were the case, then β would be holomorphic on E and therefore, since e(M) = 1 and M is hyperbolic, Lemma 1.3 would imply that β has compact support; which contradicts the choice of α.

We have β|_{E} = ρ_{1} + iρ_{2}, where ρ_{1} and ρ_{2} are real-valued pluriharmonic functions on the connected set E which vanish at infinity in M. We may assume that ρ_{1} and ρ_{2} are linearly independent (i.e. that dρ_{1} and dρ_{2} are linearly independent). For if this is not the case, then a suitable nonzero complex multiple of β|_{E} is real-valued and Lemma 2.3 applies.

By Lemma 2.2, we have ∂ρ_{1} ∧ ∂ρ_{2} ≡ 0 on E. Therefore (for ρ_{1} and ρ_{2} linearly independent), Lemma 1.4 implies that the quotient ∂ρ_{1}/∂ρ_{2} determines a holomorphic mapping

f : E → ℙ^{1}.

This mapping is nonconstant. For if f were equal to a constant a ∈ ℂ \ {0}, then the function ρ_{1} − āρ_{2} would be a nonconstant holomorphic function on E which vanishes at infinity (in M). Since M is hyperbolic, this would contradict Lemma 1.3.

According to Lemma 1.5, it suffices to show that f has at least one compact level. As a first step, we will show that f has a compact level or f has a fiber which is relatively compact in M. This will be obtained as a consequence of the following observation: the image under f of the set

Z = { x ∈ E | ρ_{1}(x) = ρ_{2}(x) = 0 } = β^{−1}(0) \ K

HYPERBOLIC KÄHLER MANIFOLDS 12

is a set of measure 0 in ℙ^{1}. For the proof of this observation, note that, by the remarks following Lemma 1.4, ρ_{1} and ρ_{2} (and hence β) are constant on the levels of f. If x is a point in Z at which dρ_{1} ≠ 0, then there is a neighborhood U of x in which the set N ≡ ρ_{1}^{−1}(0)∩U is a real-analytic submanifold of real dimension 2n − 1 and the levels of f|_{N} (which are open sets in levels of f) are complex analytic sets of real dimension 2n − 2. Hence the set f(N), which contains f(Z ∩ U), is a set of measure 0 in the real 2-dimensional manifold ℙ^{1} (by Sard’s theorem). The same argument applies near any point at which dρ_{2} ≠ 0. On the other hand, f is locally constant on the (complex) analytic set

A ≡ { x ∈ E | (dρ_{1})_{x} = (dρ_{2})_{x} = 0 } = { x ∈ E | (∂ρ_{1})_{x} = (∂ρ_{2})_{x} = 0 },

so f(A) is countable. Thus f(Z) is a set of measure 0 in ℙ^{1}. In particular, since f is an open mapping, the set E \ f^{−1}(f(Z)) is dense in E. If ζ ∈ f(E) \ (f(Z) ∪ {∞}) and L is a level of f over ζ, then L lies in some fiber of the mapping (ρ_{1}, ρ_{2}) over a point in ℝ^{2} \ {(0, 0)} and therefore, since ρ_{1}, ρ_{2} → 0 at infinity, we get L ⊂⊂ M. Hence, if L is noncompact, then \overline{L} ∩ K ≠ ∅. Fixing an open set Ω with K ⊂ Ω ⊂⊂ M, we see that only finitely many noncompact connected components L_{1}, …, L_{m} of f^{−1}(ζ) can meet M \ Ω, because all such connected components must meet ∂Ω and the collection of all connected components is locally finite in E. Therefore, if f has no compact level over ζ, then

f^{−1}(ζ) ⊂ \overline{Ω} ∪ L_{1} ∪ ⋯ ∪ L_{m} ⊂⊂ M.

In particular, by replacing K by a larger compact set if necessary (containing \overline{Ω} ∪ L_{1} ∪ ⋯ ∪ L_{m} for some ζ as above), we may assume that there exists a point

ζ_{0} ∈ ℙ^{1} \ (f(E) ∪ {∞}).

Hence the function

h ≡ (f − ζ_{0})^{−1} : E → ℂ

is a nonconstant holomorphic function on E.

Fix a C^{∞} function λ : M → ℂ and a connected compact subset K′ such that K ⊂ K′^{∘}, the set E′ ≡ M \ K′ is connected, and λ|_{E′} = h|_{E′}. Then the form

α′ ≡ ¯∂λ

is a C^{∞} compactly supported form of type (0, 1) on M satisfying ¯∂α′ = 0. Thus we may apply Lemma 1.1 to get an L^{2} harmonic form γ′ and a C^{∞} function β′ (with finite energy) such that

γ′ = α′ − ¯∂β′ = ¯∂(λ − β′) and β′ → 0 at infinity.

