TheBochner-Hartogsdichotomyforweakly1-completeKählermanifolds M R T N

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A ’ F

T ^ERENCE N ^APIER

M OHAN R AMACHANDRAN

The Bochner-Hartogs dichotomy for weakly 1-complete Kähler manifolds

Annales de l’institut Fourier, tome 47, n^o5 (1997), p. 1345-1365

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THE BOCHNER-HARTOGS DICHOTOMY FOR WEAKLY 1-COMPLETE KAHLER MANIFOLDS

by T. NAPIER (*) & M. RAMACHANDRAN (**)

Introduction.

A complex manifold M for which H^(M,0) = 0 is said to have the Bochner-Hartogs property (see Hartogs [H], Bochner [B], and Har- vey and Lawson [HL]). Equivalently, for every C00 compactly supported form a of type (0,1) with 9a == 0 on M, there is a C°° compactly supported function /? on M such that 9(3 = a. Andreotti and Vesentini [AV] proved that a strongly (n—Incomplete complex manifold of dimension n > 1 has the Bochner-Hartogs property, and Grauert and Riemenschneider [GR], that a strongly hyper-(n-l)-convex Kahler manifold of dimension n > 1 has the Bochner-Hartogs property (see Section 1). In [R], the second author proved that if the universal covering M of a compact Kahler manifold (or a Galois covering M with infinite covering group of more than quadratic growth) admits a nonconstant holomorphic function, then M satisfies the following dichotomy:

(BHD) Either M has the Bochner-Hartogs property or there exists a proper holomorphic mapping of M onto a Riemann surface.

A complex manifold which admits a continuous plurisubharmonic exhaus- tion function is said to be weakly 1-complete. The main result of this paper (Theorem 2.5) is the following:

(*) Research partially supported by a Reidler Foundation grant and by NSF grant DMS9411154.

(**) Research partially supported by NSF grant DMS9305745.

Key words: Riemann surface - g-complete - Pluriharmonic.

Math. classification: 32C.

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THEOREM. — If (M^g) is a connected noncompact weakly 1- complete Kahler manifold which has exactly one end, then M satis- fies (BHD).

If a complex manifold 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 /o on M as follows. Cutting off away from infinity, one obtains a C°° func- tion A on M. For a = <9A, there is a function /? as above and the function /o == A — f3 is the desired extension. In particular, a Riemann surface can- not have the Bochner-Hartogs property and a complex manifold cannot satisfy both of the conditions in (BHD). Moreover, a manifold with the Bochner-Hartogs property has only one end because, on a manifold with more than one end, there exists a function which is locally constant, but not constant, near infinity. A related result due to Arapura, Bressler, and the second author [ABR] is that the universal covering of a compact Kahler manifold has at most one end. In fact, as shown in [NR], a complete non- compact connected Kahler manifold M which satisfies ^(M^K) = 0 and which has bounded geometry or is weakly 1-complete has exactly one end.

It was also proved in [NR] that a complete Kahler manifold with at least three ends which has bounded geometry or is weakly 1-complete admits a proper holomorphic mapping onto a Riemann surface.

Facts concerning strictly ^-plurisubharmonic functions and Green's functions are collected in Section 1. Section 2 contains the proof of the theorem. The main step is to prove Proposition 2.3 that (BHD) holds for a weakly 1-complete Kahler manifold on which there exists, outside a compact subset, a pair of pluriharmonic functions with linearly independent differentials (this may be thought of as a generalization of a theorem of Ohsawa [01]). One may assume that M is complete and admits a positive Green's function since one can exhaust M by domains with these properties.

Hence, if [a] € H^{M,0) is a nonzero element, then one can form the L2 harmonic projection 7 and a function (3 such that 7 = 0 — 9f3. In particular, (3 is pluriharmonic outside a compact set K. One then forms a pluriharmonic function on a covering space, pushes down to obtain a second pluriharmonic function on M \ JC, and applies Proposition 2.3.

The authors would like to thank the referee for helpful comments and suggestions.

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1. Preliminaries on ^-plurisubharmonic functions.

Most of the facts discussed in this section are known, so the proofs are only sketched. Throughout this section, (M, g) denotes a Kahler manifold of dimension n and q denotes a positive integer.

Let (p be a real-valued continuous function on M. We will say that (/?

is strictly q-plurisubharmonic if (p is an element of the class ^f(q) defined by Wu [W]. We will call (p q-plurisubharmonic if the function (p + '0 is strictly 9-plurisubharmonic for every continuous strictly g-plurisubharmo- nic function '0 on M.

Remarks.

1. If (p of class G², then y? is g-plurisubharmonic (strictly g-plurisub- harmonic) if and only if, for each point XQ e M, the trace of the restriction of the Levi form

w^^w,

_%,j_J

of y to any complex subspace of T^°M of dimension q is nonnegative (respectively, positive).

2. If (p is 9-plurisubharmonic (strictly ^-plurisubharmonic), then (p is (^+l)-plurisubharmonic (respectively, strictly (g+l)-plurisubharmonic).

3. A real-valued function of class C² on a complex manifold is said to be strictly q-convex if its Levi form has at most q—1 nonpositive eigenvalues at each point. A function -0 on a complex space X is said to be strictly q-convex if, for each point x € X, there is a proper embedding of a neighborhood U of x into an open subset V of some complex Euclidean space and an extension of ^ \u to a strictly ^-convex function on V'. It follows that if (p is of class C2 and strictly ^-plurisubharmonic on M, then (/? is strictly ^-convex on M and on any analytic subset of M.

4. The set of smooth elements of ^(g) is dense in the following sense:

PROPOSITION 1.1 (Wu [W], Proposition 1). — If(p is a continuous strictly q-plurisubharmonic function on a Kahler manifold M and a is a positive continuous function on M, then there exists a C°° strictly q- plurisubharmonic function ^ such that \(p — i/}\ < a on M.

