Hyperbolic surfaces in $\mathbb{P}^3$

$Page 1: Hyperbolic surfaces in $\mathbb{P}^3$$
Vol. 58, No. 3 DUKE MATHEMATICAL JOURNAL (C) June 1989

HYPERBOLIC SURFACES IN ps

ALAN MICHAEL NADEL

CONTENTS

0. Introduction 7491. Preliminaries on meromorphic differential geometry 7502. The Wronskian 7513. Statement of Siu’s Theorem 7534. Computation of the pole order of a meromorphic connection 7545. Construction of meromorphic connections with low pole order 7556. Degeneracy of entire holomorphic curves in certain hypersurfaces 7567. An explicit family of smooth hyperbolic surfaces in p3 7608. Computation of curve genus 7639. Higher dimensional algebraic families of smooth hyperbolic surfaces

in p3 769

0. Introduction. Complex projective varieties which are hyperbolic in the senseof Kobayashi I-Ko] have attracted recent attention because of their conjectureddiophantine properties. For example, Lang l-Lal, La2] has conjectured (amongother things) that any hyperbolic complex projective varity which is defined over anumber field K can contain at most finitely many points which are rational overK; this conjecture may be regarded as a higher dimensional analogue ofthe Mordellconjecture. There are, however, very few known examples of hyperbolic varieties.The purpose of this paper is to construct smooth hyperbolic surfaces in p3.

Previously the only known examples ofsuch surfaces were the Brody-Green surfacesI-BrGr] defined in homogeneous coordinates W, X, Y, Z by

W + X + Ya + Z 4- ,(WX)d/2 -- e(YZ)/2 0

where d > 50 is even and e : 0 is sufficiently small.We consider smooth surfaces M c p3 of degree d such that each monomial in

the degree d homogeneous defining polynomial of M contains one of the homo-geneous coordinates raised to the pth power for some p > (3d + 10)/4. We showthat, for such a surface, the image of any holomorphic map C --, M is contained incertain curves of genus < 1 and degree <d2 (see Section 6); in particular, M ishyperbolic iff it contains no such curves. We obtain a result for certain higherdimensional hypersurfaces as well (Theorem 6.1).

Received January 18, 1988. Revision received October 10, 1988. Research supported by a SloanDoctoral Dissertation Fellowship.

749

750 ALAN MICHAEL NADEL

In Section 7 we give an explicit example; we show that the surface in p3 defined by

x6e(x3 -- eY3) -I- y6e+3 -I- Z6e+3 -.I- w6e(w3 "- eY3) 0

is hyperbolic for e > 3 and for e not equal to 0 and several other numbers. It followsin particular that the plane curve obtained by setting Z 0 has hyperbolic com-plement (Corollary 7.1); this provides the first known example of a smooth planecurve with hyperbolic complement. (Previously, all known examples ofplane curveswith hyperbolic complements were singular).

If {Ms}s is an algebraic family of surfaces in p3 of the type considered above,and if Mo is hyperbolic for some So, then it follows that Ms is hyperbolic for all sin a Zariski open neighborhood of so in S. The reason is that the nonexistence ofthese curves of genus < 1 and degree <d2 on a given surface is a Zariski opencondition with respect to deformations of that surface. In this way we obtain manyalgebraic families of smooth hyperbolic surfaces in p3 (see Section 9).

Recently Siu [Sil, Si2-1 introduced a new idea into nonequidimensional valuedistribution theory: meromorphic differential geometry. This paper is, to a largeextent, an application and extension of Siu’s theory. In fact, the main technical ideaof this paper is the translation of Siu’s analytic notion of degeneracy of an entireholomorphic curve (namely, vanishing of the Wronskian) into a more algebraicand more tractable notion (see Section 2). For this we formulate and prove aholomorphic analogue of the following well-known fact from Riemannian geome-try: If a geodesic in a Riemannian manifold is tangent at a point to a givencomplete totally geodesic submanifold then the geodesic lies entirely inside thatsubmanifold.

Acknowled#ements. I would like to thank Professor Siu for his constant helpand encouragement. In addition, I would like to thank the referee and S. Lu fortheir helpful comments.

1. Preliminaries on meromorphic differential geometry. This section containssome basic preliminary material on meromorphic connections. Let M be a complexmanifold and V a meromorphic connection on M. Let E be a holomorphic linebundle over M and let H(M, E) {0} be a global holomorphic section suchthat tV is holomorphic (i.e. for each Christoffel symbol F of V, tF is holomorphicon the coordinate patch of definition). In what follows we shall assume also thatthe zero set of is as small as possible ("t kills off the pole of V but nothing more").Let N M be an analytic subvariety.

Definition 1.1 We say that N is totally geodesic (relative to V) if the followingtwo conditions hold:

1. No component of N is contained identically in the pole set {t 0} of V.2. For any two holomorphie vector fields U, V defined locally on M and tangent

to N, the meromorphie vector field Vv V is also tangent to N.

HYPERBOLIC SURFACES IN p3 751

Restriction of a connection to a totally #eodesic submanifold. If N is a totallygeodesic submanifold of M then it is clear that V restricts to a meromorphicconnection VIN on N, and moreover that (tlN)(V]) is holomorphic on N.

Hessians and totally geodesic divisors. Suppose now that L is a holomorphic linebundle over M, fl is a meromorphic connection on L, and S H(M, L) {0}. LetN {S 0} be the zero set of S, assumed reduced. Suppose that no component ofN is contained identically in the pole set of . Suppose moreover that there existmeromorphic tensors A, B, C,a on M, none of whose pole sets contain identicallya component of N, such that

taS- Fa,S AaS + BaS + GaSholds. The left hand side is called the Hessian of S and is sometimes writtenHess(S)a. In invariant notation,

vvS- tv,v)S A(U)vS + B(V)vS + C(U, V)S (1)

for any local holomorphic vector fields U, V.

