Table Of ContentHYPERBOLIC HYPERSURFACES IN Pn OF FERMAT-WARING TYPE
BERNARD SHIFFMAN AND MIKHAIL ZAIDENBERG
1
0
0 Abstract. In this note we show that there are algebraic families of hyperbolic, Fermat-
2 Waring type hypersurfaces in Pn of degree 4(n−1)2, for all dimensions n ≥ 2. Moreover,
there are hyperbolic Fermat-Waring hypersurfaces in Pn of degree 4n2−2n+1 possessing
n
a complete hyperbolic, hyperbolically embedded complements.
J
6
1 Many examples have been given of hyperbolic hypersurfaces in P3 (e.g., see [ShZa] and
] the literature therein). Examples of degree 10 hyperbolic surfaces in P3 were recently found
G
by Shirosaki [Shr2], who also gave examples of hyperbolic hypersurfaces with hyperbolic
A complements in P3 and P4 [Shr1]. Fujimoto [Fu2] then improved Shirosaki’s construction
.
h to give examples of degree 8. Answering a question posed in [Za3], Masuda and Noguchi
at [MaNo]constructed thefirst examples ofhyperbolic projective hypersurfaces, including those
m with complete hyperbolic complements, in any dimension. Improving the degree estimates
[ of [MaNo], Siu and Yeung [SiYe] gave examples of hyperbolic hypersurfaces in Pn of degree
1 16(n − 1)2. (Fujimoto’s recent construction [Fu2] provides examples of degree 2n.) We
v remark that it was conjectured in 1970 by S. Kobayashi that generic hypersurfaces in Pn of
6
(presumably) degree 2n−1 are hyperbolic (for n = 3, see [DeEl] and [Mc]).
2
1 The following result is an improvement of the example of Siu-Yeung [SiYe]:
1
0 Theorem 1. Let d ≥ (m−1)2, m ≥ 2n−1. Then for generic linear functions h ,...,h
1 m
1 on Cn+1, the hypersurface
0
/
h m
at Xn−1 = z ∈ Pn : hj(z)d = 0
m ( j=1 )
X
v: is hyperbolic. In particular, there exist algebraic families of hyperbolic hypersurfaces of degree
i 4(n−1)2 in Pn.
X
ar Equivalently, Xn−1 is the intersection of the Fermat hypersurface of degree d in Pm−1 with
a generic n-plane. The low-dimensional cases n ≤ 4 of Theorem 1 were given by Shirosaki
[Shr1]. Our construction is similar to those of [SiYe] and [Shr1].
On the other hand, examples were given in [Za2] of smooth curves of degree 5 in P2, with
hyperbolically embedded, complete hyperbolic complements. Examples of hyperbolically
embedded hypersurfaces in Pn with complete hyperbolic complements were given in [MaNo]
for all n, and in [Shr1] for n ≤ 4 (with lower degrees). The following result generalizes the
result of Shirosaki [Shr1] to all dimensions.
Theorem 2. Let d ≥ m2 − m+ 1, m ≥ 2n. Then for generic linear functions h ,...,h
1 m
on Cn+1, the complement Pn\Xn−1 of the hypersurface of Theorem 1 is complete hyperbolic
Date: January 12, 2001.
Research of the first author partially supported by NSF grant #DMS-9800479.
1
2 BERNARD SHIFFMAN AND MIKHAIL ZAIDENBERG
and hyperbolically embedded in Pn. In particular, there exist algebraic families of hyperbolic
hypersurfaces of degree 4n2−2n+1 in Pn with hyperbolically embedded, complete hyperbolic
complements.
In particular, Theorem 2 provides algebraic families of curves of degree 13 in P2, of sur-
faces of degree 31 in P3, and so forth, whose complements are complete hyperbolic and
hyperbolically embedded in projective space.