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In particular, it follows that the restriction of a continuous q- plurisubharmonic (strictly ^-plurisubharmonic) function to a complex sub- manifold of dimension q is subharmonic (respectively, strictly subhar- monic).

5. The Kahler manifold (M,^) is said to be hyper-q- complete if M admits a C°° strictly g-plurisubharmonic exhaustion function. If there exists a C00 g-plurisubharmonic 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.

6. Standard arguments show that if ^p and ^p' are continuous q- plurisubharmonic functions on M, then y? + (^/, max(y?,(^'), and the com- position ;^(<^) of any nondecreasing convex function \ with y?, are all q- plurisubharrnonic.

7. Hunt and Murray [HM] and Kalka [K] studied functions which satisfy a condition which they called g-plurisubharmonicity but which is weaker than the above notion.

The following result is contained implicitly in the work of Greene and Wu [GW], Ohsawa [02], and Demailly [D2]:

THEOREM 1.2 (Demailly, Greene-Wu, Ohsawa). — Let X be an analytic subset of dimension m <: q in the Kahler manifold M and let Y be the union of the singular set Xsmg with all irreducible components ofX which are noncompact or which have dimension strictly less than q.

Then there exist neighborhoods V of X and W of Y in M and a C°°

strictly (q-^-1)-plurisubharmonic function (p on V such that ^p\x exhausts X, W C V, and ^\w is strictly q-plurisubharmonic.

The proof is an easy modification of Demailly's [D2] proof of the analogous result for strictly g-convex functions, but we include a sketch here for completeness. Similarly, as in [D2] and in the work of Col^oiu [C], a hyper-^-complete submanifold admits a hyper-g-complete neighborhood, but we won't use this fact and the proof will not be sketched.

By a theorem of Richberg [Ri], a C°° strictly plurisubharmonic function on an analytic subset of a complex space extends to a C°°

strictly plurisubharmonic function on a neighborhood. Demailly proved a version of this theorem for strictly ^-convex functions in which the function is approximated by a strictly g-convex function on a neighborhood (see also [P]). A natural modification of Richberg's proof shows that if a

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function on an analytic subset of a Kahler manifold admits local C°° strictly g-plurisubharmonic extensions, then it admits a C°° strictly ^-plurisub- harmonic extension to a neighborhood. We will only need this fact for submanifolds and the proof is simple in this case.

PROPOSITION 1.3 (Richberg). — If^p is a C°° strictly q-plurisub- harmonic function on a complex submanifold N ofM (relative to the Kahler metric Q \ N ) ) then there exists a C°° strictly q-plurisubharmonic function ^ on a neighborhood ofN in M such that ^\N⁼ ^P-

Sketch of the proof. — Let m = dim N and suppose (£/, (z\,..., Zn)) is a holomorphic coordinate neighborhood in which N is the zero set of w = ( ^ i , . . . , Zn-rn)' If^{?' is a function on a relatively compact polydisk D in U obtained by composing ip with the associated projection mapping and C > 0 is sufficiently large, then the function ^)'-\-C\w\2 is strictly ^-plurisub- harmonic on a neighborhood of N D D. By using a partition of unity, one may patch these local extensions to obtain the desired function '0. Q. E. D.

The main point of Demailly's [D2] proof of Ohsawa's theorem [02]

is essentially the following version of the theorem of Greene and Wu [GW]

on the existence of subharmonic exhaustion functions (see [D2], Proof of Theorem 2):

PROPOSITION 1.4 (Demailly, Greene-Wu). — Suppose X is a com- plex space with no compact irreducible components, Y is an analytic subset which contains the singular set Xging, U is a neighborhood ofY in X, and h is a Hermitian metric on X \Y. Then there exists a C°° nonnegative function ip on X such that

(i) y? is positive and exhaustive on X \U, (ii) ^p vanishes on a neighborhood ofY, and

(iii) (p is subharmonic (with respect to h) on X \Y and strictly subharmonic on the subset {x € X \ (p{x) > 0} of X\Y.

Proof of Theorem 1.2 ([D2]). — The statement of the theorem makes sense and is trivial for q •= 0. We proceed by induction on q. Assume that q > 0 and that the theorem holds for nonnegative integers less than q.

Let A be the union of Xsmg and all of the irreducible components of X of dimension less than ^, let B be the union of all of the noncompact irreducible components of X of dimension q, and let C be the closure

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ofX\(AuB).

Applying the induction hypothesis to A and cutting off away from A, we obtain a C°° nonnegative function a on M and a neighborhood U of A in M such that a is strictly g-plurisubharmonic on U and exhaustive on U.

Next, by Proposition 1.4, there exists a nonnegative C°° function '0 on X such that supp-0 C B \ A = B \ (A U C), ^ > 2 on B \U,

^ exhausts B \ U, and '0 is strictly ^-plurisubharmonic on the subset N = ^((O, oo)) of B \ (A U C). By Richberg's theorem (Proposition 1.3), there exists a C°° strictly g-plurisubharmonic function T on a neighborhood YI of N in M such that r = -0 on TV. Let \: R —^ R be a C°° nondecreasing convex function such that \(t) = 0 if t < 1, \(t) > t if t > 2, and ^'(^) > 0 if t > 1. Since ^^([D, 1)) is a neighborhood of the boundary of N in X, we may assume (shrinking V\ if necessary) that there is a neighborhood V^

of X \ N in M such that r < 1 on Vi D l^. Thus the function ^(r) may be extended to a nonnegative C°° g-plurisubharmonic function f3 which is denned on the neighborhood V\ U V^ of X and vanishes on a neighborhood of A U C. Moreover, since ^ > 2 on B \ £7, /3 is strictly g-plurisubharmonic and positive on a neighborhood of B \ U in M and f3 exhausts B \ U.

