Some examples of hyperbolic hypersurfaces in the complex projective space

SOME EXAMPLES OF HYPERBOLIC HYPERSURFACES IN THE COMPLEX PROJECTIVE SPACE

Hirotaka Fujimoto

Abstract. In the previous paper [6], the author constructed hy- perbolic hypersurfaces of degree 2ⁿ in the n-dimensional complex projective space for every n≥ 3. The purpose of this paper is to give some improvement of this result and to show some general methods of constructions of hyperbolic hypersurfaces of higher degree, which enable us to construct hyperbolic hypersurfaces of degree d in the n-dimensional complex projective space for every d≥ 2 × 6ⁿ.

§1. Introduction

Since S. Kobayashi conjectured that generic hypersurfaces of high de- grees in n-dimensional projective space P

( C) are hyperbolic ([8]), many reseachers constructed various examples of hyperbolic hypersurfaces in P

( C). Here, we mean by a hyperbolic hypersurface a hypersurface which is hyperbolic in the sense of S. Kobayashi. For the case where n = 3, the rst example was given by R. Brody and M. Green, who gave a hyperbolic hypersurface in P

( C) of even degree ≥ 50 ([2]). After- wards, new types of hyperbolic hypersurfaces of degree d in P

( C) were given by A. Nadel for d = 6p + 3 ≥ 21 ([10]), by J. El Goul for d ≥ 14 ([7]), by J. P. Demailly ([3]) and by Y. T. Siu–S. K. Yeung ([13]) for d ≥ 11 respectively. Moreover, J. P. Demailly-J. El Goul proved that a very generic hypersurface of degree ≥ 21 in P

( C) is hyperbolic in [4]

and M. Shirosaki constructed a hyperbolic hypersurface of degree 10 in [12].

On the other hand, for the case where n ≥ 4, in [9] K. Masuda–

J. Noguchi proved that there exists a hyperbolic hypersurface of every degree d ≥ d(n) for some positive integer d(n) depending only on n and

Received October 5, 2002.

2000 Mathematics Subject Classication: 32M05.

Key words and phrases: hyperbolicity, complex manifold.

gave some concrete examples of hyperbolic hypersurfaces in P

( C) for n ≤ 5. Moreover, Siu–Yeung gave examples of hypersurfaces in P

( C) of degree 16(n − 1)

in [13].

By improving Shirosaki’s argument in [12], the author succeeds in constructing concrete examples of hyperbolic hypersurfaces of degree 8 in P

( C), which is one of the lowest degrees among known hyperbolic hypersurfaces in P

(C) ([6]). More generally, he showed the following:

Theorem 1.1. There exists a family of hyperbolic hypersurfaces of degrees 2

in P

( C).

In this paper, we give some slightly improved version of Theorem 1.1 and, by using this, prove the following improvement of the above- mentioned result of Masuda–Noguchi:

Theorem 1.2. For every d ≥ 2 × 6

(n ≥ 3), there exists a family of hyperbolic hypersurfaces of degree d in P

( C).

Recently, in their paper [11] B. Shiman and M. Zaidenberg gave the following improvement of the above-mentioned result of Siu–Yeung:

Theorem 1.3. Let m ≥ 2n − 1. For every d ≥ (m − 1)

and generic linear functions h

, . . . , h

on C

, the hypersurface

X

:=

 

 z ∈ P

( C) :

∑

h

(z)

= 0

 



is hyperbolic. In particular, there exist algebraic families of hyperbolic hypersurfaces of degree d in P

(C) for every d ≥ 4(n − 1)

.

We note that B. Shiman and M. Zaidenberg only show the existence of hyperbolic hypersurfaces of the above type, but do not construct concrete examples.

The motivation for Theorem 1.2 was suggested by J. Noguchi. The author would like to thank him.

