BY FREELY OPERATING ARITHMETIC LATTICES

A ’ F

E ^BERHARD O ^ELJEKLAUS C HRISTINA S CHMERLING

Hyperbolicity properties of quotient surfaces by freely operating arithmetic lattices

Annales de l’institut Fourier, tome 50, n^o1 (2000), p. 197-210

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

HYPERBOLICITY PROPERTIES OF QUOTIENT SURFACES

BY FREELY OPERATING ARITHMETIC LATTICES

by E. OELJEKLAUS and C. SCHMERLING

0. Introduction.

During the last decades arithmetic quotients of bounded symmetric domains have been intensively studied. However, only few results on the hyperbolicity of these quotients are known. A complex space Y is called hyperbolic^ if the Kobayashi pseudo metric dy on Y is a metric. Brody [Br] has proved that a compact complex space is hyperbolic if and only if every holomorphic map from C into Y is constant. In this paper we shall prove that a special class of twodimensional arithmetic quotients of bounded symmetric domains is hyperbolic.

THEOREM. — Let D be a bounded symmetric domain in C² and r C Aut° D an irreducible arithmetic lattice which operates freely on D.

Then the cusp-compactification X of X == D/Y is hyperbolic.

Note that Y is called hyperbolic modulo S C Y if d y ( x ^ y ) = 0 implies that x = y or that x € S and y € S. Let TT : X —> X be the minimal resolution of the singularities of X and R := 7T~1{X\X).

COROLLARY. — X is a minimal surface of general type and hyper- bolic modulo R.

Keywords: Bounded symmetric domain — Arithmetic lattice — Surface of general type.

Math. classification: 32M15 - 32N15 - 32F45 - 32J15.

In the first section we briefly recall the relevant aspects of the theory of arithmetic quotients of bounded symmetric domains. Then we deal with the logarithmic canonical bundle. Our method of proving Kobayashi- hyperbolicity is to construct a so-called hyperbolic pseudo metric on X.

This is a hermitian pseudo metric with distance-decreasing property, which degenerates only on a proper subvariety. In our situation the push-down of the Bergman metric yields a Hermitian metric with negative holomorphic sectional curvature on X = D / F . We modify this metric by multiplying it with a suitable non-negative real function to get a pseudo metric on X which still has the distance-decreasing property for holomorphic mappings HI -^ X and degenerates exactly in the cusp points. For this purpose the vanishing order of the function has to be carefully adjusted. We apply a method which was used by Schumacher and Takegoshi in the compact case [ST]. Of main importance for us were the results of Mumford in [Mu], in particular, the estimates for the degeneration of the induced metric on X C. X along the boundary divisor R^ see also [Ko], Our considerations are also based on the explicit description of the minimal resolution X for the cusp singularities given by Hemperly [He] for arithmetic ball quotients and by Hirzebruch [Hi] for the Hilbert modular surfaces. In the situation D = II² we apply a theorem of Tai [AMRT], also proved with different methods by Mumford [Mu], which guarantees for sufficiently small F many pluricanonical holomorphic sections over X which vanish of high order along the boundary divisor jR.

The above theorem is a generalization of results of the doctorate thesis [S] of the second-named author, who would like to thank Siegmund Kosarew for encouragement and helpful discussions.

1. Neat lattices and cusp resolution.

Let D be a bounded symmetric domain in C2 equipped with the Bergman metric and Aut D the real Lie group of biholomorphic automor- phisms of D. Recall that D is either biholomorphic to the two dimen- sional ball B² = {(^,w) € C² | \z\² + |w|² < 1} or to the twofold prod- uct H² of the upper half plane. The groups AutB² = 5T/(2,1,C)/Z3 and Aut° H² = PGL(2, R) x PGL{2, R) are simple and semisimple respectively.