Since dimC < g+1, every C°° function on the subset C \ Y of C \ Xsmg is strictly (g+l)-plurisubharmonic. As in the construction of /3, we may form a nonnegative C°° (g+l)-plurisubharmonic function 7 on a neighborhood of X such that 7 is positive and strictly (g+l)-plurisubhar- monic on a neighborhood of C \ U in M, 7 exhausts C \ U, and 7 vanishes on a neighborhood of A U B = Y.

It is now easy to check that if A is a C°° increasing convex function on M and A'(^) —> oo sufficiently fast as t —^ oo, then the function (p =. a + A(/3) + A (7) has the required properties on some neighborhood V of X and some neighborhood IV of Y = A U B. Q. E. D.

A C°° strictly g-convex function y? on a complex space X of pure dimension q has no local maximum points. Wu's approximation theorem (Proposition 1.1) implies that the same is true of the restriction of a con- tinuous strictly g-plurisubharmonic function on M to an analytic subset X of pure dimension q. Similarly, we have the following:

PROPOSITION 1.5 (Maximum principle). — Ifthe restriction (p\x of a continuous q-plurisubharmonic function (p on M to a connected analytic subset X of pure dimension q assumes its maximum value m = (p{xo} at some point XQ C X, then ^\x is constant.

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Proof. — Since ^ \Xreg is subharmonic, it suffices to show that y(x) = m at some point x e Xreg- Assume that this is not the case.

Taking successive singular sets Xsing, (^sing)sing, . . . , we may assume that, for some nowhere dense analytic subset V of X, we have XQ e Vreg and (p < m on X \ Y. Every C°° function on Vreg is strictly g-plurisubhar- monic since dimY < q. So, by applying Proposition 1.3, we may form a relatively compact neighborhood U of XQ in M and a C°° strictly q- plurisubharmonic function ^ on a neighborhood of U such that ^(^o) = 0 and ^ < 0 on {U \ {xo}) D V. In particular, on some neighborhood V of Y H 9U, we have -0 < 0 and hence ^ + e^ < m for every e > 0. We also have 9? + 6-0 < m on (X H 9U) \ V provided e is sufficiently small.

Therefore (p + e'0 < m = (^ + e^)(;ro) on X n 9(7. This contradicts the maximum principle for continuous strictly g-plurisubharmonic functions, so the proposition follows. Q. E. D.

In this paper, the main tool for obtaining the Bochner-Hartogs property is the following result of Grauert and Riemenschneider [GR] (who proved a version of the vanishing theorem for higher cohomology groups) and of Siu [S], Lemma 5.10 (who proved a version in the more general setting of harmonic maps into manifolds satisfying a certain curvature condition).

THEOREM 1.6 (Grauert-Riemenschneider, Siu). — Suppose f2 is a relatively compact domain in a Kahler manifold M and Q has a C°° (n-1)- plurisubharmonic defining function (p whose differential is nonzero at every point in 9fl.

(a) If f3 is a C°° function on fl, which is harmonic on Q and which satisfies the tangential Cauchy-Riemann equations 9b/3 = 0 on 9^1, then /3 is pluriharmonic on fl.

(b) If^p is strictly (n—l)-plurisubharmonic on a neighborhood of some point in 9fl, then ^(^ °) = 0-

Sketch of the proof. — Suppose first that (3 is a function as in (a).

Let 7 = 0 / 3 and let T] = ^Fy, where * denotes the Hodge star operator.

Then rj may be thought of as a C°° form of type (n,n-l) or as a form of type (0,n-l) with values in the canonical bundle KM on H. A computation in normal coordinates then shows that rj lies in the domain of the adjoint operator <9*. The Bochner-Kodaira formula is then (see [GR]

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or [S], formula 2.1.4)

11^111^) + ll^lli^) = W\i^ + / H2 .rda,

./0^

where da is the volume element on 90, and r is the trace of the restriction of the Levi form C((p) of (p to T110^^). Here, the curvature terms drop out because 77 is of type (n, n-1) and we have used the fact that 9(3 A 9(p vanishes at each point of 90. Since (3 is harmonic, the left-hand side is equal to zero. Since (p is (7i—l)-plurisubharmonic, we have r > 0 and hence Vrj = 0. A computation in normal coordinates now shows that (3 is pluriharmonic (see [S], Proof of Lemma 5.6(d)) and (a) is proved.

Suppose now that a is a C°° compactly supported form of type (0,1) with 9a = 0. Let (3 be the C°° function on 0 which vanishes on 90 and which satisfies -,A/3 = 9*9/3 = 9*a on 0 (where A = -(d*d+dd*) is the Laplacian), let 7 = a — 9(3, and let r] = *7. Then

9*77 = - * 9 * (*7) = *97 = 0 and 9rj = - ^ \fFa - 9*9/3] = 0.

Moreover, since a has compact support, 7 = —9/3 near 90', and, since /3 vanishes on 90, 9&/3 = 0. Applying the Bochner-Kodaira formula to rj as in the proof of (a), we get VT; = 0 and it follows that 97 = 0 (and hence d7 == 0). Thus 7 is a holomorphic 1-form on 0.

On the other hand, since r > 0 at some point, the Bochner-Kodaira formula implies that |7| = |9/3| = 0 on some nonempty open subset in 90. If U is a connected neighborhood of a boundary point and / is a C°° function on U D 0 which is holomorphic on U D 0 and which vanishes on U H 90, then one may extend / to a continuous function h on U which vanishes outside 0. But then 9h == 0 in the weak sense, so h is holomorphic. It follows that h, and therefore /, must vanish identically. Letting / be a coefficient of the holomorphic 1-form 7 with respect to some local holomorphic frame, we see that 7 vanishes on a nonempty open subset of 0 and hence on all of 0. Thus /3 is holomorphic outside the support K of a. Since /3 vanishes on 90, the above discussion (with f = f3) implies that /3 vanishes on each connected component of 0\K which is not relatively compact in 0. In other words, /3 has compact support and a = 9/3. Thus (b) is proved. Q. E. D.