§2. Hyperbolic hypersurfaces of low degrees

We call a complex space M Brody hyperbolic if there is no noncon- stant holomorphic map of C into M. As was shown by R. Brody ([1]), a compact complex manifold is Brody hyperbolic if and only if it is hyperbolic in the sense of S. Kobayashi. In the following, a compact hyperbolic space means a compact Brody hyperbolic space.

We rst give the following:

Definition 2.1. We call a homogeneous polynomial Q(w) of degree d in w = (w

, w

, . . . , w

) an H-polynomial if it satises the following two conditions:

(H1) for an arbitrary holomorphic map f of C into P

(C) with a reduced representation f := (f

: f

: · · · : f

), namely, a representa- tion in terms of homogeneous coordinates on P

(C) with holomorphic functions f

’s without common zeros, if it satises

Q(f

, f

, . . . , f

) = cf

for some c ∈ C, then f is a constant.

(H2) if a holomorphic map f of C into P

( C) with a reduced rep- resentation f := (f

: · · · : f

) satises the identity

Q(0, f

, . . . , f

) = cf

for some c ∈ C and some entire function f

, then f is a constant.

Theorem 2.2. Let Q(w

, w

, . . . , w

) be an H-polynomial. Then, (i) V := {(w

: · · · : w

) : Q(w

, . . . , w

) = 0} is hyperbolic and (ii) for W := {(w

: · · · : w

) : Q(0, w

, . . . , w

) = 0} ⊂ P

(C), P

( C) \ W is Brody hyperbolic.

Proof. Take an arbitrary holomorphic map f : C → P

(C) with a reduced representation f := (f

: f

: · · · : f

) such that f ( C) ⊆ V and so Q(f

, f

, . . . , f

) = 0. By the assumption (H1) for the particular case c = 0, f is a constant, which gives the assertion (i). Next, to see (ii), take a holomorphic map f of C into P

( C) with a reduced representation f = (f

: · · · : f

) such that f (C) ∩ W = ∅. Then, the entire function Q(0, f

, . . . , f

) has no zeros. Therefore, we can nd an entire function f

such that Q(0, f

, . . . , f

) = f

. We can apply (H2) to see the assertion (ii).

For the case where n = 2 we can show the following:

Theorem 2.3 ([6, §4]). Let Q(u

, u

, u

) be a homogeneous polyno- mial of degree d ≥ 4 and consider the associated inhomogeneous poly- nomial ˜ Q(v, w) := Q(1, v, w). Assume that

(C1) the simultaneous equations

Q ˜

(v, w) = ˜ Q

(v, w) = 0

have only nitely many solutions, say P

:= (v

, w

) (1 ≤ k ≤ N), (C2) ˜ Q(P

) ̸= ˜ Q(P

) for 1 ≤ k < ℓ ≤ N,

(C3) {(u

, u

) : Q

(0, u

, u

) = 0, i = 0, 1, 2} = {(0, 0)}.

(C4) the Hessian φ := ˜ Q

Q ˜

− ˜ Q

̸= 0 at (v

, w

) (1 ≤ k ≤ N).

Then, Q is an H-polynomial.

For the readers’ convenience, we give here an outline of the proof.

To prove Theorem 2.3, we consider an algebraic curve V

: Q(u

, u

, u

) = cu

(c ∈ C)

in P

( C) for arbitrary c ∈ C. By the assumption (C3), V

has no singularities on the innite line H

:= {u

= 0 } and, by (C1), the singularities of V

are contained in the set {P

, . . . , P

}. On the other hand, by (C2), there is at most one point P

contained in V

. Moreover, we can use the assumption (C4) to conclude that a possible singularity of V

is at worst an ordinary singularity. We recall here Pl¨ ucker’s genus formula, by which the genus g(V

) is given by

g(V

) = (d − 1)(d − 2)

2 − the number of ordinary singularities ≥ 2.

We now use the following Picard theorem.

Theorem 2.4. Let V be a compact Riemann surface of genus greater than one. Then, there exists no nonconstant holomorphic map of C into V .