We shall use the following notation and lemma in our proofs of Theorems 1 and 2: We let
Gr denote the Grassmannian of complex codimension k subspaces of Cm, and we write
m,k
Q = 0 ×Cm−k ∈ Gr .
m,k k m,k
Furthermore, for
1 ≤ a ≤ c ≤ m
,
1 ≤ b ≤ c ≤ a+b
we define
Γ = {V ∈ Gr : dimV ∩Q ≥ m−c}.
m,a,b,c m,a m,b
Lemma 3. dimGr −dimΓ = (m−c)(a+b−c).
m,a m,a,b,c
Proof. Let Mat = Hom(Cm,Ca), and let M ⊂ Mat denote the surjective homomor-
m,a m,a m,a
phisms. We consider the fiber bundle
GL(a) −→ M
m,a
↓ π π(A) = kerA .
Gr
m,a
We let
Γ := π−1(Γ) = {A ∈ M : dim kerA| ≥ m−c} ;
m,a Qm,b
whence
(cid:0) (cid:1)
e dimM −dimΓ = dimGr −dimΓ .
m,a m,a m,a,b,c
Suppose A ∈ M ; i.e., A is an a×m matrix of rank a. We consider
m,a
e
I 0 I 0
A = b = b ∈ M ,
A B A′ m,a+b
(cid:18) (cid:19) (cid:18) (cid:19)
′
where B ∈ Matb,a, A ∈ Meatm−b,a. Clearly, kerA = kerA|Qm,b and thus
′
A ∈ Γ ⇔ dim(kerA) ≥ m−c ⇔ rankA ≤ c ⇔ rankA ≤ c−b.
e
It is easily seen that
e e e
codim {C ∈ Mat : rankC ≤ r} = (k −r)(l−r).
Matk,l k,l
Therefore,
′ ′
codim Γ = codim {A : rankA ≤ c−b}
Mm,a Matm−b,a
= [(m−b)−(c−b)][a−(c−b)]
e
= (m−c)(a+b−c).
HYPERBOLIC HYPERSURFACES IN Pn OF FERMAT-WARING TYPE 3
Proof of Theorem 1: Consider the Fermat hypersurface
m
F := (z : ··· : z ) ∈ Pm−1 : zd = 0
d 1 m j
( )
j=1
X
of degree d in Pm−1. Suppose that d ≥ (m − 1)2, m ≥ 2n − 1. We must show that
Xd := Fd ∩PV is hyperbolic for a generic V ∈ Grm,m−n−1.
Suppose that f = (f ,...,f ) : C → X is a holomorphic curve. By Brody’s theorem
1 m d
[Br], it suffices to show that f is constant. We write J = {1,...,m}. Let
m
I = {j ∈ J : f = 0}.
0 m j
(Of course, I may be empty.) We let I ,...,I denote the equivalence classes in J \ I
0 1 l m 0
under the equivalence relation
j ∼ k ⇔ f /f = constant.
j k
We let k = cardI and we write
α α
I = {i(α,1),...,i(α,k )},
α α
for α = 1,...,l, and also for α = 0 if k ≥ 1.
0
The result of Toda [To], Fujimoto [Fu1], and M. Green [Gr] says that for α = 1,...,l, we
have k ≥ 2 and furthermore the constants
α
µ := f /f ∈ C\{0} (1 ≤ α ≤ l, 2 ≤ j ≤ k )
αj i(α,j) i(α,1) α
satisfy
kα
1+ µd = 0.
αj
j=2
X
Geometrically, the image f(C) is contained in the projective l-plane YI given by the equa-
µ
tions
z = µ z , 2 ≤ j ≤ k ,1 ≤ α ≤ l; z = 0, 1 ≤ j ≤ k .
i(α,j) αj i(α,1) α i(0,j) 0
Here, I denotes the partition {I ,I ,...,I } of J , and µ = {µ }.
0 1 l m αj
Let YµI ⊂ Cm be the lift of YµI. Then YµI ∈ Grm,m−l. If l = 1, then YµI is a point.
Otherwise, we consider YµI ∩PV for generic V ∈ Grm,m−n−1. Applying Lemma 3 with
e e
a = m−n−1, b = m−l, c = m−2
(changing coordinates to make YµI = Qm,m−l), we conclude that YµI∩PV is either a point or
I
is empty, i.e. dim(Yµ ∩V) < 2, unless V lies in a subvariety of Grm,m−n−1 of codimension
e
s = m−(m−2) (m−n−1)+(m−l)−(m−2) = 2(m−n−l+1).
e
But given a p(cid:2)artition I, th(cid:3)e(cid:2)µ-moduli space of YµI in Grm,m−l(cid:3)has dimension
l
e
(k −2) = m−k −2l ≤ m−2l.