Since the proof of the main theorem involves related arguments on a complete Kahler manifold, we close this section with a discussion of Green's functions on Riemannian manifolds. 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). We normalize G so

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that, for each point XQ G TV,

AdistrG('^o) = -^o

where 6x0 is the Dirac function at XQ and A = —(d*d+dd*) is the Laplacian.

We will use the same notation for the corresponding integral operator G given by

{Ga){x)= ( G(x^y)a{y)dV(y) ^x € N J N

for each suitable function a on N . If a is a C°° compactly supported function, then /3 = —Ga is a C°° bounded function with finite energy (i.e.

SN l^l² ^^< °°)^an(^ ^^{= a}' Moreover, f3(xv) —^ 0 if {xy} is a sequence in N with Xy —> oo and G { ' ^ X y ) —f 0. Such a sequence {xy} always exists and will be called a regular sequence.

2. Proof of the main result.

We begin with two lemmas. The first is a special case of a result of Nishino [Ni] who proved it without the assumption that M is Kahler. In the Kahler case, one may prove it using arguments contained in the proof of [NR], Theorem 4.6.

LEMMA 2.1. — Suppose (M,^) is a connected weakly 1-complete Kahler manifold and there exists a proper holomorphic mapping of some nonempty open subset ofM onto a Riemann surface. Then M also admits a proper holomorphic mapping onto a Riemann surface.

The second lemma often helps one obtain a holomorphic mapping to a Riemann surface. An elementary proof is given here. One may also prove this fact by using holomorphic equivalence relations (see, for example, [Ka]).

LEMMA 2.2. — If uj\ and c<;2 are two linearly independent closed holomorphic 1-forms satisfying uj\ A uj^⁼ 0 on a connected complex manifold M, then the meromorphic function h = uj\/uj'2 has no points of indeterminacy in M and is locally constant on the analytic set S = [x^M | (0:1)^ = 0} U [x^M | (^)x = 0}.

Proof. — Let I be the set of points of indeterminacy of h and, for each C ^ P1, let ________

^C = (^(MV))-1^) ^

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be the fiber over ^ (see [Gu]). Since the problem is local, we may assume that there exist holomorphic functions /i and /2 on M such that dfi = uji for i = 1,2. One may see that /i and /2 are locally constant on F^ for each ^ e P1 as follows. Near each smooth point x of F^ at which df^ ^ 0, we may choose holomorphic coordinates z = ( ^ i , . . . , Zn) in which î = /2 on a connected neighborhood of x. Since d/i A dz\ = 0, we have then fi = /iî) and hence h = /{(î). In particular, h is constant along each fiber of z\ and hence each of these fibers (being of codimension 1) must be an open set in F(^. Therefore, if v is a vector tangent to F(^ at rr, then 0:2 (v) = df^(v) = 0. This is also the case if df^ = 0 at x. Thus /2 is locally constant on F^ for each C G P¹, and, by symmetry, the same is true of /i. It follows that, if XQ e J, then f^^ô)) contains the connected component H^ of F(^ containing XQ for each ^ G P1. But {^}^^pi is an infinite collection of distinct analytic sets of pure dimension n—1, so this is impossible. Therefore I = 0.

It remains to show that h is locally constant on the analytic set 5, which we may assume to be irreducible. Since /i or /2 is then constant on 5, we may assume that S lies in some irreducible component T of the zero set of /i • /2 and it suffices to show that h is constant on T. By working near a generic point of T, we may assume that there exist holomorphic coordinates z = ( ^ i , . . . , Zn) on M in which T is the zero set of z\ and, for j = 1,2, fj(z) = z^3 Qj(z) where gj is a nonvanishing holomorphic function and rrij > 0. By replacing z\ by z\u for some (local) m^1 root u of ^j, we may also assume that g^ = 1. Since d/i A df^ = 0, we then get /i = fi{zi) and î = î(î). Hence the function h(z) = f[(zi)/fî) depends only on z\ and is therefore constant on T. Thus the lemma is proved.

The main step in the proof of the theorem stated in the introduction is the following proposition, which is, in a sense, a generalization of a theorem of Ohsawa [01] and a result in [NR].

PROPOSITION 2.3. — Let {M,g) be a connected noncompact Kahler manifold of dimension n on which there exists a continuous (n—1)- plurisubharmonic exhaustion function ^. Suppose there is a compact sub- set K of M such that, on each connected component E of M \ K, there exists a pair of real-valued pluriharmonic functions p\ and p2 with linearly independent differentials dp\ and dp^. Then M satisfies (BHD).

Remarks.

1. The above condition on E holds if, for example, there exists a

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nonconstant holomorphic function on E.

2. If M satisfies the above conditions and has more than one end, then there exists a proper holomorphic mapping of M onto a Riemann surface.

Proof. — Choosing a G R sufficiently large, we may assume without loss of generality that

K = [x e M I (p(x) < a} -^ 0.

Let E be a connected component of M \ K with noncompact closure and let p\ and p^ be pluriharmonic functions on E as in the statement of the proposition. We may assume that pi and p^ extend to pluriharmonic functions on a neighborhood of E.

There are three possibilities:

(a) The analytic subset X = {x e E \ (<9pi A 9p^)x = 0} of E is nowhere dense,

(b) X = E and the set Q of points in E which lie in a compact level of the holomorphic mapping

h=9^:E^?l 9?2

(see Lemma 2.2) is nonempty, or (c) X = E and Q = 0.

