Annales de l’institut Fourier (AIF)

Colin de Verdière, Yves

Spectrum of the Laplace operator and periodic geodesics: thirty years after

Faure, Frédéric

Semi-classical formula beyond the Ehrenfest time in quantum chaos. (I) Trace formula

Grabowski, Janusz

Marmo, Giuseppe

Michor, Peter W.

Homology and modular classes of Lie algebroids

Debarre, Olivier

Pacienza, Gianluca

Păun, Mihai

AN
N
A
L
E
S
D
E
L’INSTI
T
U
T
F
O
U
R
IE
R
ANNALES
DE
L’INSTITUT FOURIER
Olivier DEBARRE, Gianluca PACIENZA & Mihai PĂUN
Non-deformability of entire curves in projective hypersurfaces of high
degree
Tome 56, no 1 (2006), p. 247-253.
<http://aif.cedram.org/item?id=AIF_2006__56_1_247_0>
© Association des Annales de l’institut Fourier, 2006, tous droits
réservés.
L’accès aux articles de la revue « Annales de l’institut Fourier »
(http://aif.cedram.org/), implique l’accord avec les conditions
générales d’utilisation (http://aif.cedram.org/legal/). Toute re-
production en tout ou partie cet article sous quelque forme que ce
soit pour tout usage autre que l’utilisation à fin strictement per-
sonnelle du copiste est constitutive d’une infraction pénale. Toute
copie ou impression de ce fichier doit contenir la présente mention
de copyright.
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.cedram.org/
Ann. Inst. Fourier, Grenoble
56, 1 (2006) 247-253
NON-DEFORMABILITY OF ENTIRE CURVES
IN PROJECTIVE HYPERSURFACES
OF HIGH DEGREE
by Olivier DEBARRE, Gianluca PACIENZA & Mihai PĂUN
Abstract. — In this article, we prove that there does not exist a family of
maximal rank of entire curves in the universal family of hypersurfaces of degree
d > 2n in the complex projective space Pn. This can be seen as a weak version of
the Kobayashi conjecture asserting that a general projective hypersurface of high
degree is hyperbolic in the sense of Kobayashi.
Résumé. — Dans cet article, nous démontrons qu’il n’existe pas de famille de
rang maximal de courbes entières dans la famille universelle des hypersurfaces
de degré d > 2n dans l’espace projectif complexe Pn. Cela peut se voir comme
une version faible de la conjecture de Kobayashi affirmant qu’une hypersurface
projective générale de haut degré est hyperbolique au sens de Kobayashi.
Let X be a hypersurface in the complex projective space Pn. The Koba-
yashi conjecture claims that X is hyperbolic, provided that X is general
and d = deg(X) > 2n−1. By the Brody criterion ([1]), the hyperbolicity of
X is equivalent to the fact that every holomorphic map C → X is constant.
The conjecture has been proved for n = 3, d > 21 and X very general ([6],
[8] – see [3] for an account; for a recent extension of Demailly’s approach
to dimension 4, see [9]). An important and recent progress in the direction
of the conjecture for all n > 3 was made by Y.-T. Siu in [11]: he obtains a
confirmation of the conjecture under the assumption d n.
Consider the universal family X ⊂ Pn×PNd of hypersurfaces in Pn with
fixed degree d (the number Nd is equal to
(
n+d
d
)
− 1). We will denote by
Xt the fiber of X over the parameter t ∈ PNd .
Keywords: projective hypersurfaces, entire curves, Kobayashi hyperbolicity.
Math. classification: 14J70, 32Q45.
248 Olivier DEBARRE, Gianluca PACIENZA & Mihai PĂUN
Theorem 1. — Let U ⊂ PNd be an open set and let Φ : C×U → X be
a holomorphic map such that Φ(C×{t}) ⊂ Xt for all t ∈ U . If d > 2n, the
rank of Φ cannot be maximal anywhere.
Of course the theorem is an immediate consequence of the Kobayashi
conjecture (and hence, if the degree d is big enough, of the result of Siu). So,
the theorem above may be rephrased as follows : the Kobayashi conjecture
may possibly fail only if there is an entire curve on a general hypersurface
X which is not preserved by a deformation of X.
The question above is motivated by the “picture” in the algebraic sit-
uation: the existence of an algebraic cycle on the general member of the
family implies its deformation on the nearby fibers. However, dealing with
transcendental objects (e.g., entire curves) seems to be much more com-
plicated. For entire curves tangent to a holomorphic foliation of dimension
one, a substitute for the Hilbert scheme was found by M. Brunella in [2].