HYPERBOLIC KÄHLER MANIFOLDS 13

The function τ ≡ λ − β′ : M → ℂ is pluriharmonic. Moreover, τ is nonconstant because β′ vanishes at infinity and, by Lemma 1.3, the nonconstant holomorphic function h|_{E′} = λ|_{E′} cannot approach a constant at infinity.

If τ_{1} ≡ Re τ is nonconstant, then the maximum principle implies that we may choose a point x_{0} ∈ E′ ⊂ E at which

τ_{1}(x_{0}) > max_{K′} τ_{1}.

Furthermore, we may choose x_{0} to lie in the dense subset E′ \ f^{−1}(f(Z)). The level L of f through x_{0} is then relatively compact in M. Hence K′ ∪ L is a compact set and the restriction of τ_{1} to this set attains its maximum at some point x_{1} ∈ K′ ∪ L. But then

τ_{1}(x_{1}) = max_{K′∪L} τ_{1} ≥ τ_{1}(x_{0}) > max_{K′} τ_{1}.

Thus τ_{1}(x_{1}) = max_{L} τ_{1} and hence τ_{1} is constant on L. Therefore

L ⊂ { x ∈ M | τ_{1}(x) ≥ τ_{1}(x_{0}) }.

But the set on the right-hand side is a (closed) subset of E′, the closed subset L of E is relatively compact in M, and \overline{E′} ⊂ E, so L must be a compact level of f. If τ_{1} is constant, then the imaginary part τ_{2} ≡ Im τ is nonconstant and the same argument applied to τ_{2} again yields a compact level of f. Therefore, by Lemma 1.5, M admits a proper holomorphic mapping onto a Riemann surface.

3. Hyper q-convexity and the proof of the technical lemma

As in [NR2], we will say that a real-valued C^{2} function φ on a Kähler manifold (M, g) is q-plurisubharmonic (strictly q-plurisubharmonic) if, for each point x_{0} ∈ M, the trace of the restriction of the Levi form L(φ) to any complex subspace of T_{x_{0}}^{1,0}M of dimension q is nonnegative (respectively, positive). The Kähler manifold (M, g) is said to be hyper-q-complete if M admits a C^{∞} strictly q-plurisubharmonic exhaustion function. If there exists a C^{∞} exhaustion function which is strictly q-plurisubharmonic on the complement of some compact subset of M, then (M, g) is said to be strongly hyper-q-convex. The following fact is contained implicitly in the work of Richberg [Ri], Greene and Wu [GrW], Ohsawa [O], Colţoiu [Cl], and Demailly [D2] (see [NR2]):

Proposition 3.1 (Richberg, Greene-Wu, Ohsawa, Colţoiu, Demailly). Suppose M is a Kähler manifold of dimension n and X is a nowhere dense analytic subset with no compact irreducible components. Then there exists a C^{∞} strictly (n − 1)-plurisubharmonic function φ on a neighborhood V of X such that φ exhausts X.

HYPERBOLIC KÄHLER MANIFOLDS 14

The following lemma is a consequence of the work of Grauert and Riemenschneider [GR] (for a relatively compact domain E), of Gromov [Gv3] and of Li [L] (for E = M), and of Siu [Si] (for a harmonic mapping of a relatively compact domain into a manifold satisfying certain curvature conditions).

Lemma 3.2 (Grauert-Riemenschneider, Li, Siu). Let (M, g) be a connected complete Hermitian manifold of dimension n, let E be a (not necessarily relatively compact) domain with smooth compact (possibly empty) boundary in M, let φ be C^{∞} real-valued function on M such that dφ ≠ 0 at every point in ∂E and such that E = { x ∈ M | φ(x) < 0 }, and, for each point x ∈ ∂E, let

τ(x) = tr (L(φ)|_{T_{x}^{1,0}(∂E)}).

Assume that g is Kähler on E and that τ ≥ 0 on ∂E. Then we have the following:

(a) If β is a C^{∞} function on \overline{E} such that β is harmonic on E, β satisfies the tangential Cauchy-Riemann equation ¯∂_{b}β = 0 on ∂E, and there is a sequence of positive real numbers R_{m} → ∞ and a point p ∈ M such that

∥∇β∥_{L^{2}(B_{p}(R_{m})∩E)}^{2} = ∫_{B_{p}(R_{m})∩E} |∇β|^{2} dV = o(R_{m}^{2}) as m → ∞,

then β is pluriharmonic on E.