This concludes that the holomorphic map f : C → V

is a constant and Q satises the condition (H1).

To see the condition (H2), we consider the following algebraic curve V ˜

: Q(u

, u

) = cu

(c ∈ C)

in P

( C). By the same argument as the above, we can prove the genus of ˜ V

is greater than one. Therefore, Q satises the condition (H1) and so it is an H-polynomial.

By a generic homogeneous polynomial of degree d in n + 1 variables we mean an arbitrary polynomial in a Zariski dense set in the space of all nonzero homogeneous polynomials of degree d in n + 1 variables which is canonically identied with the space C

, where N (d, n) = (

d

) . Proposition 2.5. A generic homogeneous polynomial of degree d(≥

4) satises the condition in Theorem 2.3 and so is an H-polynomial.

For the proof, refer to the original paper [5].

For the case n ≥ 3, we can prove the following:

Theorem 2.6. Let Q(u

, u

, . . . , u

) be an H-polynomial of degree

d

and P (u

, u

) a homogeneous polynomial of degree d

( ≥ 3) such

that P (u

, u

) and ˜ P (w) := P (1, w) satisfy the conditions;

(P1) P (0, u

) ̸≡ 0,

(P2) P ˜

(w) has only simple zeros α

, α

, . . . , α

, (P3) P (α ˜

) ̸= ˜ P (α

) for 1 ≤ k < ℓ ≤ d

− 1.

For m ≥ 2, if d

:= md

and 2

d

− 2 + 1 m < 1, then

R(u

, u

, . . . , u

, u

) := P (u

, u

) − Q(u

, u

, . . . , u

)

is an H-polynomial.

Remark. We can easily show that a generic homogeneous polyno- mial P (w

, w

) of degree d

satises the conditions (P1), (P2) and (P3).

For the proof of Theorem 2.5, we recall the following result from the second main theorem of holomorphic curves in P

( C):

Theorem 2.7. Let f be a holomorphic map of C into P

( C) which is nondegenerate, namely, whose image is not included in any hyperplane in P

( C), and let H

, . . . , H

be hyperplanes in P

( C) which are located in general position. Assume that there are positive integers m

, . . . , m

such that each pull-back f

(H

) of H

considered as a divisor does not have positive multiplicity smaller than m

. Then,

∑

( 1 − n

m

)

≤ n + 1.

For the proof, refer to p.112 in [5].

As a special case n = 1, we have the following theorem for meromor- phic functions on C.

Corollary 2.8. Let φ be a nonconstant meromorphic function on C. Assume that there are mutually distinct values α

, . . . , α

and positive integers m

, . . . , m

such that f (z) − α

does not have zeros of multiplicity smaller than m

respectively. Then,

∑

( 1 − 1

m

)

≤ 2.

Proof of Theorem 2.6. Consider a holomorphic map f : C → P

( C) with a reduced representation f := (f

: · · · : f

) satisfying the iden- tity

R(f

, . . . , f

) = cf

. If f

≡ 0, we have

Q(0, f

, . . . , f

)

= ef

and so

Q(0, f

, . . . , f

) = e

f

for some constant e, e

. Hence, f is a constant by the assumption (H2) for Q. Otherwise, setting φ := f

/f

, we have

(1) P (φ) ˜ − c = Q

( 1, f

f

, . . . , f

f

)

.

On the other hand, if ˜ P (w) − c has a multiple zero w

, then w

is equal to some α

and its multiplicity is not larger than two by the assumption (P2). Moreover, by (P3), there is at most one α

with c = P (α

). These imply that ˜ P (w) −c has at least d

−2 simple zeros β

, β

, . . . , β

, where β

̸= α

for any j, k. Take a zero of φ − β

. Since the right hand side of (1) has no zero of multiplicity smaller than m, φ(z) − β

has also no zero of multiplicity smaller than m. Therefore, if φ is not a constant, then we have

(d

− 2) (

1 − 1 m

)

≤ 2

by Corollary 2.8. This contradicts the assumption and we conclude that φ is a constant. We write f

= c

f

for some constant c

. Then, we have

c

f

= Q(f

, f

, . . . , f

)

and so

c

f

= Q(f

, f

, . . . , f

)

for some nonzero constants c

and c

. By the condition (H1) for Q, we can conclude that f is a constant. Therefore, R satises the condition (H1).