Claim. N is totally geodesic.

Proof of claim. If U and V are tangent to N then all terms in (1) other than{vuv)S clearly vanish along N. Therefore, vuv)S also vanishes along N. HenceVv V is tangent to N. This proves that N is totally geodesic, as desired.

2. The Wronskian. In this section we formulate and prove a holomorphicanalogue ofthe following well-known fact from Riemannian geometry: If a #eodesicin a Riemannian manifold is tangent at a point to a 9iven complete totally 9eodesicsubmanifold then the 9eodesic lies entirely inside that submanifold.

Let M be a complex manifold of complex dimension m > 2, V a symmetric holo-morphic connection on M with Christoffel symbols F’a, L a holomorphic linebundle over M, and fl a holomorphic connection on L. (The results of this sectionare local in nature, and that is why we work with holomorphic connections ratherthan meromorphic ones.)

Let A M be a holomorphic map from the unit disc A whose Wronskian (relativeto V) vanishes identically:

f, ^ f, ^...^ f(m)=_ O. (2)

Here f’ is the velocity vector off and ftk+X):= V..ftk). Let z denote the Euclideancoordinate on A. By (2) we can write

f(k) hk_xf(k-x) + + hxf’ (3)


for some k (2 < k < m) where hi,..., hk- are meromorphic functions. By replacingA by a subdisc (not necessarily centered at the origin) if necessary, we can assumewithout loss of generality that h,..., hk- are in fact holomorphic.

Fix S e H(M, L) (0} and consider the following condition on S:

Hess(S)a AtS + Bt@S + CtjS (4)

where A,, B,, C,a are holomorphic tensor fields on M, and where the Hessian isdefined as follows:

Hess(S)a !!aS- F!,,S.

For the rest of this section we will assume that (4) holds. Geometrically this meansthat the subvariety {S 0} is totally geodesic (see Section 1).

LEMMA 2.1 For i= 2, 3 the following is true: I,,S- (,c,)iS is a linearcombination of S, !f,S, (!y,)2S,..., (y,)’- S with coefficients which are holomorphicfunctions of z.

Proof. We note that

Hess(S)(U, V)= NvvS- tvuv)S

for any vector fields U, V. Take U f’, V ft-) and combine this formula with(4) to obtain

!f,f,,-,,S- !y,,,S A(f’)!y,,-,,S + B(ft’-))y,S + C(f’,fti-))S.

This formula gives us the desired result for 2, and gives us also the inductivestep to get from 1 to i. The lemma follows. QED

The main result of this section is the following.

THEOREM 2.1 Let S be a global holomorphic section of L over M which satisfies(4), and let f: A M be a holomorphic map which satisfies (2) and (3). Assume that

1. S(f(O)) O, and2. (S)(ft’)(0)) 0 for 1, 2,..., m 1.

Then S(f(z)) 0 for all z e A.

Proof. We must show that f*S is the zero section of f*L. (For simplicity ofnotation, we omit the f* and write simply S instead of f*S.) By (3) we can write

By Lemma 2.1, we see that

k-1

f,k,S Z hi!y’’’S"i=1

(y,)ks is a linear combination of S, y,S, (y,)k-x S with holomorphic coefficients.

HYPERBOLIC SURFACES IN i3 753

Therefore S (or more precisely f’S) is a holomorphic solution to a kth orderlinear homogeneous holomorphic ordinary differential equation. Furthermore, wehave by assumption the zero initial conditions S(f(O))=0, S(ft(O))= 0(1 < < k 1). By Lemma 2.1 we can convert these initial conditions into the morefamiliar

(y,)SIz=o 0 for 0, 1,..., k 1.

Hence by uniqueness of solutions to ODE’s we obtain f*S O. QED

Relation to Cartan’s Proof. In Cartan’s wonderfully simple proof[Cal, Ca2] ofthe defect relations for entire holomorphic curves in projective space, the followingfact played an important role:

Let f: A V be a holomorphic map from the unit disc into an n-dimensionalcomplex vector space V. Assume that the (classical) Wronskian off vanishes identically:

f ^ f, ^...^ f(.-l= O.

Let W be the proper subspace of V spanned by f(O), f’(O), ftn-2)(O). Then f(z) Wfor all z A.

Our Theorem 2.1 is an obvious generalization of this fact in which a holomorphicconnection is used in place of the linear structure of V, and a totally geodesicsubmanifold is used in place of the subspace W.

3. Statement of Siu’s Theorem.in a form suitable for our purposes.

In this section we present Siu’s Theorem [Sill

THEOREM 3.1 (Siu) Let M be a smooth complex projective variety of complexdimension m and let V be a meromorphic connection on M. Let # 0 be a globalholomorphic section of some holomorphic line bundle L over M such that tV isholomorphic. Denote by KM the cannonical bundle of M and set v (m(m- 1))/2.Assume that KM (R) L is ample. Let f: C - M be a holomorphic map. Then eitherf(C) is contained in the pole set {t 0} of the connection or else f’ ^ f" ^ ^ftm =-0.

Remark 1. Siu considers more generally the defect 6s(D for divisors D on Mwhich are totally geodesic relative to V. Theorem 3.1 actually corresponds to thespecial case of Siu’s theorem where the divisor D is zero, and its conclusion isqualitative rather than quantitative.

Remark 2. Siu makes the additional assumption that the holomorphic mapf: C --. M has greater than algebraic growth (i.e., is transcendental). In the specialcase of Siu’s theorem which we state above, this hypothesis is not needed becauseone may always replace f(z) by f(exp(z)).