We will show that, if (a) or (c) holds, then M admits an exhaustion by C°° domains which are strongly hyper-(n-l)-convex at each boundary point in E\ and if (b) holds, then E admits a proper holomorphic mapping onto a Riemann surface. Briefly, in the case (a), we will work on a relatively compact (weakly) hyper- (n-l)-convex domain f^ in M. Combining y?, p{ + p|, and a strictly (n-l)-plurisubharmonic function near X (obtained from Theorem 1.2), we will obtain a strictly (n-l)-plurisubharmonic function near E D 9^1. In the case (&), we will show that Q = E and hence, by Stein factorization, one obtains the desired mapping. Finally, in the case (c), we will again work on a hyper-(n-l)-convex domain f2 and we will apply a standard patching argument to strictly (n-l)-plurisubharmonic functions on neighborhoods of fibers of h (or of a suitable holomorphic function if h is constant) to obtain a strictly (n—l)-plurisubharmonic function as in the case (a).

Suppose first that E satisfies the condition (a) and let Y be the union of all of the compact irreducible components of X of dimension n—1. The

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maximum principle (Proposition 1.5) implies that ^p(Y) is a countable set, so we may choose b G (a, oo) \ ip(Y) so large that K is contained in some connected component ^ of {x € M \ (p(x) < b}. By Theorem 1.2 (Demailly, Greene-Wu, Ohsawa), there exists a positive C°° function '0 on E \ Y which is strictly (n-l)-plurisubharmonic on a neighborhood o f X \ y i n £ ' . Therefore, since p^ + pj is strictly (n-l)-plurisubharmonic on E \ X and since ^-l(&) H Y = 0, the function 7 = r • (p^ + p|) 4- '0, where r is a sufficiently large positive constant, is strictly (n-l)-plurisubharmonic on a relatively compact neighborhood U of ^~l(b) D E in E \ Y. Applying Wu's approximation theorem [W] (Proposition 1.1) to the function 7—log(6—y?) on f2 D U, we get a C°° strictly (n-l)-plurisubharmonic function A on Ur\Q which approaches infinity at 9f^. Finally, let c > 0 and let \: R —> R be a C°° nondecreasing convex function such that \(t) = 0 for t < c, X'(t) > 0 for t > c, and -^(t) —> oo as t —> oo. If c is sufficiently large, then the function ^(A) extends to a C°° nonnegative (n-l)-plurisubhar- monic function on ^ U (M \ E) which vanishes on M \ E, which is strictly

(n-l)-plurisubharmonic near E D <9f2, and which exhausts f2 H E.

Assuming now that (b) holds, we show that Q = E. We first observe that Q is open for point-set topological reasons. For if XQ C Q and LQ is the (compact) level of h through a-o, then there is a relatively compact neighborhood U of LQ in E\ (h~l(h(xo)) \Lo). Since 9U is compact, there is a neighborhood V of h(xo) in P1 such that h^^V) D 9U = 0. Hence the level through each point in the neighborhood h~^(y) D U does not meet 9U and must, therefore, be a compact subset of U.

Next, let b > a be so large that there is a connected component f^ of the set {x € M \ y(x) < b} such that Q D ^ -^ 0, K C f^, and f^ D ^ is connected. If a;o ^ 0 H ^; then the irreducible component A of /^(/i^o)) containing a;o ls a compact subset of 0 D £J. For if {^} is a sequence in Qr\^l\{xo} converging to XQ and, for each ^, L^ is the (compact) level of h through Xy, then, by the Remmert-Stein-Thullen theorem (see [Gu]), A lies in the closure of |j L^. On the other hand, ^)\L^ is constant for each v and

v

^(^v) ~⁹' ^(^o) where a < (p(xo) < b. Therefore, since y = a or b at each boundary point of fl, H £', |j Ly must lie in some compact subset of Q. D E

v

and the claim follows.

Since h extends to a holomorphic mapping on a neighborhood of E (pi and /?2 extend by the choice of a), the set of critical values of ^f(^n^) is finite and the inverse image of this finite set is an analytic subset B of Q, D E. The above discussion implies that, if LQ is the level of h through

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a point XQ C Q H f2 and Z/o H Q is smooth, then LQ C Q D f2. Therefore Q H ^ \ Z? is a closed subset of the connected set fl, r\ E \ B and hence, since Q is open, we must have equality. In particular, Q D f2 is dense in fl.r\ E. Applying the above again, we see that if LQ is any level of h which meets f2, then every irreducible component of LQ that meets Q must be a compact subset of f2 D E. It follows that Lo is compact. Thus ^2 D E C Q and, since the choice of b was arbitrary, we get Q = E. Therefore every level of h is compact and, by Stein factorization [St], we obtain a proper holomorphic mapping of E onto a Riemann surface.

Finally, assuming that X = E and Q = 0 (i.e. that E satisfies the condition (c)), we apply a modification of a construction due to Ohsawa [01] for the case of a weakly 1-complete surface. We also assume for now that the mapping h: E —^ P1 is nonconstant.

We first show that the union C of the collection C of all compact irreducible components of fibers of h is a nowhere dense analytic subset of E. For if KQ is a compact subset of E and LQ is a level of h, then any compact irreducible component Co of LQ which meets KQ must lie in the compact subset (fi~^l((p(Ko)). Moreover, since Co 7^ LQ^ Co must meet some irreducible component HQ of the analytic set H = {x € E | {h^}x = 0}

and, since h is locally constant on H^ we have HQ C LQ. Only finitely many irreducible components of H meet (^-1((^(Ao)), so the collection of all levels LQ with such an irreducible component Co is finite. It follows that C is locally finite in E and hence that C is an analytic set.