Now, to check the condition (H2) we consider a holomorphic map f of C into P

( C) with a reduced representation f = (f

: · · · : f

) and a holomorophic function f

satisfying the identity

R(0, f

, . . . , f

) = P (0, f

) − Q(0, f

, . . . , f

)

= e

f

for some constant e

. Here, we may assume that e

̸= 0 and f

̸= 0.

Because, otherwise, the condition (H2) is reduced to (H1). We now use the assumption (P1), whence we have

e

f

− e

f

= Q(0, f

, . . . , f

)

for some constant e

. Consider the polynomial F (X, Y ) := e

X

− r

Y

. Obviously, F can be factored into the product of d

linearly independent linear forms with multiplicity one. In the similar manner as in the check of (H1), we can show that there is a nonzero constant e

such that

e

f

= Q(0, f

, . . . , f

).

By the assumption (H2) for Q, we can conclude that f is a constant.

The proof of Theorem 2.6 is completed.

Theorems 2.3 and 2.5 give the following:

Theorem 2.9. For any m ≥ 2 and d ≥ 4, there is a family of hyper- bolic hypersurfaces of degrees md(≥ 8) in P

(C), which is parametrized with md − 1 + (d + 1)(d + 2)/2 analytically independent parameters.

Proof. By Theorem 2.3 there are H-polynomials of degree d ≥ 4 in variables u

, u

, u

, which are parametrized with (d+1)(d+2)/2 analyti- cally independent parameters. On the other hand, generic homogeneous polynomials in variables u

, u

satisfying the condition (P1), (P2) and (P3) of degree md are parametrized by md + 1 analytically independent parameters. Therefore, the homogeneous polynomial R given in Theo- rem 2.6 are parametrized by md + (d + 1)(d + 2)/2 analytically indepen- dent parameters, because the terms of degree md with respect to u

are duplicate. By Theorem 2.6, these polynomials are H-polynomials and by Theorem 2.2 the zero locus of these polynomials are hyperbolic hyper- surfaces in P

(C). Since two homogeneous polynomials dene the same hypersurface if and only if one is identical with a nonzero constant multi- ple of the other, a family of hyperbolic hypersurfaces in P

( C) obtained in this way is parametrized with md − 1 + (d + 1)(d + 2)/2 analytically independent parameters. The proof of Theorem 2.9 is complete.

Using Theorem 2.6 repeatedly and with similar arguments as above, we have the following theorem:

Theorem 2.10 ([5, Theorem 2.4]). For each n ≥ 3 there is a family

of hyperbolic hypersurfaces of degree 2

in P

(C), which is parametrized

with 2

+ 6 analytically independent parameters, and a hypersurface

W of degree 2

in P

( C) such that P

( C)\W is Brody hyperbolic.

We can also construct many hyperbolic hypersurfaces in the complex projective space. For example, by Theorem 2.3, we can construct an H-polynomial of degree 5 in three variables and, by the use of the case m = 3 of Theorem 2.6 repeatedly, hyperbolic hypersurfaces of degree 5 × 3

in P

(C), which are used later.

§3. Hyperbolic hypersurfaces of high degrees

In this section, we construct some examples of hyperbolic hypersur- faces in P

( C) of high degrees.

A polynomial F (x

, x

, . . . , x

) in x

, x

, . . . , x

is called a weighted homogeneous polynomial with weights (d

, d

, . . . , d

) if F (t

, t

, . . . , t

) is a homogeneous polynomial in t

, . . . , t

.