Throughout this paper F denotes a lattice in G:= Aut° D, i.e. F is a discrete subgroup of G and the quotient G/T has finite volume with respect to the invariant Haar measure on G. The group r operates properly

discontinuously on D. In particular, the groups F^ = {7 € f\^z = z} are finite for all z e D. The quotient D/F carries a complex structure such that the natural surjection D —> D/T is a holomorphic covering, ramified in z C D if and only if I\ ^ {e}. A lattice F in POL(2, R) x PGL(2, R) is called reducible, if F is commensurable with the direct product A = Fi x Fa of discrete subgroups Fi,!^ C PGL(2^R'), i.e. the intersection F D A is a cofinite subgroup (a subgroup of finite index) of both F and A. Otherwise r is called irreducible.

At first we consider the case D = HI2. Let K = 'Q(Vd) be a totally real quadratic number field, d 6 N square-free, and OK the ring of integers ofK.

The embedding

a = r + 5Vd t — ^ r — 5Va =: a, r, s € Q, of K into M induces an embeddding

SL(2,K)-> SL{2,R) x5^rL(2,R), A ^ (A, A).

Now SL(2,K) can be viewed as a subgroup of S'L(2,1R) x 5'L(2,R), and we define FK to be the image of SL(2,OK) under the canonical surjection SL(2,R) x SL{2,R) -> PGL(2,M) x PG?L(2,R).

By a theorem of Selberg every irreducible non cocompact lattice F in PGL(2,R) x PGI^.R) is commensurable to some FK as above, up to an inner automorphism of PGZ^.R) x PGL(2,R). Therefore these lattices are arithmetic in the sense of Borel [Bo].

Every lattice in AutB² is irreducible. Note that there are non- arithmetic lattices in AutIB2. However, if K = Q(\/^d), d G N square- free, then every lattice F in G := AutB² which is commensurable to G H SL(3, OK) /^3, is arithmetic.

If the quotient X = D/T of D by an irreducible arithmetic lattice F is not compact, then it can be compactified by finitely many points (cusps) to a normal projective algebraic variety X, the so-called cusp-compactification ofX.

Next we roughly describe the cusp-compactification of quotients of bounded symmetric domains in C² by irreducible arithmetic lattices.

This construction is a special case of a general construction in arbitrary dimensions given by Baily and Borel [BB]. For details we refer to [He], [Ho]

in the case D = B² and to [Fr2] for D = E².

The holomorphic action of G = Aut° D on D extends to a topological action on the topological closure D of D in C2, and for x e 9D the isotropy group Px = {g € G \ gx = x} is a minimal parabolic subgroup of G with unipotent radical L^. Let F be an irreducible arithmetic lattice in G. A point x € 9D is called a rational boundary point (with respect to F) if r D Ux is a cocompact lattice in Ux. The set of rational boundary points is invariant under the r-action on D, and the F-orbit Yx = {^x 17 e F} of a rational boundary point is called a cusp. The assumptions on F imply that the set Ay of cusps is finite. The complex structure on X := D/T can be extended to a (uniquely determined) complex structure on X := X U Ar such that X is a normal complex space, called the cusp-compactification of X. Every cusp is a singular point of X. If F does not operate freely on D, then there are also finitely many singular points in X corresponding to z € D with I\ 7^ {e}. In every cusp K, G Ay we fix a point x^ and define I\ :== r D PX^' Let r' be a cofinite normal subgroup of r. The cusp K decomposes into finitely many F'-cusps ^ = ^ U . . . U /^ , T\ = [F : F^F'], and the holomorphic surjection X' = D/V —> X extends to a holomorphic map a : X' —^ X with cr"^) = {^' € Ay \ ^ ' C K.} and ^cr"1^) = r^.

The vector space Vn of holomorphic r-automorphic forms of weight n on D is finite dimensional, and each basis of Vn provides a holomorphic embedding of X into some PN for ne N sufficiently large. In particular, X is a projective algebraic variety, and there is a very ample holomorphic line bundle 0{n) on X such that the global holomorphic sections of 0(n) are in 1-1-correspondence with the elements of Vn.