(b) If E is a hyperbolic end of M, then τ ≡ 0 on ∂E.

Remark. If φ′ is another smooth defining function for E, then

L(φ′)|_{T^{1,0}(∂E)} = (φ′/φ)|_{∂E} · L(φ)|_{T^{1,0}(∂E)}.

So the conditions τ ≥ 0 and τ = 0 are independent of the choice of the defining function.

Sketch of the proof. For part (a), let γ = ¯∂β and let η = \overline{∗γ}, where ∗ denotes the Hodge star operator. Then η may be thought of as a C^{∞} form of type (n, n − 1) or as a form of type (0, n − 1) with values in the canonical bundle K_{M} on \overline{E}. A computation in normal coordinates implies that it suffices to show that \overline{∇}η = 0 (see [Si, Proof of Lemma 5.6(d)]). Applying the Gaffney construction [Ga], one gets a sequence of C^{∞} functions {λ_{m}} on M such that, for each m, λ_{m} ≡ 0 on M \ B_{p}(R_{m}), λ_{m} ≡ 1 on B_{p}(R_{m}/2), and |dλ_{m}| ≤ 3/R_{m} on M. Since ∂E is compact, we may also assume that ∂E ⊂ B_{p}(R_{m}/2). A straightforward computation in normal coordinates then shows that, because λ_{m}^{2}η = η = \overline{∗¯∂β} on a neighborhood of ∂E and λ_{m}^{2}η has compact support in \overline{E}, the form λ_{m}^{2}η lies in the domain of the

HYPERBOLIC KÄHLER MANIFOLDS 15

adjoint operator ¯∂^{∗}. The Bochner-Kodaira formula is then (see [GR] or [Si, formula 2.1.4])

∥\overline{∇}(λ_{m}^{2}η)∥_{L^{2}(E)}^{2} = ∥¯∂(λ_{m}^{2}η)∥_{L^{2}(E)}^{2} + ∥¯∂^{∗}(λ_{m}^{2}η)∥_{L^{2}(E)}^{2} − ∫_{∂E} |γ|^{2} · |∇φ|^{−1} · τ dσ; (3.3)

where dσ is the volume element on ∂E.

Remark. Here, the curvature terms drop out because η is of type (n, n − 1) and we have used the fact that ¯∂β ∧ ¯∂φ vanishes at each point in ∂E. The authors mistakenly left out the normalizing factor |∇φ|^{−1} in the proof of the analogous version [NR2, Theorem 1.6].

If K is a compact subset of E, then, for m sufficiently large, we have

∥\overline{∇}η∥_{L^{2}(K)}^{2} = ∥\overline{∇}(λ_{m}^{2}η)∥_{L^{2}(K)}^{2} ≤ ∥\overline{∇}(λ_{m}^{2}η)∥_{L^{2}(E)}^{2}.

Therefore, since τ ≥ 0, it suffices to show that

∥¯∂(λ_{m}^{2}η)∥_{L^{2}(E)}^{2} + ∥¯∂^{∗}(λ_{m}^{2}η)∥_{L^{2}(E)}^{2} → 0 as m → ∞.

Since β is harmonic and g is Kähler on E, we have ¯∂η = 0 and ¯∂^{∗}η = 0. We also have |η| ≤ |∇β|. Therefore, for some positive constant C, we have

∥¯∂(λ_{m}^{2}η)∥_{L^{2}(E)} = ∥2λ_{m} ¯∂λ_{m} ∧ η∥_{L^{2}(E)} ≤ \frac{C}{R_{m}} ∥∇β∥_{L^{2}(B_{p}(R_{m})∩E)} = \frac{o(R_{m})}{R_{m}} → 0

and

∥¯∂^{∗}(λ_{m}^{2}η)∥_{L^{2}(E)} = ∥ − 2λ_{m} ∗ (∂λ_{m} ∧ ∗η)∥_{L^{2}(E)} ≤ \frac{C}{R_{m}} ∥∇β∥_{L^{2}(B_{p}(R_{m})∩E)} = \frac{o(R_{m})}{R_{m}} → 0.

Thus part (a) is proved.

Assume now that E is a hyperbolic end and let β be the harmonic measure of the ideal boundary of M with respect to E. Thus β is a C^{∞} function on \overline{E}, β is harmonic on E, β ≡ 0 on ∂E, 0 < β < 1 on E, and lim sup_{x→∞} β(x) = 1. Moreover, β has finite energy and therefore, by part (a), β is pluriharmonic on E. With the notation from the proof of part (a), we see that each of the terms in equation (3.3) approaches 0 as m → ∞. In particular,

∫_{∂E} |γ|^{2} · |∇φ|^{−1} · τ dσ = 0.