The set ^(C) is discrete, so we may choose a number b € (a, oo)\(^((7) so large that there is a connected component ^ of {x € M \ (p(x) < b}

such that K C ^2 and fl, D E is connected. We may also choose a relatively compact neighborhood W of y?"1^) H E in E \ C. For each point x e ^^-l(&) H £¹, the analytic set h^^h^x)) \ C has no compact irreducible components and hence, by Theorem 1.2, there is a C°° strictly (n-l)-plurisubharmonic function 71 on a neighborhood V\ of h~^{h(x))\C in E. Moreover, we have h^^D^) D W C V\ for any sufficiently small neighborhood D[ of h(x) in P1. There is also a nonnegative C°° function Ai with suppAi C D[ and Ai =. 1 on some neighborhood D\ of h{x).

We may, therefore, choose C°° strictly (n-l)-plurisubharmonic functions 7i 5 • • • 5 7m on open sets V\,..., Vm in E, respectively; open sets D\,..., Dm and D^ ..., D^ in P1; and nonnegative C°° functions A i , . . . , \m on P1

such that, for each j = 1,... ,m, we have Xj = 1 on Pj, suppA^- C Dp h^^nW C Vj, and (^(^n^ C /i-^Pi) U .. .U/i-^D^). Moreover,

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by Lemma 2.2, h is locally constant on the analytic set S = {x e E \ {9p2)x = 0}, so h(S H ^-l(&)) is a finite set and we may assume that, for each j, \j{h) is constant near each point of W D S.

We now show that, for a sufficiently large positive constant s, the C°°

function

m

7EE5•(p2)2+]^A,(fa)7, j=i

is strictly (n-l)-plurisubharmonic on some neighborhood of (p~^l{b)^\E. It is easy to see that 7 is strictly (n—l)-plurisubharmonic on a neighborhood oi S^[ip~^l(b)^\E, since, near each point of this set, each of the nonnegative functions A i ( / i ) , . . . , \m(h) is constant and at least one of the functions is positive. Given a point x near ^^(b^^E and a tangent vector v e T^'°M, we have

m

/;(7)(^^)=25|^p2(^)|²+^7,(^(A,)(^,^) j=i

m m

+^2Re[9A,(^)^(^]+^A,(^))/;(7,)(^,^).

.7=1 J=l

If x is not near 5', then, since {Op2)x 7^ 0 and <9pi A <9p2 = 0, we may choose holomorphic coordinates ( ^ i , . . . ,Zn) near x in which p2 = 2Re^i, Pi = Pi(^i), and

^ ^ c ^ ^ ^

9/92 9^1 QZ\ \ l J

r^j

Thus 9/i = Q—^i and hence |<9p2^)|2 = |^i(v)|2 ^ go|^(v)|2 for some positive constant qo. Since A i , . . . , \m > 0, since m a x ( A i ( ^ ) , . . . , \m(h)) >

0 near ^^-l(&) H £', and since, for each j, 7^ is strictly (n-l)-plurisubhar- monic on the neighborhood Vj of suppA^-(^) H ^'~^l(&), there exist positive constants q\ and q^ (independent of s) such that, for every e > 0, for every point x e y^"1^) H £1, and for every collection of orthonormal tangent vectors e i , . . . , e^-i G r^'°M, we have

n-l n-l

E ^M^^ ^) ^ (^o - gi • (1 + (2e)-1)) . ^ |^(e,)|2 - ,6gi + 92.

Z=l 2=1 "

Thus if we choose 6 < Iq^/qi and 5 > (2go)-lgl •(l+(2e)~1), then 7 will be strictly (n-l)-plurisubharmonic near points in (^(^nE which lie outside an (arbitrarily small) neighborhood of 5 as well as those which lie near S.

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Proceeding now as in the case (a), one gets a C°° nonnegative (n-1)- plurisubharmonic function on f2 U (M \ E) which vanishes on M \ E, which exhausts £'Df2, and which is strictly (n-l)-plurisubharmonic near Er\9fl..

If h is constant, then <9(cipi + c^p^) = 0 for some pair of constants ci,C2 G C which are not both zero. The function / = cipi + 02 p2 on £J is holomorphic and nonconstant; because the functions 1, pi, and p^ are linearly independent. One may now proceed as above by using / in place of the mapping h.

Thus for each of the (finitely many) connected components E of M\K which have noncompact closure, either E admits a proper holomorphic mapping onto a Riemann surface (and hence a C°° plurisubharmonic exhaustion function), or there exists an arbitrarily large relatively compact domain Q in M and a C°° nonnegative (n—1)-plurisubharmonic function on 0 U (M \ E) which vanishes on M \ E, which exhausts E D 0, and which is strictly (n-l)-plurisubharmonic near E D 9fl.. If all of these connected components of M \ K have the former property, then M admits a C°°

plurisubharmonic exhaustion function and a complete Kahler metric and Lemma 2.1 implies that there is a proper holomorphic mapping of M onto a Riemann surface. If at least one of these connected components has the latter property, then H^{M,0) = 0. For if a is a C°° compactly supported form of type (0,1) and Qa = 0, then we may choose a C°°

relatively compact domain 0 which contains the support of a and which admits a C°° (n—1)-plurisubharmonic defining function which is strictly (n-l)-plurisubharmonic near at least one connected component of 90. The theorem of Grauert and Riemenschneider [GR] and Siu [S] (Theorem 1.6) then implies that ^(0, 0) = 0 and hence that there is a C°° compactly supported function (3 on 0, and therefore on M, such that 9(3 = a. Thus M has the property (BHD). Q. E. D.

To apply Proposition 2.3, it will be convenient to have the following lemma, which is essentially a special case of [NR], Theorem 2.6.