Hereafter we assume that r operates freely on D. Then X = D/Y is a complex manifold, and the natural surjection D —>• D/Y is an unramified holomorphic covering. For the minimal resolution TT : X —> X of the singularities o f X the divisors Tr"1^), K € Ar, have normal crossings. Their structures have been described in detail by [He] and [Ho] for D = B², and by [Hi] for D = H2:

a) D = B2 : E^ := Tr"^) is a smooth elliptic curve for every K € Ar.

b) D = H2: RK := Tr"1^) is either a rational curve BQ with an ordinary double point and self intersection number B^ < — 1 or a cycle of r ^ 2 smooth rational curves B o , . . . , Br-i with the following structure:

a) r == 2: ^1,^2 intersect transversally in two points, and B2 ^ —2, Bj <, -3.

/?) r > 3: BiBi^ = BoBr-i = 1, B² <-2, B²^ < -3, 0 <.i <, r-2.

For the following considerations we have to put an even stronger

assumption on r, namely that F is a neat subgroup of G. Recall that an element g e GL{n, C) is called neat, if the subgroup of C* generated by the eigenvalues of g is torsion-free. A subgroup H C GL(n, C) is called neat if every element in H is neat. In our situation we call F a neat subgroup of G if every element of F has a neat representative in 677(2,1,C) (resp. in 5L(2,]R) x S'L(2,IR)). A neat arithmetic lattice F in G operates freely on D because every finite subgroup of F is trivial.

For every finitely generated number field with ring of integers o and every 7 e SL(n,o}, 7 7^ e, the group SL{n,o) contains a cofinite neat normal subgroup Ai with 7 ^ Ai, see [Bo]. It follows that every subgroup A C SL(n,C), which is commensurable with SL(n,o), has a similar property: If 7 e A, 7 ^ e, and Ai is a cofinite neat subgroup of SL(n,o) with 7 ^ Ai, then A H ^}g^gA^g-1 is a cofinite neat normal subgroup of A. This implies

LEMMA 1. — Let D be a bounded symmetric domain in C2 and r an arithmetic subgroup of G = Aut° D. For every finite set M C F with e ^ M the group F contains a cofinite neat normal subgroup r⁷ with

M n r = 0. Q

2. The logarithmical canonical bundle.

Beside the minimal resolution X -^ ~X we take more generally into account all resolutions TT : X -^ ~X of the singularities of ~X such that R :=

Tr'^X^) has normal crossings, and we identify X with TT'^X) C X. The canonical line bundle of X can be extended to a holomorphic line bundle KX(\ogR) over X, which is characterized by the following property: For every open polycylinder A C X satisfying

A n R = {(z, w) e A I z = 0} (resp. A H R = {{z, w) e A | zw = 0}) a holomorphic section of KX(logR) over A has the form

/ \(^z A j \ / .fdz/\dw\

a{z, w)[ — A dw , resp. a ( z , w) ———— ) ,

\ z / \ zw /

where a(z,w) is holomorphic in A. The bundle KX(logR) is called logarithmical canonical bundle of X (with respect to K). From the above property it follows immediately that KX(\ogR) is isomorphic to the holomorphic line bundle K (g) L, where K is the canonical line bundle of X

and L is the holomorphic line bundle associated to the boundary divisor R. Note that dim^X,!/¹) = 1 for every n € N. For each irreducible neat arithmetic lattice r in G and any n e N4" the pull-back 7r*(0(n)) of the ample line bundle 0{n) on X is isomorphic to the line bundle (KX^ogR))^ on X, see [He] and [Mu], Prop. 3.3 and 3.4, and therefore isomorphic to X71 (g) I/1. We note for later use:

LEMMA 2. — For every XQ^ x\ ^ X C X , XQ ^ rci, and n € N sufficiently large, there exists some s G H°(X^ Kn 0 L71"1) with s(xo) ~^- 0 and s(x\} = 0.