COROLLARY 3.1. Let M c P" be a smooth hypersurface of de[/ree d and let V bea meromorphic connection on with respect to which M is totally [/eodesic. Supposethat V is holomorphic, where # 0 is a [/lobal holomorphic section of the line bundle(gp,([/) for some [/. If 1/2(n 1)(n 2)[/< d n 1 and iff: C - M is holomorphicthen either f*t =- 0 or else f’ ^ f" ^ ^ fin-l) =_ O.

Proof. Since M is totally geodesic relative to V, we can restrict V to M andapply Theorem 3.1. We remark that KM is equal to d n 1 times a hyperplanesection. QED

4. Computation of the pole order of a meromorphic connection. The goal of thissection is to estimate the pole order of a meromorphic connection on projectivespace in terms of its Christoffel symbols relative to one given inhomogeneouscoordinate system. Let V be a meromorphic connection on " and let F,a be theChristoffel symbols of V relative to the inhomogenous coordinates X X". LetZ, Z" be homogenous coordinates on " so that Xi= Zi/Z (1 <i< n).Assume that A,tj(Z,..., Z") and B(Z,..., Z") are homogenous polynomials ofdegree g such that

E, A,lJIB

as rational functions on " for each fixed 0, fl, a.

THEOREM 4.1. Let be the [/lobal holomorphic section of the line bundle (9,(9 + 3)defined by the homo[/enous polynomial

(Z)3B(z,..., Z).

Then V is holomorphic. In particular, the pole order of V is < [/+ 3.

Proof. We must check that V is holomorphic on each inhomogenous co-ordinate patch.

First we note that tV is holomorphic on the coordinate path q/= {Z # O} c "on which X, X" are defined because B(1, X, X")Fp is a holomorphicfunction on q/for each fixed , fl, a.Next we check tV on some other coordinate patch, say := {Z" :/: O} = z,. Onwe have coordinates Y,..., Y" defined by Y := Z-/Z" (1 < < n). Let G,v be

the Christoffel symbols of V relative to the coordinates Y1, Y". We recall thefollowing transformation law for Christoffel symbols:

(The Einstein summation convention is in effect here.) We must show that

(y)3B(y yn, 1)G,v

HYPERBOLIC SURFACES IN 755

is regular on for each fixed/, v, z. For this purpose we look at each term in thetransformation law.We know that

B(y, yn, 1)F’

is regular on because

Fa A,a/B(y1, yn, 1)

by assumption, and the B’s cancel. All that remains is to check that the followingtwo expressions become regular on after being multiplied by (y1)3:

We note that yi Xi-1/X. and hence that

0 ifj#n,i-- 1

1X"

ifj=i- 1

Si-1

(X.)2ifj n.

In particular, t3Y/t3X is smooth on and vanishes to at least first order alongY 0. By similar reasoning (but with the roles of X and Y reversed) we see thatc3Xi/OY (resp. t32Xi/c3YOYk) has pole order at most 2 (resp. 3) along YX 0 and isotherwise smooth on .The proof now follows. QED

5. Construction of meromorphic connections with low pole order. Let M P" bea smooth hypersufface of degree d. In order to apply Siu’s theorem we would liketo construct a meromorphic connection V on P" with low pole order relative to d(more specifically, with pole order <2(d- n- 1)/(n- 1)(n- 2)) such that M istotally geodesic relative to V. Unfortunately, for the generic hypersurface M, nosuch connection V can be found. We therefore must restrict our attention tohypersurfaces of a special kind--namely, hypersuffaces for which each monomialin the homogenous defining equation contains some homogenous coordinate raisedto a high power.More specifically, fix positive integers d, k (d > k) and let Po, P,

C[-X X] be polynomials of degrees < k in the inhomogenous coordinatesXx,..., X on [. Let So Po and let S (Xi)a-Pi for 1 < < n.


In this section we construct a meromorphic connection V on pn with respect towhich every hypersurface in pn of the form

aiSi 0, (5)i=0

where ao, an are constants (not all zero), is totally geodesic. Furthermore, thepole order of V is < k(n + 1) + n + 3.For the construction, fix , fl (1 < , < n) and consider the following system of

equations:

So doSo (3,,So E.. oSn F

(6)

where t3o 1 and t3 c3/cX for 1,..., n.Our choice of the system (6) is explained as follows. If S is any linear combination

of So,..., Sn then (6) gives

r ,s +which is the same as (1) (with C,a Fa and A, 0 B,).We assume that the determinant ofthe square matrix which appears in this system

is not identically zero; we denote this determinant by B.Thus we can solve for F2,a for each choice of , fl to obtain a symmetric

meromorphic connection V on pn with respect to which every hypersurface (5) istotally geodesic. (We do not use Fa to define V.)To estimate the pole order of V we use Cramer’s rule to write

where C[X,A,,a, B Xn] are certain determinants, with B as defined previously.Since the ith equation appearing in (6) is divisible by (X)a-k-2 (for 1, n; weadopt the convention that the top equation is the 0th equation), we see that A,aand B are both divisible by (X1X2...Xn)d-k-2; after removing this commonfactor, we may assume that A,a and B have degree <(n + 1)k + n. Therefore, bySection 4, V has pole order < (n + 1)k + n + 3. This section is now complete.

6. Degeneracy of entire holomorphic curves in certain hypersurfaces. Green andGriffiths [GrGr-I conjectured that if M is a complex projective variety of generaltype and if f: C - M is a holomorphic map then the image f(C) cannot be Zariskidense in M. Green [Gr] proved this conjecture for Fermat hypersurfaces of highdegree. However, a complete proof of this conjecture seems to be a long way off.


In this section we prove the Green-Griffiths conjecture for a certain class of hyper-surfaces in pn which we now proceed to describe.

Fix positive integers d, k (d > k). Let 6e(d, k) be the complex vector subspace ofthe vector space of all degree d homogenous polynomials in Z, Z which isdefined as follows:

6e(d, k) is spanned by monomials of the form

(Z)o(Z ), (Z").

where d > d k for some e {0, n}. Here do d. are semipositive integerswhich satisfy do + + d, d.