Hence if τ > 0 at some point in ∂E, then |γ| = |¯∂β| = 0 on V ∩ ∂E for some nonempty connected open subset V of M which meets ∂E. If f is a C^{∞} function on V ∩ \overline{E} which is holomorphic on V ∩ E and which vanishes on V ∩ ∂E, then one may extend f to a continuous function h on V which vanishes outside E. But then ¯∂h = 0 in the weak sense, so h is holomorphic. It follows that h, and therefore f, must vanish identically. Shrinking V and letting f be a coefficient of the holomorphic 1-form γ̄ with respect to some local

HYPERBOLIC KÄHLER MANIFOLDS 16

holomorphic frame, we see that γ vanishes on a nonempty open subset of E and hence on the entire end E. Thus β is a real-valued holomorphic function on E and must therefore be a constant function. This contradicts the choice of β, so the claim (b) follows.

Remark. Part (b) also holds if M is weakly 1-complete or if M has bounded geometry. In the former case, one replaces M by a sublevel of the exhaustion function. In the latter case, when M is parabolic, one produces a pluriharmonic function β as in [NR1]. Similarly, Part (b) holds if E ⊂⊂ M and ∂E is not connected. For one may take β to be a harmonic function which is equal to 1 on one boundary component and 0 on the other components.

Proof of Lemma 2.2. Let n = dim M. Assume that ∂ρ_{1} ∧ ∂ρ_{2} is not everywhere 0. To obtain a contradiction, we will construct an end with an (n − 1)-plurisubharmonic defining function and we will then apply Lemma 3.2. Let

Z = { x ∈ E | ρ_{1}(x) = ρ_{2}(x) = 0 } and A = { x ∈ E | (∂ρ_{1} ∧ ∂ρ_{2})_{x} = 0 } (A is a nowhere dense complex analytic set), and fix open sets Ω_{1}, Ω_{2}, and Ω_{3} such that

K ⊂ Ω_{1} ⊂⊂ Ω_{2} ⊂⊂ Ω_{3} ⊂⊂ M

and such that A ∩ Ω_{3} \ \overline{Ω}_{1} is empty or has no compact irreducible components (one can form these sets by choosing open sets Ω_{1} ⊂⊂ Ω′_{3} ⊂⊂ M and setting Ω_{3} = Ω′_{3} \ H, where H is a finite subset of E \ \overline{Ω}_{1} which contains a point from each irreducible component of A which meets Ω′_{3} \ \overline{Ω}_{1}). Let Ω_{0} = Ω_{3} \ \overline{Ω}_{1}, let Z_{0} = Z ∩ Ω_{0}, and let A_{0} = A ∩ Ω_{0}. By applying Proposition 3.1 (Richberg, Greene-Wu, Ohsawa, Colţoiu, Demailly) to A_{0} and forming a suitable extension of the resulting function to Ω_{0} (possibly after shrinking the neighborhood of A_{0} on which the function is initially defined), one obtains a positive C^{∞} exhaustion function α on Ω_{0} which is strictly (n − 1)-plurisubharmonic on a neighborhood of A_{0}. Suppose

ψ = α + γ · (ρ_{1}^{2} + ρ_{2}^{2}),

where γ is a C^{∞} positive function on Ω_{0}. Then ψ exhausts Ω_{0} because ψ ≥ α. Moreover, at each point x_{0} ∈ Z_{0}, we have

L(ψ)_{x_{0}} = L(α)_{x_{0}} + γ(x_{0}) · L(ρ_{1}^{2} + ρ_{2}^{2})_{x_{0}}.

Therefore, since α is strictly (n − 1)-plurisubharmonic on a neighborhood of A_{0} and since ρ_{1}^{2} + ρ_{2}^{2} is (n − 1)-plurisubharmonic on E and strictly (n − 1)-plurisubharmonic on E \ A, we may choose γ so that the trace of the restriction of L(ψ) to each (n − 1)-dimensional complex subspace of T_{x_{0}}^{1,0}M is positive for each point x_{0} ∈ Z_{0}. The restriction of ψ to some neighborhood V of Z_{0} is then strictly (n − 1)-plurisubharmonic.