Proof. — The bundle 0(n) is very ample for n ^> 0. In particular, there exists some t C H°(X,0(n)) with t(7r(.ro)) 7^ 0 and ^(7r(a;i)) = t(y) = 0 for every cusp point y € X\X. Then 7r*(t) € ^(X.K71 (g) J71) vanishes on J?U{.ri} and 7r*(t)(a;o) 7^ 0. Define s := 7L-jt)- for some d -^ 0 in H°{X,L). D For a cofinite normal subgroup F' of F let TT' : X' —r X be a resolution of the singularities of the cusp-compactification X of X' = P/F' such that the divisor K := X'\X' has normal crossings. We denote by K ' the canonical line bundle of X' and by L' the holomorphic line bundle on X' associated to the divisor R ' . Let X' be the minimal resolution with canonical bundle K1.

LEMMA 3. — Let F be an irreducible arithmetic lattice in Aut° HI². Then F contains a cofinite neat normal subgroup F' with the following properties:

1) X' is a minimal surface of general type.

2) For every finite set B C X' C X' there exist no <E N and s e

^(X', K^⁰ (g) L'-¹) with s(x) + 0 for every x € B.

Proof. — We may (and will) assume that IHP/F is not compact and, by Lemma 1, that F is neat. Let F be commensurable with YK = SL(2, oj<) for some totally real quadratic number field K. For n 6 N let Fj<(n) be the principal congruence subgroup of FK with respect to n. The cusp- compactification ofIHP/F^n) does not contain any rational or elliptic curve if n is sufficiently large [Fri]. Choose F' C Fj<(n) to be a cofinite normal subgroup of F. Then X' is a minimal projective algebraic surface with only finitely many rational (and no elliptic) curves, all of them contained in R ' . Therefore X' is either a minimal surface of general type or a

minimal 7^3-surface, but the latter case would contradict Lemma 2, since dimi^X',!/⁷¹) = 1 for every n e N.

To prove 2) note that the pluricanonical map fk maps X' holo- morphically and birationally onto fk{X') C P^ for k > 5. It suffices to study the case B == {xo}. Since XQ is not contained in any (—2)-curve, /fe(^o) ^ fk(^)' For m ^> 0 there is an irreducible hypersurface S C P^

of degree m with /^(^'o) ^ 5'? fk(K) C 6'. This implies the existence of a holomorphic section t e H°(X1'.K'^) which vanishes exactly on f^1^).

Define no := m/c and s := -jr for some d' ^ 0 in ^°(X', L'). D Finally we will study the case D = B² in more detail, assuming that F is a neat arithmetic lattice in AutB2. Let F' be a cofinite normal subgroup of r. Since E' := X'\X' is a disjoint union of finitely many smooth elliptic curves £^/, ^! € Ar', the unramified holomorphic covering map X' —r X of degree [F : F7] extends to a holomorphic map p : X' —> X, ramified along E ' . We can easily calculate the ramification order of p over £^, /^ € Ay. Let E^^...^E^ be the irreducible components of p"^!^) with r^ = [F : r^r'] and F^ :== F' H I\. The map p is ramified of order [r : r'] • [r : r^r']~1 = [r\ : r'J along these curves and the ramification divisor of p equals

D= E E^-^i- ¹ )^-

/<eAr ^K=I

Let ^ := E^=i^,^ ^{for K} ^ ^^ Since K ' = p^K) (g) [P] (cf.

[BPV], p. 41) and since p*(Z) = (^^Ar^^^^ we obtain

LEMMA 4. — Let r be a neat arithmetic lattice in Aut B2 and let r' be a cofinite normal subgroup ofT. Then

p*(jr

0 r

-

) = K^ ® 0 [E^]

-^-^]

/<eAr

for aJ2 n € N. D

3. Degeneration of the Bergman metric along the boundary divisor.

In 1977 Mumford generalized the Proportionality Theorem of Hirze- bruch to the case of non-compact quotients of bounded symmetric domains

by neat arithmetic lattices. For our setting we extract the following result from the Main Theorem and Proposition 3.4 in [Mu], which yields a descrip- tion of the invariant Bergman metric h on X along the boundary divisor R = X\X on X.