In short, 6e(d, k) is the vector space spanned by monomials of degree d whichcontain the (d k)th power of one of the homogeneous coordinates.Any S 6e(d, k) can be decomposed into a sum

S= So + + s. (7)

where St 6e(d, k) is divisible (as a polynomial) by (Zi)d-k. This decomposition isunique if k < d/2; we will always assume that this is the case.

Definition 6.1. Let S be an element of ff’(d, k) and let S So + + S. be adecomposition as in (7). We say that S is nondegenerate if the following determinantis not identically zero:

doSo .s.Here 8/t3Z for 0, n.The main result of this section is the following theorem.

THEOREM 6.1 (Main Theorem). Let M c be a smooth hypersurface of degreed. Suppose that M is the zero set of some nondegenerate S (d, k). Assume that

d-n-l>(n-- 1)(n-2)(k(n+ 1)+n+3)

2

Let C M be holomorphic. Then the image f(C) is not Zariski-dense in M.

Remark. The proof of this theorem will provide reasonably precise informationon the position off(C) in M. Namely, f(C) is contained either in the pole set {t 0}of the meromorphic connection V, or in the intersection M M’ of M with a


hypersurface M’ c P" of the form

aoSo + + a.S, 0

where ao, a, are complex numbers, not all zero, and not all equal (i.e., M’ : M).Before proving the theorem we must first digress to discuss the equivalence of

two definitions of nondegeneracy.Equivalence of two notions of nondegeneracy. Let S 5e(d, k) be given and

let S So + + S, be a decomposition as in (7). An alternative definition ofnondegeneracy is the following:

Definition 6.2.identically zero:

We say that S is nondegenrate if the following determinant is not

So 8 So O.So

s. os(This definition of nondegeneracy differs from the previous one only in the fact

that the first column of the determinant matrix has no Oo’S.)We shall now prove that the two defintions of nondegeneracy are equivalent by

showing that the determinant which appears in Definition 6.2 is equal to (1/d)Ztimes the determinant which appears in Definition 6.1. Let F: C"+ C"+ be thepolynomial map defined by F := (So,..., S,). Since F is homogenous of degree d,Euler’s Theorem gives F (1/d)Z’O,F. Therefore,

F ^ OxF ^ ^ t3.F (Z’c3,F) ^ OxF ^ ^

1 0=-dZ c3oF ^ OlF ^ ^ O.F.

The equivalence of the two definitions has now been established.Nondegeneracy in the sense of Definition 6.2 is needed in Section 5 in order to

construct meromorphic connections.

Proof of theorem. Fix a decomposition S So + + S, as in (7). Because wehave assumed that S is nondegenerate, the construction in Section 5 gives us ameromorphic connection V on P" and a global holomorphic section :-0 of(gp,(k(n + 1) + n + 3) such that tV is holomorphic and such that every hypersurfaceof the form

aoSo + + a,S, 0


is totally geodesic relative to V. (Here the a are constants, not all zero.) In particular,M is totally geodesic relative to V.

Let D (t 0} M be the zero divisor of the restriction of to M. We note thatKt- ((n- 1)(n- 2)/2)D is an ample divisor on M. (Here Kt is the canonicaldivisor of M.) This assertion follows from the hypothesis that d is sufficientlylarge relative to n and k, since Kt (d- n- 1) (hyperplane section) whileD (k(n + 1) + n + 3) (hyperplane section). (Here denotes linear equivalence ofdivisors on M.)Now suppose that C M is a nonconstant holomorphic map. If f(C) c D then

we are done. Assume now that f(C) k D. Since Kt ((n 1)(n 2)/2)D is ampleand f(C) 4: D, Corollary 3.1 implies that

f, ^ f,, ^ ^ f,,-l O. (8)

Pick zoC such that f(zo)M-D. Let V be the vector space of all S’H(P", (gp,(d)) such that the following conditions hold:

1. S’ is a complex linear combination of So,..., S,.2. S’(f(zo)) O.3. S’(fe)(Zo) 0 for 1, 2, n 2.

Here is any meromorphic connection on 9,,(d) which is holomorphic in aneighborhood of f(zo). Note that S V. Since there are n + 1 linearly independentsections So,..., S, from which to form S’ (see condition 1 above) but only n- 1constraints (given by conditions 2 and 3) we see that dimc V > 2. Pick one particularS’ e V such that S and S’ are linearly independent. By the main result of Section 2,we know that

S’(f(z)) 0 for all z C.

Define M’ := {S’ 0} c P". Thenf(C) M c M’ and the proofis complete. QED

Definition 6.3. A complex quasi-projective variety M is said to be hyperbolic iffevery holomorphic map C M is constant.

Remark. This definition of hyperbolicity is different from Kobayashi’s defini-tion. However, when M is complete (=compact), a theorem ofBrody [Brl, Br2, Lal]implies that the two definitions are equivalent.

Main Corollary (of proof). Let M c 13 be a smooth hypersurface of degree d.Suppose that M is defined by an equation of the form S 0 for some nondegenerateS 5e(d, k), where d > 4k + 10. Suppose that f: C M is a nonconstant holomor-phic map. Then the image f(C) is contained in an irreducible curve of degree < d2

and genus < 1. In particular, ifM contains no such curves then M is hyperbolic.

Proof. By the proof of the theorem, the image off is contained in some reducedcurve C of degree <d2 (namely, the reduction of either the pole set of V or theintersection M M’). Since C is irreducible, the image off must in fact lie in some


irreducible component Co of C. Clearly Co also has degree <d2. Finally, Comust have genus < 1, for otherwise the Picard theorem would imply that f isconstant. QED

We remark that Lang [Lal] has conjectured that any complex projective varietywhich contains no rational curves and no nontrivial images ofabelian varieties mustbe hyperbolic. We have thus proved Lang’s conjecture for certain hypersurfaces inp3.