LEMMA 2.4. — Let {M,g) be a connected noncompact complete hyperbolic Kahler manifold, let MQ be a C°° relatively compact domain in M, and let E be a connected component of M \ MQ with noncompact closure. Then there exists a pluriharmonic function r on M such that 0 < T < 1, T has finite energy, and, for every regular sequence {xy}

in M approaching oo (see Section 1), r(xv) —> 1 if Xy € E for y ^> 0 and r{xy) —^ 0 ifXy G M \ E for v > 0.

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The construction of r is contained within the proof of [NR], Theo- rem 2.6. That r < 1 is not proved explicitly, but one can easily verify this property by forming an exhaustion of M by C°° relatively compact domains and writing the Green's function on M as the limit of the corresponding sequence of Green's functions.

We are now ready to prove the main result.

THEOREM 2.5. — Let (M,g) be a connected Kahler manifold of dimension n which has exactly one end and which admits a continuous (n—1)-plurisubharmonic exhaustion function y?. Assume that at least one of the following conditions is satisfied:

(i) ^p plurisubharmonic,

(ii) (M, g) is complete and hyperbolic, or (m) (p is of class C°°.

Then M satisfies (BHD).

Remark. — An example of Cousin [Co] shows that one cannot remove the requirement that M have only one end (see, for example, [NR], Example 3.9).

Proof. — Assuming that there exists a C°° compactly supported form a of type (0,1) such that 9a = 0 and [a] ^ 0 in H^{M, 0), we will show that there is a proper holomorphic mapping of M onto a Riemann surface.

We first assume that (i) or (ii) holds. Fix a C°° relatively compact domain MQ in M such that suppa C MQ and M \ MQ is connected.

If (p is plurisubharmonic (i.e. (i) holds), let a > sup^ ip and let ^ be the connected component of {x C M \ ^p(x) < a} containing MQ.

Then, by a theorem of Nakano [N] and Demailly [Dl], the (weakly 1- complete) domain f^ admits a complete Kahler metric g ' . Moreover, (^,p') is hyperbolic (in fact, the Green's function vanishes at the boundary) and if a is sufficiently large, then f2 \ MQ is connected. It suffices to show that fl, admits a proper holomorphic mapping onto a Riemann surface (for a suitable choice of a). For we may then form an exhaustion of M by such domains. Applying a theorem of Narasimhan [Na], Corollary 1 to the Cartan-Remmert reductions, we get the required mapping on M. If (p is not plurisubharmonic, we set (f^(/) = {M^g).

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Thus, in either case, (^,</) admits a continuous (n-l)-plurisubharmonic exhaustion function and a positive symmetric Green's function G(x,y), and the Kahler metric g ' is complete. Therefore, since <9*a is a C°° function with compact support on f^, the function (3 ^ -2G(<9*a) is a C°° bounded function with finite energy and —-A/3 = 0*9(3 = 9*a.

Moreover, {3{xy} —> 0 for any regular sequence {x^} in f2 approaching oo (see Section 1). The form 7 = a - 9f3 is a harmonic form in L^ i(^,^');

the harmonic projection of a. By the Gaffney theorem [G], 7 is closed (and coclosed). In particular, 7 is a holomorphic 1-form on f^ (since 9^ = 0) and f3 is pluriharmonic on the connected set E =. fl, \ MQ. Since [a] ^ 0 in H^(M, 0) and suppa C Mo CC ^, we have [a] ^ 0 in H^, 0).

It follows that f3 is not constant on E. For, if f3 were constant on E^

then, since f3 vanishes at infinity along any regular sequence, we would have f3 = 0 on f^ \ Mo. Hence 7 would vanish on E and, therefore, on f2, since 7 is harmonic. But we would then get a = 9f3, where f3 is a C°° function with compact support in f^; which contradicts the nonvanishing of [a\. Thus (3\E is not constant (and hence is nowhere locally constant).

If 7 = 0, then /3 \E is a nonconstant holomorphic function on the end E. Proposition 2.3 then implies that Q satisfies (BHD) and hence, since H\ (f^, 0) 7^ O, there is a proper holomorphic mapping onto a Riemann surface. Thus we may assume that 7 is not everywhere zero. Let p = Re f3\E and p ' = lmf3\E- If the functions 1, p, and p ' are linearly independent on £', then, again, Proposition 2.3 gives one the required mapping. If 1, p, and p1

are linearly dependent on E, then, since p and p1 vanish at infinity along any regular sequence, the functions p and p ' must be linearly dependent.

Therefore, after multiplying a by a suitable nonzero complex constant, we may assume that f3\E= P (i.e. f3\E is real-valued) and hence that 7^= —9p.

To obtain a second pluriharmonic function (i.e. to apply Proposition 2.3) we will use Lemma 2.4 to obtain a pluriharmonic function on a covering space of Q with two ends and then push down to E.

We first observe that, for any point XQ G E, the mapping 7Ti(£1, xo) —^

7Ti(f2,a;o) is not surjective. For if the mapping is surjective, then, since the C°° closed real 1-form 6 = —7 — 7 on Q is equal to dp on E, it follows that the function po given by

[x

po{x) = p{xo) + 9 V . r e f 2

JXQ

is a well-defined C°° function with dpQ = 6 = dp on E and po(«^o) = p(xo).

Hence po = P on £1, so f3 — po is a C°° function on 0 which vanishes on

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E = fl. \ MQ and which satisfies

<9(/3 - po) = 9(3 - 9po = a - 7 + 7 = a.

But this contradicts the assumption that [a] ^ 0 in 1^(^,0), so the mapping cannot be surjective.