LEMMA 5 (Mumford). — Let D be a bounded symmetric domain in C² and F a neat irreducible arithmetic lattice in G = Aut° D. The holomorphic tangent bundle T and the canonical line bundle of X = D/F can be extended to holomorphic vector bundles t and KX(\ogR) on X such that h and (det/i)"¹ define singular metrics on t and KX(\ogR) respectively, in the following sense:

For every open polycylinder A C X satisfying

A H R = {{z, w) <E A z = 0} (resp. A D R = {(z, w) C A | zw = 0}) and for every local basis oft and of KX(\ogR) over A the following is true:

1) There are constants C > 0 and m, n € N such that :

IM^)1 < ^.(logM)

^

(det^-^w) < C. (log H)²-, {resp. \h^w)\ < C . (log H+log H)²⁷", (det^-^w) ^ C - (log \z\ +^^1)2"

for every (z, w) € A H X.

2) For every holomorphic section s € ^f°(A H X,T) the following statements are equivalent:

i) ⁵ = «5|Anx for some s € H°(^t) ii)

^,w)(^) ^ C. (log H)^ (resp./i(^)(5,s) ^ C-(log M+log H)2771

for some (7 > 0 and some m € N.

in

5(^,w) = a ( ^ w ) - (^Q-) +&(>,w)—(resp.s(>,w) = a ( z , w ) - [z—\

+b{z,w)'(w—} with a,beO(A).

Proof. — (See [Mu] Main Theorem) For the equivalence i)<^ ii) see [Mu], Prop. 1.3. For the equivalence ii)<=> iii) we refer to [Mu] Prop. 3.4. D

From Lemma 5 we immediately deduce the following estimates for the Bergman metric h in a neighbourhood of the boundary divisor R:

COROLLARY. — Let A be an open polycylinder in X with A D R = {(z, w) e A I z = 0}(resp. A D R = {(z, w) C A | z • w = 0}).

For every local basis of T over A there exist constants C > 0 and m € N such that

1^-MI ^ ^(logl^.H-^resp. \h^w)\ < C.(log M+log H)2- .H-^H-2

for every {z, w) e A D X. D In the situation D = H2 we shall apply a general theorem of Tai [AMRT], Theorem 2, later also proved with different methods by Mumford [Mu], Theorem 4.1 and Proposition 4.2, which, in this situation, reads as follows:

Jfr is an irreducible neat arithmetic lattice in Aut° D, then for some cofinite normal subgroup F' ofF every desingularization of X' = D/Y' is of general type.

The unramified covering X' —> X can be extended to a meromorphic map p : X' —f X. A finite sequence of suitable blow ups of X' leads to a desingularization X' of X' and a birational holomorphic map T : X' —r X1

such that p := por is holomorphic. Note that X'\X' has normal crossings.

A central step in the above cited proofs of Taps Theorem is a verification of the following statement, see [AMRT], p. 301 and [Mu], p. 270-272.

LEMMA 6. — Let m € N and s G H°(X, K^ (g) L^) with s{x) = 0 for every x € X\X. There exists a cofinite normal subgroup F' in F such that p* {s) C ^ { X ' . K ' ^ ) . D

4. Construction of a hyperbolic pseudo metric.

Now we construct a continuous Hermitian pseudo metric on the cusp- compactification X of X = D/T for a neat irreducible arithmetic lattice F in Aut° D. More precisely, we construct such a pseudo metric on a suitable desingularization X of X and push it down to X and X. This pseudo metric will be distance-decreasing for holomorphic mappings from the unit

disk E into X and vanishes exactly in the cusp points. By an averaging process such a pseudo metric can be constructed for every freely operating irreducible arithmetic lattice.

The standard Bergman metric on D is a Kahler-Einstein metric with Kahler-Einstein coefficient A = —1. The ball B2 has constant negative holomorphic sectional curvature Ky := —|, and the holomorphic sectional curvature of H2 is bounded from above by K^2 := — ^ .

PROPOSITION. — Let D be a bounded symmetric domain in C² and r an irreducible arithmetic lattice in Aut° D which operates freely on D.