HYPERBOLIC KÄHLER MANIFOLDS 17

Fix constants a and b with a > b > max_{∂Ω_{2}} ψ. For ϵ > 0 sufficiently small, the set { x ∈ Ω_{0} | ρ_{1}^{2} + ρ_{2}^{2} ≤ ϵ and ψ(x) ≤ a } is compact (and possibly empty) and is contained in V. Choosing a C^{∞} nondecreasing convex function χ : ℝ → ℝ which vanishes on the interval (−∞, − log (a − b)] and which approaches +∞ at +∞, we obtain a C^{∞} (n − 1)- plurisubharmonic function φ on the open set

N = { x ∈ M \ Ω_{2} | ρ_{1}^{2} + ρ_{2}^{2} < ϵ } ∪ { x ∈ V ∩ Ω_{2} | ρ_{1}^{2} + ρ_{2}^{2} < ϵ and ψ(x) < a }

by defining

φ(x) = \{ \begin{matrix}ρ_{1}^{2} + ρ_{2}^{2} − log(ϵ − (ρ_{1}^{2} + ρ_{2}^{2})) & if x ∈ N \ Ω_{2} \\ ρ_{1}^{2} + ρ_{2}^{2} − log(ϵ − (ρ_{1}^{2} + ρ_{2}^{2})) + χ(−log(a − ψ(x))) & if x ∈ N ∩ Ω_{2} \end{matrix}

Moreover, \overline{N} ⊂ E, φ → ∞ at ∂N, φ ≥ − log ϵ on N, and, since ρ_{1}^{2} + ρ_{2}^{2} vanishes at infinity in M, M \ N is compact and φ → − log ϵ at infinity in M. Therefore, if c is a regular value of φ with c > − log ϵ and E_{0} is the (unique) connected component of the set { x ∈ N | φ(x) < c } with noncompact closure in M, then E_{0} is a C^{∞} domain in M which has nonempty compact boundary and which admits a C^{∞} (n − 1)-plurisubharmonic defining function φ − c with nonvanishing differential at each boundary point.

Now observe that φ is strictly (n − 1)-plurisubharmonic on N \ A because, on this set, φ − (ρ_{1}^{2} + ρ_{2}^{2}) is (n − 1)-plurisubharmonic and ρ_{1}^{2} + ρ_{2}^{2} is strictly (n − 1)-plurisubharmonic. Moreover, (∂E_{0}) \ A ≠ ∅ because dim_{ℝ} A ≤ 2n − 2 < dim_{ℝ} ∂E_{0}. Since (M, g) is hyperbolic, this contradicts Lemma 3.2, part (b). Thus we must have ∂ρ_{1} ∧ ∂ρ_{2} ≡ 0 on M.

Remarks. 1. In place of Lemma 3.2, part (b), one may instead apply part (a) to the harmonic measure and then apply Lemma 2.1.

2. By the remarks following the proof of Lemma 3.2, Lemma 2.2 also holds if M is weakly 1-complete or M has bounded geometry.

4. Compact Kähler manifolds and their coverings

Theorem 0.2 and Corollary 0.3 are proved in this section.

Proof of Theorem 0.2. Suppose (M, g) is a connected compact Kähler manifold of dimension n and π : M̃ → M is a connected infinite Galois covering with covering group

Γ = π_{1}(M)/π_{1}(M̃).

We equip M̃ with the complete Kähler metric g̃ = π^{∗}g lifted from M. We have e(M̃) = 1, 2, or ∞ (see Cohen [Co] or Scott and Wall [SW]). According to a theorem of Gromov [Gv2], if e(M̃) = ∞, then M̃ admits a proper holomorphic mapping to a Riemann

HYPERBOLIC KÄHLER MANIFOLDS 18

surface (see also [NR1]). By another theorem of Gromov [Gv1], if e(M̃) = 1, then either Γ has more than quadratic growth or Γ contains ℤ^{2} as a subgroup of finite index. In the former case, the work of Varopoulos [V1], [V2] and the work of Chavel and Feldman [CF] (see [Ra]) imply that (M̃, g̃) admits a positive Green’s function which vanishes at infinity. Hence one may apply Theorem 0.1 in this case. Thus it remains to address the latter case in which Γ contains ℤ^{2} as a subgroup of finite index.