7. An explicit family ofsmooth hyperbolic surfaces in 13. The goal ofthis sectionis to give an explicit example of a smooth hyperbolic surface in 3. Let e > 3 be aninteger and let d 6e. Let Eo (-d/3)((d + 3)/d)2e+1 and let e be a nonzerocomplex number such that e2e+ 1/2Eo, Eo. The main result of this section is thefollowing.

THEOREM 7.1.Z by

The surface M p3 defined in homogenous coordinates W, X, Y,

Xd(X3 ..}. ey3)+ yd+3 + Za+3 + wd(w3 + ey3) 0 (9)

is hyperbolic.

Remark. The condition that eEe/l 1/2Eo, Eo is precisely equivalent to thecondition that the surface defined by (9) be nonsingular.

COROLLARY 7.1. The smooth plane curve C c 2 defined in homogenous co-ordinates W, X, Y by

Xd(X3 .-Ji- eY3) + yd+3 + wd(w3 ..]_ e, y3) 0 (10)

has hyperbolic complement.

Proof of corollary. Let M be as in the theorem. The projection M 2 givenby M [l/V, X, Y, Z] [W, X, Y-I e p2 exhibits M as a (d + 3)-fold cover of 2,branched only over C. Thus every holomorphic map f: C 2 C lifts (since Cis simply connected) to a holomorphic map f: C M which (according to thetheorem) must be constant. QED

The rest of this section will be devoted to proving the theorem.

The meromorphic connection. In this subsection we will write down explicitly ameromorphic connection V on 3 with low pole order and with respect to whichM is totally geodesic. For this purpose we must fix a choice of inhomogenouscoordinates. Our choice is as follows: we set Y 1 and take W, X, Z to be ourinhomogenous coordinates. With respect to these coordinates, the Christoffel


symbols for V will be given by

Fw (d + 3)(d + 2)W3 + ed(d- 1)(d + 3)W + edW

(d + 3)(d + 2)Xa + ed(d- 1)(d + 3)X4 + edX

(d + 2)Z

All other Christoffel symbols are defined to be zero.If we denote by s any linear combination of Xd(X3 + e), 1, Zd+3, Wd(W3 + e)

then it is easily seen that

(Here , fl, tr { W, X, Z}, and the Einstein summation convention is in effect for a.)Therefore, the surface in p3 given (in the inhomogenous coordinates W, X, Z) bys 0 is totally geodesic relative to V.We note that the least common multiple of the denominators of the above

Christoffel symbols is WXZ(W3 + ed/(d + 3))(X3 + ed/(d + 3)). As in Section 4 wehave

TV is holomorphic where T is the global holomorphic section of (9p3(12) given by thefollowin9 homogenous polynomial (in W, X, Y, Z):

T WXy3Z(W3 +ed y3 X3 +d+3 d+3

The pole of V has degree < 12 while the canonical divisor of M is equal to dtimes a hyperplane section. Since d 1 > 12, we will be able to apply Siu’s theorem.

Remark. We have used here the notation T rather than because below we willlet denote the global coordinate on C. No confusion should arise.

Hyperbolicity. With the preliminaries on meromorphic connections complete,we are now ready to show that M is hyperbolic. Suppose that f: C M is anynonconstant holomorphic map; we will obtain a contradiction. The map f is givenby entire holomorphic functions W(t), X(t), Y(t), Z(t) (t C) which are neversimultaneously zero. These entire functions satisfy equation (9). Moreover, theseentire functions satisfy the following (not necessarily disjoint) conditions:

1. No three of them are proportional.2. No one of them vanishes identically.3. No one of W(t), X(t), Z(t) is a constant multiple of Y(t).


Proof that these conditions hold:First we consider condition 1. If it so happens that three out of our four entire

functions are proportional to one another then we find by equation (9) that all fourare in fact proportional. This contradicts the assumption that f is nonconstant.Next we consider condition 2. Suppose that Y(t) O. Then equation (9) gives

X(t)+s + Z(t)/s + W(t)/s -= 0

which immediately implies that IX(t), Z(t), W(t)-I is a constant curve in p2 (sincethe Fermat curve is smooth) and contradicts condition 1. The remaining cases ofcondition 2 are readily seen to be special cases of condition 3.

Finally we consider condition 3. Suppose for example that X(t) cY(t) for someconstant c. By (9) we get an equation of the form

BY(t)+3 + Z(t)a+s + W(t)a(W(t)3 + eY(t)3) 0

for some constant B. By Theorem 8.2 we conclude that [W(t), Y(t), Z(t)] is aconstant curve in p2, contradicting condition 1 above. The case W(t)= c Y(t) ishandled in entirely the same fashion, as is the case Z(t) c Y(t) except that we mustinvoke Theorem 8.1 rather than Theorem 8.2. QED

It follows from Corollary 3.1 that we are in one of the following situations:(A) f lies in the pole set of V. That is, f*T =_ O.(B) f is autoparallel. That is, f’ ^ f" O.We consider each of these two situations in turn. Suppose first that we are in

situation (A). Then our entire functions W(t), X(t), Y(t), Z(t) satisfy

edy(t)3 X(t)3+ Y(t)3 -0W(t)X(t)Y(t)3Z(t) W(t)3 + d + 3 d + 3

In particular, one of our entire functions vanishes identically (contradicting condi-tion 2) or else one of W(t), X(t) is a constant multiple of Y(t) (contradictingcondition 3).

Suppose now that we are in situation (B). Then, as in Section 6, we see that f liesin a hypersurface M’ : M defined by an equation of the form

(11) aXd(X3 + e,Y3) + bYd+3 + cZd+3 -}- (wd(w3 + e,Y3) O.