α 0
α=1
X
4 BERNARD SHIFFMAN AND MIKHAIL ZAIDENBERG
Since m ≥ 2n−1, we have s ≥ m−2l+1 and thus for generic V ∈ Grm,m−n−1, YµI ∩PV
is at most a point for all (I,µ). Since f(C) ⊂ YI ∩PV for some YI, it follows that f must
µ µ
be constant.
Proof of Theorem 2: Suppose that d ≥ m2 − m + 1, m ≥ 2n. Since by Theorem 1, Xn−1
is hyperbolic for generic hj, it suffices to show that any entire curve f : C → Pn\Xn−1 is
constant (see e.g., [Za2]).
We proceed as in the proof of Theorem 1. Suppose that f = (f1,...,fm) : C → Pn\Xn−1
is a holomorphic curve. As before, let I = {j ∈ J : f = 0}, and let I ,...,I denote the
0 m j 1 l
equivalence classes in J \I under the equivalence relation
m 0
j ∼ k ⇔ f /f = constant.
j k
Since d > m(m−1), by [To, Fu1, Gr] we have
kα
k ≥ 2 and 1+ µd = 0 for 2 ≤ α ≤ l,
α αj
j=2
X
after permuting theI andusing ourprevious notation. (Also, k ≥ 1, but 1+ k1 µd 6= 0.)
α 1 j=2 1j
The proof of this result proceeds by considering the map (f ,...,f ) : C → F ⊂ Pm, where
0 m d
f = − m fd = eϕ. (We let I = {j : f /f = constant} =6 ∅.) The betterPestimate for d
0 j=1 j 1 j 0
arises from the fact that f has no zeros.
0
As bePfore, the image f(C) is contained in the projective l-plane YI, and YI ∩ PV is
µ µ
either a point or is empty, unless V lies in a subvariety of Grm,m−n−1 of codimension s =
I
2(m−n−l+1). But this time, the µ-moduli space of Yµ in Grm,m−l has dimension
l
e
k −1+ (k −2) = m−k −2l+1 ≤ m−2l+1.
1 α 0
α=2
X
Since m ≥ 2n, we have s > m−2l+1 and hence for generic V ∈ Grm,m−n−1, YµI ∩PV is a
point or is empty for all (I,µ).
Remark: Note that the algebraic family of degree d = (m− 1)2 hyperbolic hypersurfaces
in Pn constructed in Theorem 1 has dimension (n+1)m−1, as does the family of Theorem
2. (Recall that in Theorem 1, m ≥ 2n − 1, whereas in Theorem 2, m ≥ 2n.) By the sta-
bility of hyperbolicity theorems (see [Za2]), in the corresponding projective spaces of degree
d hypersurfaces, both families possess open neighborhoods consisting of hyperbolic hyper-
surfaces, with hyperbolically embedded complements in the second case. We note finally
that the best possible lower bound for the degree of a hypersurface in Pn with hyperbolic
complement should be d = 2n+1 (see [Za1]), and the degree 2n−3 hypersurfaces in Pn are
definitely not hyperbolic because they contain projective lines. (In fact, starting with n = 6,
these lines are the only rational curves on a generic hypersurface of degree 2n−3 in Pn; see
[Pa].)
Acknowledgement: We would like to thank Jean-Pierre Demailly and Junjiro Noguchi for
useful discussions. The first author also thanks Universit´e Joseph Fourier, Grenoble, and
the University of Tokyo for their hospitality.
HYPERBOLIC HYPERSURFACES IN Pn OF FERMAT-WARING TYPE 5
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Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA
E-mail address: [email protected]
Universit´e Grenoble I, Institut Fourier, UMR 5582 CNRS-UJF, BP 74, 38402 St. Martin
d’H`eres c´edex, France
E-mail address: [email protected]