Proposition 3.1. With a given polynomial

F := ∑

i1,...,im

a

x

· · · x

in x

, . . . , x

associate the weighted homogeneous polynomial F

(x

, x

, . . . , x

) := ∑

i1,...,im

a

x

x

· · · x

in (x

, x

, . . . , x

) with weights (1, d

, . . . , d

) for some positive integers d

, where d := max {i

d

+ · · · + i

d

: a

̸= 0}. Assume that

(i) F

(0, x

, . . . , x

) consists of only one nonzero monomial, namely, we can write

F

(0, x

, . . . , x

) = cx

. . . x

for some nonzero constant c and nonnegative integers j

, . . . , j

. (ii) if F (φ

, . . . , φ

) = 0 for meromorphic functions φ

on C, then at least one of the φ

’s is a constant.

Then, for arbitrary H-polynomials Q

(w

, . . . , w

) of degree d

(1 ≤ i ≤ m), the hypersurface

V :=

{

w = (w

: . . . : w

) : w

F (

Q

(w)/w

, . . . , Q

(w)/w

)

= 0 }

in P

( C) is hyperbolic.

Proof. Consider a holomorphic map f of C into V (⊂ P

( C)) with a reduced representation f := (f

: f

: · · · : f

). If f

≡ 0, then we have

Q

(0, f

, . . . , f

)

· · · Q

(0, f

, . . . , f

)

= 0.

We note here that d = j

d

+ · · · + j

d

(> 0). Therefore, Q

(0, f

, . . . , f

) ≡ 0 for some i

by the assumption (i), whence f is a constant by the assumption (H1) for Q

. We now assume that f

̸≡ 0. Then,

F (φ

, . . . , φ

) = 0

for meromorphic functions φ

:= Q

(f

, f

, . . . , f

)/f

, whence some φ

is a constant by the assumption (ii). So, we can write

Q

(f

, f

, . . . , f

) = cf

for some constant c. This concludes that f is a constant by (H1). The proof of Proposition 3.1 is completed.

We give an example satisfying the assumptions of Proposition 3.1.

Proposition 3.2. Set F (x, y) := x

+ y

+ x

y

+ 1 for positive integers p, r, s. Assume that

(2) p ≤ t := min(r, s), 6

p + 2 t < 1.

Then, F (x, y) satises the assumptions (i) and (ii) of Proposition 3.1 for arbitrary positive integers d

and d

.

For the proof, we use the following consequence of Theorem 2.6:

Theorem 3.3. Let f

, f

, . . . , f

be nonzero holomorphic functions on C which satisfy the identity

f

+ f

+ · · · + f

= 0 for some integer p ≥ n

. Consider the partition

{0, 1, . . . , n} = I

∪ I

∪ · · · ∪ I

such that i and j are in the same class I

if and only if f

/f

is a constant.

Then ∑

i∈Iℓ

f

= 0 for every ℓ.

For the proof, refer to [5, Proposition 3.4.7].

Proof of Proposition 3.2. By denition, the weighted homogeneous polynomial with weights (d

, d

) associated with F is given by

F

(x

, x

, x

) := x

x

+ x

x

+ x

x

+ x

, where d := rd

+ sd

(> max(pd

, pd

)). Since

F

(0, x

, x

) = x

x

,

the assumption (i) holds. To see (ii), take nonconstant meromorphic functions φ, ψ with

(3) F (φ, ψ) = φ

+ ψ

+ φ

ψ

+ 1 = 0.