Here a, b, c, are constants, not all zero and not all equal. Furthermore, we canassume without loss of generality that 0 since we can always replace equation(11) by itself minus times equation (9).We now investigate individually each of several cases. Suppose first that a 0. Thenequation (11) gives

b Y(t)+ 3 + cZ(t)+ 3 =_ 0

HYPERBOLIC SURFACES IN :3 763

(where b and c are not both zero) and we conclude that either Y(t) 0 (contradictingcondition 2) or else Z(t) is a constant multiple of Y(t) (contradicting condition 3).

Suppose next that a 0. We may assume without loss of generality that in facta 1. By subtracting equation (11) from equation (9) we see that our entire functionssatisfy

(1 b)Y(t)a+3 + (1 c)Z(t)d+3 + W(t)a(W(t)s + eY(t)3) =- O.

If on the one hand c 1 then by Theorem 8.2 we conclude that I-X(t), Y(t), Z(t)] isa constant curve in p2, contradicting condition 1 above. If on the other hand c 1then equation (11) gives

X(t)a(X(t)s + e Y(t)3) + b Y(t)+3 + Z(t)a+ 3 O,

and by Theorem 8.2 we conclude that IX(t), Y(t), Z(t)] is a constant curve incontradicting condition 1 above.

This concludes the proof that M is hyperbolic. QED

8. Computation of curve genus. In this section we carry out several computa-tions which are needed in Section 7. The goal of this section is to prove that certainplane curves are hyperbolic. In each case our method involves showing first thatthe given curve is irreducible and second that the genus of the given curve is at leasttwo. The second step is accomplished by applying either the genus formula for planecurves with singularities or a branched covering method (essentially the Riemann-Hurwitz formula).Throughout this section a plane curve will be a reduced but not necessarily

irreducible curve in p2. By the genus of an irreducible plane curve we will mean thegenus of its normalization. A given plane curve will be called hyperbolic if everyholomorphic map from C into the curve is constant; this will be the case if the curveis irreducible and has genus at least 2.

THEOREM 8.1. Fix B C and d > 18 in 6Z. Then the plane curve C c 1)2 definedin homogenous coordinates X, Y, Z by

Xd(X3 _].. Z3) + ya(y3 + Z3) ._ BZa+3 0

is hyperbolic.

THEOREM 8.2. Fix B C and d > 12 in 67/. Then the plane curve C p2 definedin homogenous coordinates X, Y, Z by

Xd(X3 -t-- Z3) - ya+3 + BZd+3 0

is hyperbolic.

The rest of this section will be devoted to proving Theorems 8.1 and 8.2.


Proof of Theorem 8.1. When B 0 this result follows from Lemma 8.2.Now assume that B - 0 and consider the plane curve C’ defined in homogenouscoordinates U, V, W by

u2e(u --]-- W) -- v2e(v + W) --1- BW2e+1 O.

Here e d/6. By Lemma 8.1 we know that C’ is hyperbolic. Furthermore, we havea nontrivial morphism C C’ defined by

C[X, Y,Z][X3, y3, z3]__ [U, 1/, W]C’.

Therefore, C is hyperbolic. QED

Proof of Theorem 8.2. When B 0 this result follows from Lemma 8.4.Now assume that B : 0 and consider the plane curve C’ defined in homogenouscoordinates U, V, W by

u2e(u-- W)-Ji-- V2e+l + BW2e+l O.

Here e d/6. By Lemma 8.3, C’ is hyperbolic. Furthermore, we have a nontrivialmorphism C C’ defined by

C IX, Y Z] IX3, y3, Z3] C’.

Therefore, C is hyperbolic. QED

LEMMA 8.1. Fix B v 0 in C and d > 6 in 27/. Then the plane curve C c 12 definedin affine coordinates X, Y by

P(X, Y) := Xd(X + 1)+ Yd(Y + I) + B O

is irreducible and has genus at least 2.

Proof. Once C is known to be irreducible, its genus is given by the followinggenus formula [Na, p. 126]:

genus(c)=d(d- 1)6p

2 pc

Here 6p depends only on the analytic structure of the signularity of C at p, andvanishes if C is smooth at p.

First we analyze the singularity structure of C. It is easily checked that C is smoothat infinity. The signular points (X, Y) of C are found by solving simultaneously the

HYPERBOLIC SURFACES IN p3

following three equations:

P(X, Y) 0

Xa-((d + 1)X + d)= 0

Y-((d + 1)Y + d)= O.

The results appear in the chart below. We let Bo da( 1/(d + 1))a/l.

765

value of B

no

2Bo

all other values

singular points p (X, Y)

0,d+ 1,0

d+l’d+lno singular points

analytic germ ofsingularity

U2 + Vd+ 0

node

value of dip

d+2

We will now verify the information which appears in the last two columns of thechart.We deal first with the case B Bo. By symmetry in X, Y it suffices to consider

only the point p (0, -d/(d + 1)). We shall show that the singularity of C at p isanalytically isomorphic to the sigularity

U2 -{.- Vd+l 0

at U V 0; it will then follow from [Na, Lemma 2.1.7, p. 122] that 6v (d + 2)/2,as desired. We note that at p (0, -d/(d + 1)) we have Pr 0, Prr -: 0. Therefore,by Taylor expansion we see that

P(X,Y)=Xd(X+I)+ Y+d,,+,,1, g Y +d+lwhere g(?) is a polynomial with {/(0) : 0. Define new holomorphic coordinates U,V centered at p by

d y+U= Y+d+l d+l

V X(X + 1)TM.

Then C is given locally at p by U2 + Va+ 0, as claimed.


We now deal with the case B 2Bo and p (-d/(d + 1), -d/(d + 1)). At this pwe have Px Pr Pxr 0 and Pxx Prr : 0. Therefore, a Taylor expansion ofP about p shows that C has a node at p. For a node it is known that 6v 1[Na, chart on p. 123].Assuming the information in our chart, we now complete the proof of the lemma.