By replacing M by a finite covering, we may assume that Γ = ℤ^{2}. Choosing loops σ_{1} and σ_{2} in M mapping to (1, 0) and (0, 1), respectively, in ℤ^{2}, there exists, for j = 1, 2, a unique (real) harmonic 1-form θ_{j} on M which integrates to δ_{ij} on σ_{i} for i = 1, 2 and which integrates to 0 on the image of π_{1}(M̃) in π_{1}(M). The liftings π^{∗}θ_{1} and π^{∗}θ_{2} integrate to pluriharmonic functions τ_{1} : M̃ → ℝ and τ_{2} : M̃ → ℝ, respectively, which satisfy

τ_{1}((r, s) · x) = τ_{1}(x) + r and τ_{2}((r, s) · x) = τ_{2}(x) + s ∀ (r, s) ∈ Γ = ℤ^{2}, x ∈ M̃. In particular, φ ≡ τ_{1}^{2} + τ_{2}^{2} is a C^{∞} plurisubharmonic exhaustion function on M̃.

Remark. The theorem now follows from [NR2], but a direct proof is provided below.

If H_{c}^{1}(M̃, O) ≠ 0, then there exists a C^{∞} compactly supported form α of type (0, 1) on M̃ such that ¯∂α = 0 and such that α ∉ ¯∂C_{c}^{∞}(M̃). Since e(M̃) = 1, we may choose a connected compact set K such that α ≡ 0 on M̃ \ K and such that M̃ \ K is connected. Fix a regular value a of φ with a > max_{K} φ, and let Ω be the connected component of { x ∈ M̃ | φ(x) < a } containing K. We may choose K and a so that Ω\K is connected. By a theorem of Nakano [Nk] (see also Demailly [D1]), Ω admits a complete Kähler metric g′. Furthermore, since φ−a is a negative (pluri)subharmonic function on Ω which approaches 0 at each point in ∂Ω, (Ω, g′) is hyperbolic and the Green’s function vanishes at ∂Ω.

Observe that α ∉ ¯∂C_{c}^{∞}(Ω) ⊂ ¯∂C_{c}^{∞}(M). Therefore, if ∂Ω is connected (i.e. e(Ω) = 1), then Theorem 0.1 implies that Ω admits a proper holomorphic mapping with connected fibers onto a Riemann surface. If ∂Ω is not connected, then, by Lemma 1.2, there exists a pluriharmonic function ρ on Ω such that 0 < ρ < 1 on Ω, ρ → 1 at some boundary component C, and ρ → 0 at (∂Ω) \ C ≠ ∅. In particular, ρ has compact fibers. Moreover, since the functions dτ_{1} and dτ_{2} are linearly independent, dρ and dτ_{i} must be linearly independent for i = 1 or for i = 2. Therefore, Lemma 2.1 implies that some open subset of Ω admits a proper holomorphic mapping onto a Riemann surface. Hence, by Lemma 1.5, Ω again admits a proper holomorphic mapping with connected fibers onto a Riemann surface. Exhausting M̃ by such domains, we get a proper holomorphic mapping (with connected fibers) of M̃ onto the limit Riemann surface.

HYPERBOLIC KÄHLER MANIFOLDS 19

The following observation, which is well known (at least for the case of a Galois covering), was used in [NR1] (along with the main result of [NR1]) to show that a connected noncompact Kähler manifold whose fundamental group is an amalgamated free product along ℤ admits a finite covering manifold which maps holomorphically onto a Riemann surface [NR1, Theorem 4.1]:

Proposition 4.1. Suppose (M, g) is a connected compact Kähler manifold for which some connected covering space admits a surjective proper holomorphic mapping onto a Riemann surface. Then some finite covering of M admits a surjective holomorphic mapping onto a (compact) Riemann surface.

Proof. By hypothesis, there exists a connected covering space π : M̃ → M, a Riemann surface X̃, and a surjective proper holomorphic mapping Φ̃ : M̃ → X̃. Clearly, we may assume that M̃ is noncompact. By Stein factorization, we may also assume that Φ̃ has connected fibers. Observe that if ρ : M̃′ → M̃ is the covering space with fundamental group equal to the image of the fundamental group of some generic fiber of Φ̃, then the composite mapping Φ̃′ : M̃′ → X̃ has compact levels. For this is clear for the nonsingular fibers, because Φ̃ is locally C^{∞} trivial over the complement of the (discrete) set C of critical values in X̃. Given a level F′_{0} over a point x_{0} ∈ C, we may form a coordinate unit disk Δ centered at 0 = x_{0} such that Δ ∩ C = {x_{0}}. Let F_{0} = ρ(F′_{0}) = Φ̃^{−1}(0), let U = Φ̃^{−1}(Δ) (a connected neighborhood of F_{0}), and let U′ be the connected component of ρ^{−1}(U) containing F′_{0}. Then the restriction mapping U′ \ F′_{0} → Δ^{∗} is a C^{∞} fiber bundle with compact levels and hence, aftering Stein factoring, we get U′ \ F′_{0} →^{α} Y →^{β} Δ^{∗}; where α is a proper holomorphic submersion with connected fibers and β is a covering map. In particular, we have Y = Δ or Δ^{∗}, and hence the bounded holomorphic function α extends to a holomorphic mapping γ : U′ → Δ (the image must lie in the interior Δ of \overline{Δ} by the maximum principle). Since Φ̃′(F′_{0}) = {0} and the image of the covering map Δ → Δ^{∗} does not contain 0, we must have Y = Δ^{∗} (and γ(F′_{0}) = 0). Hence the covering map Y → Δ^{∗} is a finite mapping and therefore the covering map U′ → U must have finite fibers. Thus F′_{0} is compact and the claim follows. Therefore, by replacing M̃ by the covering space M̃′ and Φ̃ by the Stein factorization of Φ̃′, we may assume that, for a generic fiber F, the mapping π_{1}(F) → π_{1}(M̃) is surjective. In particular, X̃ = Δ or ℂ because π_{1}(M̃) maps surjectively onto π_{1}(X̃) and π_{1}(F) maps to 0.