For every XQ e X C X there exists a continuous function (f) : X —> R>o with (t){xo) > 0 and the following properties:

1) The curve R = X\X is contained in N^ := {x € X \ (f)(x) = 0}, and (f) is smooth on X\N(p.

2) The product (f)' h defines a continuous pseudo metric on X which is identically zero on R.

3) There exists a € R, 0 < a < —KD, such that for every holomor- phic map f : E —> X and every ^ € f~l(X\N^) the equation

(1) Q^of) o^eKrk)

{ )

w w

holds.

Proof. — It suffices to prove the proposition for some cofinite neat normal subgroup r' C r, replacing {xo} by some finite set B C X' = D / T ' C X ' . In fact, if p : X' —> X is the meromorphic extension of the natural holomorphic covering map X' —> X and if 0 satisfies the above conditions on X' with B := p~l{xo), then

^: x -^ p>o, ^x) := / n ^

V^/ep-¹^)

n = [r : r'], has the desired properties for X and XQ. Therefore we assume that r is an irreducible neat arithmetic lattice, see Lemma 1.

We study the cases D = B² and D = HI² separately.

I. D = B2

We fix no € N such t(xo) ^ 0 for some t e ^{X.K⁷¹⁰ (g) I/¹⁰-¹)) and choose a cofinite normal subgroup r' of F with [I\ : r^] > 2no for all cusps K G Ar, see Lemma 2 and Lemma 1. We define

s :=p*(t) e ^{X^p^K^ <g)L⁷¹⁰-¹)).

Note that p : X' —^ X is holomorphic. In the terminology of Lemma 4 let d € ^(^(^^Art^]^"¹^) ^{be a} holomorphic section with zero-divisor Z^GAr^ ^: ^l^- Applying Lemma 4 we note that

(/>(x) := (\s(x)\2 . \d(x)\2 . (det/i^))-^)^ for ^ e X>\

(f)(x) := 0 for x e R ' = X'VC', is a non-negative real function on X ' .

Let A C X' be an open polycylinder with Anj^' = {(z, w) € A|^ == 0}.

Lemma 5 yields for all (z, w) C A H X':

10(^)1 ^(logl^l)

^!

^

with suitable constants G > 0 and n € N. In particular, (f) satisfies condition 1) on X ' . For the pseudo metric 0 • h the corollary to Lemma 5 yields

|<^w) 'hi,(z,w)\ < C ' (log 1^1)^ . |^

for all (2^ w) (E A H X' and suitable constants (7 > 0 and m e N. Therefore the pseudo metric ( f ) ' h satisfies condition 2) on X ' .

Finally,

^log^o/) ^ _ 4 ^^(det^/i))

V 3 / 7 ^^7 ^;

9C9C 7 5^

for every C € /-1(X'\A^). Hence for a := ^ < j = -^ equation (1) holds.

II. D = E2

According to Lemma 3 and the above remarks we assume that there exist no e N and s e H°(X, K^ 0 L-1) with s(xo) ^ 0. Let d € H°(X, L71^1) with zero-divisor (no + 1)-R. Then sd e ^(X.K710 0 L710), and Lemma 6 yields a^cofinite normal subgroup F' c F such that t := p*(sd) e H^^X', K^⁰)^ where p : X¹ -. X is the holomorphic map used in Lemma 6.

Let d' € H°(X^ L^⁰) with zero-divisor n^R'. We define a non-negative real function (j) on X' as follows:

^^(^'(^^•(det^))-^)^! for xeX^

(t>(x) := 0 for x € ^' = X'VX'.

Applying Lemma 5 in both cases A D ^^/ == {(z,w) e A | ^ = 0} and A n I?' = {(2;, w) € A [ 2: • w = 0} we see that 0 satisfies condition 1) on X ' . It follows from the corollary to Lemma 5 that ( j ) ' h also satisfies condition 2) on X⁹ since ²^^ > 2 and

|(^,w)./^(^w)| < C.(log|^|)2 m.|^|( 2(n o+l)+ 2 n o)4^T-2^

resp. |^-^(2;,w)| < ^ • ( l o g l ^ l + l o g l w D ^ - l ^ w l1^ ?

for every (z, w) € A H X' and for suitable constants C > 0 and m € N.