First we show that C is irreducible. In the case B Bo the irreducibility followsfrom Lemma 8.6, while in the case B 2Bo the irreducibility follows from Lemma8.5. If B :/= Bo, 2Bo then C is smooth and hence irreducible. The genus is now easilyobtained from the genus formula. QED

LEMMA 8.2. Fix an inteoer d > 1. Then the plane curve C c p2 defined in

inhomo#enous coordinates X, Y by

P(X, Y):= Xa(X3 + 1)+ ya(y3 + 1)= 0


Proof. One easily checks that the only singular point of C is the point(X, Y)= (0, 0). Moreover, the singularity is ordinary (i.e., there are d distincttangents). The irreducibility of C then follows from Lemma 8.5 while the genusformula [Fu, p. 199] gives

(d + 2)(d + 1)_ d(d- 1)genus(C) 2. QED2 2

LEMMA 8.3. Fix B 0 in C and d > 4 in 27/. Then the plane curve C c d)2 definedin inhomo#enous coordinates X, Y by

P(X, Y):= Xa(X + 1) + ya+l + B 0


Proof. For values of B for which the curve C is smooth, the lemma is clearlytrue. We therefore consider only values ofB for which C is singular. We now searchfor singular points (X, Y) of C. One easily checks that C is smooth at infinity. Thesingular points (X, Y) are precisely the solutions to the following three simultaneousequations:

P(X, Y) 0

Xn-X((d + 1)X + d)= 0

(d + 1)Ya O.

Thus the only singular point is p=(X, Y)=(-d/(d + 1),0) when B(-d/(d + 1))a(-d/(d + 1) + 1). At this p (and for this B) we have Pxx # O. Therefore,


a Taylor expansion of P(X, Y) about p gives

P(X,Y)= X+d+ 1 9 X+d+ 1 +

where 9(?) is a polynomial with O(0) :- 0.We introduce new holomorphic coordinates U, V centered at p as follows:

U= X+d+ i X+d+ 1

V--Y.

In these coordinates, C takes the form U2 -- Vd+l --O. Therefore, 6p (d + 2)/2[Na, p. 122] and the genus formula gives

d(d- 1) d + 2genus(C) > 2. QED

2 2

LEMMA 8.4. Fix an integer d > 1. Then the plane curve C c p2 defined inhomogenous coordinates X, Y, Z by

Xd(X3 + Z3) _. ya+ 3 0

is irreducible and has 9enus at least 2.

Proof. Irreducibility is proved in Lemma 8.7. We consider here the genus. Wewill express the normalization ( of C as a branched cover - pl of Pl such thatn has at least 3(d + 3) distinct branch points and n has degree d + 3. The proof isthen concluded as follows. Let 09 be a nontrivial meromorphic 1-form on P withprecisely two poles (counted with multiplicity). Then n*o9 has 2(d + 3) poles and atleast 3(d + 3) zeros. Since n*o has more zeros than poles, we conclude thatgenus(() > 2, as desired.We now define n to be the composition

where v is the normalization map and where r/([X, Y, Z]) [Y, Z] P.We now set Z 1 and work in affine coordinates X, Y. Then C and r/are given

respectively by

P(X, Y):= Xa + Xd+3 + ya+3 0

/(X, Y)= Y.


It is clear that r/has degree d + 3, and hence that zr also has degree d + 3. To findthe branch points of r/we use Lagrange multipliers. In other words, we look forpoints (X, Y) on C at which Vr/= (constant)VP. That is, we must find 3(d + 3)distinct solutions (X, Y) to the following three simultaneous equations:

P(X, Y) 0

dX- + (d + 3)Xa-2 0

(d + 3)yd+2 =/= O.

The solutions are as follows. For each of the 3 cube roots of unity we set

d )1/3X=- d+3

For each such X there are (d + 3) distinct choices of Y given by

y (_Xa Xd+3)l[(d+3).

(There are d + 3 choices of roots.) We thus have 3(d + 3) distinct points (X, Y)which are easily seen to satisfy the three simultaneous equations above. We aredone. QED

LEMMA 8.5. Let C 1)2 be a plane curve (reduced, but possibly reducible) whichpossesses precisely one singular point p. Suppose moreover that p is an ordinary point.

If C is reducible then it is the union of lines.

Proof. Suppose that C is reducible, say C C1 w C2 where C1 and C2 arenonempty plane curves of degrees d and d2 respectively. Let m denote themultiplicity of C at p (i 1, 2). We compute the intersection product C. C2 in twodifferent ways. Firstly, by Bezout’s theorem, Cx. C2 dx d2. On the other hand, C1and C2 intersect only at the point p, and with distinct tangents; hence C1.C2mlm2[Fu]. Therefore, mlm2 did2. But, of course, ms < d (i 1, 2), and weconclude that m d (i 1, 2). Consequently, Cx and C2 are unions of lines, andthe lemma follows. QED

LEMMA 8.6. Suppose that C c p2 is a plane curve (reduced but possibly reducible)such that each singularity of C is analytically isomorphic to one of the formU2 vEn+ +1. Then C is irreducible.

Proof. If C were reducible then it would contain a reducible singularity (i.e., onewhose analytic germ is reducible) at the point where two components meet. But byassumption the singularities of C are not reducible. QED


LEMMA 8.7. The plane curve defined in homogenous coordinates X, Y, Z by

P(X, Y, Z)"= Xd(X3 + Z3) + yd+3 0

is irreducible provided d 3f > 3.

Proof. It suffices to show that P is an irreducible polynomial. This can be provedby Kummer theory. Another proof is as follows. Suppose that P QR where Q andR are polynomials in X, Y, Z. Consider the images of P, Q, R under the ringhomomorphism

c IX, Y, z] --, c Is, t]

defined by sending

Y-.