Now by a result of Kollár [K, Proposition 1.2.11] (one gets the proposition for compact Kähler manifolds by using the Barlet space in place of the Douady space in order to

HYPERBOLIC KÄHLER MANIFOLDS 20

get the required family of normal cycles in the proof), for a generic fiber F of Φ̃, the normalizer of the group Λ = im (π_{1}(M̃) → π_{1}(M)) = im (π_{1}(F) → π_{1}(M)) is of finite index in π_{1}(M). Therefore, by passing to the associated finite covering of M, we may assume that π : M̃ → M is a Galois covering and hence that M = M̃/Γ, where Γ = π_{1}(M)/Λ.

Each automorphism σ ∈ Γ maps fibers (of Φ̃) to fibers, because Φ̃ is constant on every connected compact analytic set. Thus σ descends to an automorphism of X̃. The image Θ of the homomorphism Γ → Aut (X̃) acts properly discontinuously on X̃ and the kernel is a finite normal subgroup of Γ. If X̃ = ℂ, then Θ′ = Θ is torsion free and, if X̃ = Δ, then, by a well known theorem of Selberg, there exists a torsion-free normal subgroup Θ′ of finite index in Θ. Thus we get a commutative diagram of holomorphic mappings

[figure: M̃/Γ′ = M′; π′; π; M̃; Φ̃; M; Φ′; X′ = X̃/Θ′; X̃]

where Γ′ is the inverse image of Θ′, π′ : M′ → M is a connected finite covering space, X′ is a Riemann surface, and Φ′ is a surjective holomorphic map.

For example, the main result of [NR1] and Proposition 4.1 together give the following:

Corollary 4.2. Suppose (M, g) is a connected compact Kähler manifold for which there is a connected covering space with at least three ends. Then some finite covering of M admits a surjective holomorphic mapping onto a Riemann surface.

Theorem 0.2 and the above together give Corollary 0.3 for the cases (i) and (iii). Finally, we address the case of a covering with two ends (in particular, case (ii)). By Lemma 1.2, Lemma 1.5, and Lemma 2.1, the conclusion of Theorem 0.1 also holds if e(M) = 2 and there exists a nowhere locally constant holomorphic function (or two pluriharmonic functions with locally linearly independent differentials) on the complement of a compact set. Similarly, one gets the corresponding versions of Theorem 0.2, Corollary 0.3, and Corollary 4.2. We also have the following fact which gives Corollary 0.3 for the case (ii):

Theorem 4.3. Suppose (M, g) is a connected compact Kähler manifold for which there exists a surjective homomorphism π_{1}(M) → ℤ whose kernel is not finitely generated. Then M admits a surjective holomorphic mapping onto a Riemann surface X and the homomorphism factors through π_{1}(X).

HYPERBOLIC KÄHLER MANIFOLDS 21

Proof. Let π : M̃ → M be a connected Galois covering space with

π_{∗}(π_{1}(M̃)) = Λ = ker (π_{1}(M) → ℤ).

Then ℤ = Γ = π_{1}(M)/Λ acts (fixed-point freely and properly discontinuously) by deck transformations on M̃. The main step is, of course, to show that M̃ admits a proper holomorphic mapping onto a Riemann surface X̃. One can then pass to the quotient X̃/ℤ.