We write

φ := f

f

, ψ := f

f

with entire functions f

, f

and f

having no common zero. Consider the holomorphic map

:= ( f

: f

: f

) : C → P

( C) and hyperplanes

H1:={w0= 0}, H2:={w1= 0}, H3:={w2= 0}, H4:={w0+ w1+ w2= 0}

in P

( C), which are located in general position. Then, the pull-backs

∗

(H

) of H

(j = 1, 2, 3), considered as divisors, are given by zeros of entire functions f

counted with multiplicities. So, the multiplicities of zeros are divided by p, whence they have no positive multiplicities smaller than p. Now, consider the divisor

(H

), which is given by zeros of the function

H := f

+ f

+ f

.

On the other hand, we can rewrite the identity (3) as (4) H = −f

f

f

.

Take a point z

∈ f

(H

). If f

(z

) ̸= 0, then we have f

(z

) = 0 or f

(z

) = 0. In any case, H has a zero with multiplicity at least t at z

. Assume that f

(z

) = 0. If f

(z

) = 0, then f

(z

) ̸= 0, because f

, f

, f

have no common zero. This implies ∑

j=0

f

(z

)

̸= 0, which contradicts z

∈ f

(H

). Then, f

(z

) ̸= 0 and, similarly, f

(z

) ̸= 0.

This is impossible. In fact, the left hand side of (4) is holomorphic but the right hand side is not holomorphic by the assumption p < r + s. In conclusion,

(H

) does not have positive multiplicity smaller than t.

Then, there are constants c

, c

, c

with (c

, c

, c

) ̸= (0, 0, 0) such that c

φ

+ c

ψ

+ c

= 0.

Because, otherwise, Theorem 3.3 gives 3

( 1 − 2

p )

+ (

1 − 2 t

)

≤ 3,

which contradicts the assumption. If c

= 0, then φ is a constant mul-

tiple of ψ. Combining this with the identity (3), we can conclude φ and

ψ are constants. Otherwise, we have c

f

+ c

f

+ c

f

= 0. Since

p ≥ 4 by the assumption, is a constant by Theorem 3.3. The proof of Proposition 3.2 is completed.

By Propositions 3.1 and 3.2, we have the following:

Proposition 3.4. Let Q

(w) be H-polynomials of degree d

(i = 1, 2) in n + 1 variables w = (w

, w

, . . . , w

) and p, r, s positive integers satisfying the condition (2). Then, the zero locus of the polynomial

R(w) := Q

(w)

w

+ Q

(w)

w

+ w

− Q

(w)

Q

(w)

is a hyperbolic hypersurface in P

( C) of degree d := rd

+ sd

.

This gives the following detailed statement of Theorem 1.2:

Theorem 3.5. For each n ≥ 3 we can take a positive integer d(n) such that there are hyperbolic hypersurfaces of degree d for every d ≥ d(n) in P

( C). Here, for example, we can take

(5) d(n) := 9(2

+ 5 × 3

) + 2

(5 × 3

− 1) + 5 × 3

(2

− 1), which is not larger than 2 × 6

.

For the proof of Theorem 3.5, we give the following Lemma:

Lemma 3.6. Let d

and d

be mutually prime positive integers. For an arbitrarily given positive integer m

, every integer d with

(6) d ≥ m

(d

+ d

) + d

(d

− 1) + d

(d

− 1) can be written as d = rd

+ sd

with r, s ≥ m

.

Proof. We denote by n

the right hand side of (6) and consider an arbitrary integer d ≥ n

. Take integers t, ℓ such that d − n

= td

+ ℓ and 0 ≤ ℓ < d

. We claim that ℓ can be written as

(7) ℓ = rd

+ sd

with integers r, s satisfying the conditions |r| < d

and |s| < d

. To see this, we take integers r

and s

such that ℓ = r

d

+ s

d

by the use of the assumption that d

and d

are mutually prime. We write these numbers as

r

= ud

+ r

, s

= vd

+ s

with integers u, v, r

and s

, where 0 ≤ r

< d

and 0 ≤ s

< d

. Here, we may assume that ℓ > 0 and (r

, s

) ̸= (0, 0), because otherwise (7) is obvious. Then, we have

(8) ℓ = (u + v)d

d

+ r

d

+ s

d

.