Finally

^log^o/) ^ 2np ^Mdetg*^)) 9C9C 4 n o + l 9CaC ^

for every < e f-^X^N^). Hence for a := 4^- < \ = -K^z equation (1) holds for (f) instead of (f). Let T : X' —> Xf be the birational holomorphic map introduced at the end of Section 3. Then (f) := cj) o r~¹ obviously satisfies the statement of the proposition for X ' ' . D Recall that the Kobayashi pseudo metric dy on a connected complex space V dominates every continuous pseudo metric p on Y which is distance-decreasing with respect to holomorphic mappings from the unit disk E into Y:

dy{y, z) ^ p ( y , z) for all y , z € V.

In particular, if yo e Y and p(yo, z) > 0 for all z ^ yo in some neighborhood of yo, then dy(^/o, z) > 0 for all z -^ yo in Y.

THEOREM. — Let D be a bounded symmetric domain in C2 and r C Aut° D an irreducible arithmetic lattice operating freely on D. Then the cusp-compactification X of D/T is hyperbolic.

Proof. — Let XQ C D/T C X. By the above remark it suffices to show the existence of a continuous pseudo metric p^o on X which is distance- decreasing with respect to holomorphic mappings E —> X and satisfies Pxo{xo,y) > 0 for all y ^ XQ in some neighborhood U of XQ. In fact, this implies that d^(x,y) = 0 if and only if x = y or x,y e R. In particular, the image of every holomorphic map C —> X is contained in R, and every holomorphic map C —^ X has to be constant. Let KD and a be defined as above and (j): X —> R>o a function which satisfies the statement of the proposition. It is sufficient to show that for some r > 0 the pseudo metric r • 0 • h on X is distance-decreasing with respect to holomorphic mappings f ' . E - ^ X .

We apply a method which was used in the compact case in [ST] and verily the inequality

«/«) := | w(0) • (/m)(i - ici ² ) ² ^ —^y ^ ^

for every ^ 6 E. Recall that a < —KD and

^log^o/) ^(detCrfe)) tf.^<r\

9^ (co) = (-a)———9^———(co) = -a(/ )(co)-

Following the standard proof of the Lemma of Ahlfors we note that there exists Co € f~l(X\N^) with

u/(Co) = max{u/(C) | C € E}, <o = Co(/), and therefore

Q2 log Uf

^^T^

^ipg^o/) ^loga^) , c> ² 2

= -"ac^r"^"^^ ⁰ ^^^^ ^)lc=co -

The sectional curvature condition for the metric h yields

a^Wh) (^h}(^

—^-^-—(Co) >. -^D{J h)[(,o)

and obviously ^ log(l - ICI^k^o = (1-^y Finally we obtain

0 ^ -a((/*/i)(Co)) - KD(rh)(Co) - ^ _2, ^ -

hence

- ^ ( o + ^ ^ ^ ^ ^ ^ l - I C o l2)2^ ! . This proves

^(C) < ^(Co)

• ^(^(Co)) l< < ^ _ L ^ ^ V f * ^ ^ ^ <</ - V 1 I/- |2^2 ^ ll^lloo

=

^-L ^ ^ ' o^"^^^^))^ (^))(Co)(l-ICol ) < ~(~r~K~\

COROLLARY. — Let TT : X —^ X be the minimal resolution of the singularities of X. Then X is a minimal surface of general type and hyperbolic modulo R = TT'^^X). D

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[BB] W. BAILY, A. BOREL Compactification of arithmetic quotients of bounded sym- metric domains, Ann. of Math., 84 (1966), 422-528.

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Manuscrit recu Ie 26 octobre 1998, accepte Ie 8 avril 1999.

E. OELJEKLAUS & C. SCHMERLING, Universitat Bremen

Fachbereich Mathematik und Informatik Postfach 330440

D-28334 Bremen.

oel@math.uni-bremen.de schmerling@math.uni-bremen.de