We first observe that standard arguments imply that M̃ admits a proper pluriharmonic function into ℝ. For example, given a loop σ in M which maps to 1 ∈ ℤ, there is a unique real harmonic 1-form θ on M which integrates to 1 on σ and to 0 on elements of Λ. The lifting π^{∗}θ integrates to a pluriharmonic function ρ : M̃ → ℝ which is proper because ρ(m · x) = ρ(x) + m for all x ∈ M̃ and m ∈ ℤ = Γ.

Fix a regular value c of ρ, a connected component N of ρ^{−1}(c), and a connected covering space π̂ : M̂ → M̃ with

π̂_{∗}(π_{1}(M̂)) = im (π_{1}(N) → π_{1}(M̃)).

Then π̂ maps a relatively compact neighborhood V_{1} of some connected component N_{1} of π̂^{−1}(N) biholomorphically onto a neighborhood V of N in M̃, and, since π_{1}(M̃) = Λ is not finitely generated, N_{1} is a boundary component of some connected component P̂ of M̂ \ π̂^{−1}(N) which has at least one other boundary component N_{2}. By replacing ρ by −ρ and c by −c, if necessary, and by shrinking V_{1}, we may assume that, for some connected component P of \{ x ∈ M̃ | ρ(x) < c \}, P̂ is a connected component of π̂^{−1}(P) and P̂ \ \overline{V}_{1} is connected. We may also assume that V_{1} ∩ P̂, and hence V ∩ P, is connected. By a theorem of Nakano [Nk] (and Demailly [D1]), there exists a complete Kähler metric on P and, therefore, on P̂. Moreover, P̂ is hyperbolic and the Green’s function vanishes at ∂P̂, because ρ ∘ π̂|P̂ − c is a negative subharmonic function which vanishes at the boundary. Since the boundary component N_{1} is compact, we may apply Lemma 1.2 to P̂ to get a nonconstant bounded real-valued pluriharmonic function τ̂ with finite energy on P̂. In particular, since ρ(x) → −∞ as x → ∞ in \overline{P}, the functions 1, ρ ∘ π̂|P̂, and τ̂ are linearly independent. Hence τ̂ determines a pluriharmonic function

τ = τ̂ ∘ (π̂|_{V_{1}∩P̂})^{−1} : V ∩ P → ℝ

such that 1, ρ|_{V ∩P}, and τ are linearly independent. Since ρ|_{V ∩P} has a compact level, Lemma 2.1 implies that some open subset of M̃ admits a proper holomorphic mapping onto a Riemann surface. On the other hand, each of the sublevels Ω of the plurisubharmonic exhaustion function ρ^{2} on M̃ admits a complete hyperbolic Kähler metric whose Green’s function vanishes at ∂Ω. Therefore, by Lemma 1.5, Ω admits a proper holomorphic mapping with connected fibers to a Riemann surface. Exhausting M̃ by such domains, we get a proper holomorphic mapping Φ̃ : M̃ → X̃ with connected fibers of M̃ onto the limit Riemann surface X̃.

As in the proof of Proposition 4.1, the action of ℤ on M̃ descends to a properly discontinuous action of ℤ on X̃. This action is also fixed-point free because ℤ is torsion free. Thus Φ̃ descends to a surjective holomorphic mapping Φ : M → X of M onto the Riemann surface X = X̃/ℤ.

Remarks. 1. The above theorem generalizes part of a result of Arapura [Ar].

2. The theorem can also be proved using techniques of Gromov and Schoen [GS].

Proof of Corollary 0.3. The conclusion for the cases (i) and (iii) follows from Theorem 0.2, Proposition 4.1, and Corollary 4.2. For the case (ii), suppose e(M̃) = 2 and π_{1}(M̃) is not finitely generated. Then the covering group Γ = π_{1}(M)/π_{1}(M̃) contains ℤ as a subgroup of finite index. The inverse image H of ℤ in π_{1}(M) is a subgroup of finite index and the image of π_{1}(M̃) in π_{1}(M) is a normal subgroup of H. Thus we may form the finite covering space M′ with im (π_{1}(M′) → π_{1}(M)) = H and the Galois covering π′ : M̃ → M′. We then get a surjective homomorphism π_{1}(M′) → Γ′ = π_{1}(M′)/π_{1}(M̃) = ℤ and the kernel π_{1}(M̃) is not finitely generated. Theorem 4.3 now gives the claim.

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Department of Mathematics, Lehigh University, Bethlehem, PA 18015

  E-mail address: tjn2@lehigh.edu

Department of Mathematics, SUNY at Buffalo, Buffalo, NY 14214

  E-mail address: ramac-m@newton.math.buffalo.edu