Here, since 0 ≤ ℓ < d

, we easily see u + v ≤ 0. For the case u + v = 0, we have (7) for r := r

and s := s

. If u + v ≤ −2, then

ℓ = |ℓ| ≥ 2d

d

− (r

d

+ r

d

) ≥ 2d

d

− (d

− 1)d

− (d

− 1)d

> d

. This is a contradiction. For the case u + v = −1, we rewrite (8) as

ℓ = r

d

+ (s

− d

)d

= (r

− d

)d

+ s

d

.

Since |s

− d

| < d

when s

> 0 and |r

− d

| < d

when r

> 0, we have (7).

By (7), we have

d = n

+ td

+ rd

+ sd

= m

(d

+ d

) + d

(d

− 1) + d

(d

− 1) + rd

+ sd

= (m

+ d

− 1 + r)d

+ (m

+ d

− 1 + s)d

.

Since r := m

+ d

− 1 + r and s := m

+ d

− 1 + s are not less than m

, we have Lemma 3.6.

Proof of Theorem 3.5. To this end, for each n( ≥ 3) we set d

(n) := 2

and d

(n) := 5 × 3

. As is mentioned in the previous section, we can nd H-polynomials Q

and Q

of degree d

(n) and d

(n) respectively.

Dene d(n) by (5). By Lemma 3.6, we can write every d ≥ d(n) as d = rd

(n) + sd

(n) with r, s ≥ m

:= 9, because d

(n) and d

(n) are mutually prime. Set p := 8 and consider the polynomial R dened in Proposition 3.4. Then, these r, s, p satisfy the condition (2) and hence we can apply Proposition 3.1 to nd a homogeneous polynomial R of degree d such that

V := {R = 0}

is a hyperbolic hypersurface in P

( C).

References

[1] R. Brody, Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc. 235 (1978), 213–219.

[2] R. Brody and M. Green, A family of smooth hyperbolic hypersurfaces in P3, Duke Math. J. 44 (1977), 873–874.

[3] J. P. Demailly, Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials, Proc. Sympos. Pure Math., Vol. 62, Part 2, Amer. Math. Soc., Providence, RI, 1997, 285–360.

[4] J. P. Demaily and J. El Goul, Hyperbolicity of generic surfaces of high degree in projective 3-space, Amer. J. Math. 122 (2000), 515–546.

[5] H. Fujimoto, Value distribution theory of the Gauss map of minimal surfaces in R^m, Aspect of Math. E21, Vieweg, 1993.

[6] , A family of hyperbolic hypersurfaces in the complex projective space, Com- plex Variables 43 (2001), 273–283.

[7] J. El Goul, Algebraic families of smooth hyperbolic surfaces of low degree in P³C, Manuscripta Math. 90 (1996), 521–532.

[8] S. Kobayashi, Hyperbolic manifolds and holomorphic mappings, Marcel Dekker, 1970.

[9] K. Masuda and J. Noguchi, A construction of hyperbolic hypersurface of Pⁿ(C), Math. Ann. 304 (1996), 339–362.

[10] A. Nadel, Hyperbolic surfaces in P³, Duke. Math. J. 58 (1989), 749–771.

[11] B. Shiman and M. Zaidenberg, Hyperbolic hypersurfaces in Pⁿ(C) of Fermat- Waring type, Proc. Amer. Math. Soc. 130 (2001), 2031–2035.

[12] M. Shirosaki, A hyperbolic hypersurface of degree 10, Kodai Math. J. 23 (2000), 376–379.

[13] Y. T. Siu and S. K. Yeung, Defects for ample divisors of abelian vaarieties, Schwarz lemma, and hyperbolic hypersurfaces of low degrees, Amer. J. Math. 119 (1997), 1139–1172.

Department of Mathematics

Kanazawa University

920-0934 Kanazawa, Japan