Zt-- sf+l

Then

Q(1, t, sf+l)R(1, t, s:+) P(1, t, s:+)

1 + Sf+l -[- f+l

But 1 + sf/ + t:/ is irreducible since it defines (in inhomogenous coordinates)the Fermat curve, which is nonsingular and hence irreducible. It follows that at leastone of Q, R, say Q, is a function of X alone. Then

Q(O, o, 0)R(0, Y, 0) Q(0, Y, 0)R(0, Y, 0)

P(0, Y, O)

yd+3.

Hence Q has degree 0 while R has degree d + 3. This proves that P is an irreduciblepolynomial. QED

9. Higher dimensional algebraic families of smooth hyperbolic surfaces in p3.

The purpose of this section is to construct multidimensional algebraic families ofsmooth hyperbolic surfaces in p3. To put this section in proper perspective, wenote that hyperbolicity is not known in general to be a Zariski open condition with


respect to algebraic deformations of complex projective varieties. (Hyperbolicity is,however, a "classically-open" condition with respect to deformations [Brl, Br2].)Fix positive integers e, k, d such that

1. e>32. d=6e+33. d>4k+10.

We note that k can be taken arbitrarily large. The choice of these three conditionsis explained as follows: Conditions 1 and 2 enable us to use the example from Section7 while condition 3 enables us to use the Main Corollary from Section 6.

Let 6e(d, k) denote (as in Section 6) the vector space of all homogenouspolynomials of the form

(Z)"-eo + (Z)-e + (Z)-e + (Z3)-ea

where Po, P1, P2, P3 C[Z, Z1, Z2, Z3] are homogenous of degree d. We note that

dimc 5a(d, k)= 4|k+4]/\k\ /

Define q/to be the set of all S (d, k) for which the following conditions hold:1. S is nondegenerate (cf. Definition 6.1).2. The surface S 0 is nonsingular.3. The surface S 0 is hyperbolic.

Claim. q/is Zariski open in 6e(d, k).

Proof of claim. Conditions 1 and 2 are clearly Zariski open conditions. And bythe Main Corollary of Section 6, condition 3 is equivalent to the nonexistence ofcurves of genus < 1 and degree < d2 on the surface S 0; the nonexistence of suchcurves is a Zariski open condition. The claim is proved.

Claim. q/is nonempty.

Proof of claim. Consider the following polynomial S in (d, k):

S :-" (Z)6e((zO)3 -Jr- 8(Zl)3) -- (Zl)6e*3 -- (Z2)6e*3 -]-- (Z3)6e((z3)3 "Ji- (Zl)3).

For e sufficiently small, S satisfies conditions 1 and 2 in the definition of q/abovebecause S is a perturbation of the Fermat polynomial. And by Section 7, the surfaceS 0 is hyperbolic for e not equal to 0 and several other numbers. The claim isproved.

Finally, since two polynomials cut out the same surface iff they are proportional,we obtain an effectively parameterized algebraic family {Mts]} ofsmooth hyperbolicsurfaces Mts] {S 0} c i)3 parameterized by [S] c q//C*. (Here [S] denotes theequivalence class of polynomials which are proportional to S.) The dimension of


the parameter space q//C* of this family is given by

which can be made arbitrarily large by taking k large. This section is now complete.

REFERENCES

[Brl] R. BRODY, Intrinsic metrics and measures on compact complex manifolds, doctoral thesis,Harvard, 1975.

[Br2] ,Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc. 235 (1978), 213-219.[BrGr] R. BRODV ANO M. GREEN, ,4 family of smooth hyperbolic hypersurfaces in pa, Duke Math. J.

44 (1977), 873-874.[Call H. CARTAN, Sur les systP.mes de fonctions holomorphes a varidtds lindaires lacunaires et leurs

applications, Ann. Sci. ]cole Norm. Sup. (3) 45 (1928), 255-346.[Ca2] , Sur les zdros des combinaisons lindaires de p fonctions holomorphes donndes, Mathe-

maticaa (Cluj) 7 (1933), 5-31.I-Fu] W. FULTON, Algebraic Curves, W.A. Benjamin Inc., New York, 1969.I-Gr] M. GREEN, Some Picard theorems for holomorphic maps to algebraic varieties, Amer. J. Math.

97 (1975), 43-75.[GrGr] M. GREEN AND P. GR1FFITHS, Two applications of algebraic geometry to entire holomorphic

mappings, The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, Calif., 1979),Springer-Verlag, New York, 1980, 41-74.

I-Ko] S. KOBAYASHI, Hyperbolic manifolds and holomorphic mappings, Marcel Dekker, New York,1970.

[La ] S. LANG, Hyperbolic and Diophantine analysis, Bull. Amer. Math. Soc. (NS) 14 (1986), 159-205.I-La2-1 ,Hi#her dimensional Diophantine problems, Bull. Amer. Math. Soc. $0 (1974), 779-787.[Na-I M. NAMaA, Geometry of Projective Algebraic Curves, Marcel Dekker, New York, 1984.[Sill Y.-T. SIu, Defect relations for holomorphic maps between spaces of different dimensions, Duke

Math. J. 55 (1987), 213-251.[Si2] ,Nonequidimensional value distribution theory and meromorphic connections, preprint.[Si3] ,Nonequidimensional value distribution theory and subvariety extension, Lecture Notes

in Math. 1194, Springer-Verlag, 158-174.

DEPARTMENT OF MATHEMATICS, HARVARD UNIVERSITY, CAMBRIDGE, MASSACHUSETTS 02138CURRENT ADDRESS: SCHOOL OF MATHEMATICS, INSTITUTE FOR ADVANCED STUDY, PRINCETON, NEW

JERSEY 08540

Documents

Hyperbolic surfaces in $\mathbb{P}^3$