Fix a point XQ e E, let F be the image of 7Ti(£, 2:0) m 7Ti(^ ^0)5 and let TT : n —)• ^2 be a connected covering space with 7r^(7ri(^,.ri)) = F for some point x\ € Tr'^.ro). Since F is a proper subgroup and E is a C°°

domain, TT maps a neighborhood of the closure £1 of the connected component £1 of £ = Tr"1^) containing a;i isomorphically onto a neighborhood of E and f^\£i is not relatively compact in ^ (i.e. ^2 has at least two ends).

In particular, E\ is a hyperbolic end of f^ with respect to the complete Kahler metric 7r*(/. Therefore, by Lemma 2.4, there exists a pluriharmonic function r on f2 such that 0 < r < 1, r has finite energy, and, for every regular sequence {xy} in 0 approaching oo,

. _ f l i f ^ e £ i f o r z / > 0

^Im^T^}-^ i f ^ e ^ \ £ i f o r ^ > 0 .

Since TT maps £1 isomorphically onto £, the restriction r [^ determines a pluriharmonic function on E. If the differential of this function and the differential of the pluriharmonic function p are linearly independent on E, then, by Proposition 2.3, f2 satisfies (BHD). Therefore, since p is nonconstant, it now suffices to assume that there exist real constants r and s such that p o TT == rr + s on E\ and to obtain a contradiction.

The first observation is that if -7 = 7r*7, then 7 is a closed form of type (0,1) on ^ which is equal to the form —9{rr + s) on the nonempty open set E\ and hence on the entire set f2. Therefore, on the nonempty open set E \ E\, we have

—9{p o 7r) == 7 = —9{rr + 5).

Hence the restriction of the function (p o 7r) — (rr + s) to E \ E\ is real- valued and holomorphic and is therefore locally constant. Thus if E^ is a connected component of E which is not equal to £'1, then, for some real constant s', we have p o TT = rr + s ' on £'2. Now since 7r(£'i) = Tr(E^) = E, we may choose a regular sequence {Xy} in E and sequences {^} and {zy}

in £'1 and E^ respectively, such that Tr(^) = Tr(^) = o*^ for each v. The sequences {^} and {zy} are then regular sequences in f^, because the lifting of the function v = —G(xo^') to ^ is a negative subharmonic function and v(y^),v(z^) —> 0. Therefore r{yy) —^ 1 and r { z ^ ) —> 0. Since p vanishes at infinity along any regular sequence, we get

0 == limp(.Ti,) = lim(rr(^) + s) = r + s

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and

0 = limp(a^) = \\m(rr{z^) + s ' ) = s⁷.

Therefore, for each point x (E E and each pair of points y € £'1 riTr""1^) and z € J^nTr"1^), we have r{y) —1= r~lp{x) = r{z). Since 0 < r < 1 on f^, this is not possible. Thus the proof is complete for the cases (i) and (ii).

Finally, suppose (p is of class C°° (i.e. the condition (iii) holds). As in the case (i), we fix Mo, a, and f2. Here, we choose a to be a regular value of (p and we let g ' be the restriction of g to f2. It suffices to show that there exists an arbitrarily large choice of a for which f^ admits a proper holomorphic mapping ^ onto a Riemann surface. For if ^ is such a mapping, then (p is constant on each level of ^. Hence, near a generic point of ^2, there exist holomorphic coordinates ( ^ i , . . . , Zn) in which ip is a function of z\ and we get

W = -^l—dz.dz, = (2^1i)-lA(^l^l > 0.

oz\oz\

Thus it will follow that (p is plurisubharmonic on each of the domains f2.

Letting a —>• oo, it will then follow immediately that (p is plurisubharmonic on M as in the case (i) and the proof in the case (iii) will be complete.

As in the proof of Theorem 1.6 (Grauert-Riemenschneider and Siu), since the boundary of Q. is regular, we have 7 = a — 9 {3 on Q where 7 is harmonic and f3 vanishes on 9f2. The metric g ' is not complete, but the proof of Theorem 1.6 shows that 7 is closed. In particular, {3 is pluriharmonic on the connected set E == f2 \ Mo. Moreover, since [a] ^ 0, (3 is nonconstant onE.

If 9^1 is not connected, then we may form a C°° function r on f2 which is harmonic on Q and locally constant, but not constant, on 9f2. By Theorem 1.6 (a), r is then pluriharmonic on ^ and, since f3 vanishes on 9f2, dr and df3 are linearly independent on E. Therefore, by Proposition 2.3, f2 admits a proper holomorphic mapping onto a Riemann surface. Thus we may assume that 90. is connected.

If a' is a regular value in the interval (sup^ ^, a) which is sufficiently close to a and ^/ is the connected component of {x C M \ (p(x) < a'}

containing Mo, then f^ CC 0 and the set E ' == Q' \ Mo is connected. As above, we may write 7' = a — 9f3' on f^' where 7' harmonic and ft' vanishes on Q^l'. If df3 and d{3' are linearly independent on -E", then Proposition 2.3 implies that 0' satisfies (BHD). If df3 and df31 are linearly dependent, then {3 is constant on 9^1'. Since /3 is nowhere locally constant in ^ \ f^ and /3 = 0 on <9^2, the maximum principle implies then that f3 is equal to a nonzero

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constant on Q^t'. Hence the restriction of the real part or the imaginary part o f / 3 t o ^ \ f ^ i s a positive or negative pluriharmonic function which vanishes at points in 9f2. It follows that ^ admits a plurisubharmonic exhaustion function and hence, a proper holomorphic mapping onto a Riemann surface.

Therefore, by the above remarks, M satisfies (BHD). Q. E. D.

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Manuscrit recu Ie 28 mai 1996, revise Ie 20 fevrier 1997, accepte Ie 16 juillet 1997.

T. NAPIER, Lehigh University

Department of Mathematics Bethlehem, PA 18015 (USA)

&

M. RAMACHANDRAN, SUNY at Buffalo

Department of Mathematics Buffalo, NY 14214 (USA).