Very roughly, the proof goes as follows. First of all, we consider the
(non-zero) section of the holomorphic bundle Λ1+NdΦ∗TX given by the
Jacobian of Φ (in fact, for some technical reasons, we will work with a
sequence of reparametrizations of Φ, but we skip this point here, to keep
the discussion clear). In order to use the positivity of the canonical bundle
of the hypersurfaces, we take the wedge product of the previous section with
an appropriate family of meromorphic vector fields on X , and thus get a
section σ of (a twist of) Φ∗K−1X . Next, we show that the laplacian of the
logarithm of the norm of this section dominates a positive multiple of the
norm of σ, and use negative curvature arguments to derive a contradiction,
as soon as the degree d satisfies the numerical hypothesis of our theorem.
Proof of the theorem. The proof uses two ingredients: the first is that
the vector bundle TX ⊗ p∗OPn(1) is generated by its global sections (where
p is the projection Pn×PNd → Pn). The second relies on some negative cur-
vature arguments, very much in the spirit of the Kobayashi-Ochiai theorem
([10]).
We recall the following proposition, due to Siu ([11]).
Proposition 2 (Siu). — The vector bundle TX ⊗ p∗OPn(1) is globally
generated.
The proof of this proposition is given in [11]; we reproduce it here, for
the convenience of the reader. Observe that the global generation of the
restriction TX|Xt ⊗ OXt(1) of the same bundle to a fiber Xt has been
previously proved by Voisin ([12], Prop. 1.1) who deduced from it important
results about the algebraic hyperbolicity of a (very) general hypersurface
ANNALES DE L’INSTITUT FOURIER
NON-DEFORMABILITY OF ENTIRE CURVES 249
(for an account of the subsequent developments of Voisin’s approach, see
[4]).
Proof. — Consider global coordinates (Zj)06j6n (resp. (aα)|α|=d) on
Cn+1 (resp. on CNd+1). The equation of the manifold X in Pn × PNd can
be written as ∑
aαZ
α = 0
where we use here the multi-index notation Zα =
∏
Z
αj
j . Consider the
open set U0 = {Z0 6= 0} × {ad0...0 6= 0} in Pn × PNd . For the rest of the
proof, we will work on U0, with the induced nonhomogeneous coordinates.
Consider a multi-index α ∈ Nd and an integer j such that αj > 1. On
the set U0, consider the vector field
Vα,j =
∂
∂aα
− zj
∂
∂âα
where zj = Zj/Z0, âk = ak if k 6= j, and âj = αj − 1. The vector field
Vα,j is tangent to X0 = X ∩U0, as a quick verification shows. On the other
hand, we can extend it to the whole manifold X as a meromorphic vector
field and its pole order is equal to 1. Remark that Vα,j is a meromorphic
section of the kernel of the differential of the first projection p|X : X → Pn.
We also have a “lifting” property for the vector fields, as follows. Consider
a vector field
V0 =
n∑
j=1
vj
∂
∂zj
on Cn, where vj =
∑
k
v
(j)
k zk + v
(j)
0 is a polynomial of degree at most one
in the zj-variables. There exists a vector field
V =
∑
|α|6d
vα
∂
∂aα
+ V0
that is tangent to X0 and that extends to the whole manifold X as a
holomorphic section of the tangent bundle. Indeed, if we want V to be
tangent to X0, the condition to be satisfied is∑
α
vαz
α +
∑
α,j
aαvj
∂zα
∂zj
= 0
and the complex numbers vα are simply chosen such that the coefficient
of the monomial zα in the above equation is equal to zero. The extension
property is also quickly verified, as well as the global generation of the bun-
dle TX⊗p∗OPn(1) by the vector fields already constructed. The proposition
is thus proved. 
TOME 56 (2006), FASCICULE 1
250 Olivier DEBARRE, Gianluca PACIENZA & Mihai PĂUN
Consider a holomorphic map Φ : C× U → X over the base U ⊂ PNd as
in the theorem. We suppose that Φ has maximal rank. If d > 2n, we are
going to derive a contradiction.
As U is an open set, we can shrink it and suppose that it is equal to a
polydisc B(δ0)Nd . We will consider the following sequence of maps
Φk : B(δ0k)Nd+1 → X
given by Φk(z, ξ1, . . . , ξNd) = Φ(zk
Nd ,
1
k
ξ1, . . . ,
1
k
ξNd). The technical reason
for which we need to change the radius of the disc will be clear in a moment.
Notice that the initial map Φ1 = Φ is of maximal rank, thus the section
(1) JΦ(z, ξ) =
∂Φ
∂z
∧ ∂Φ
∂ξ1
∧ · · · ∧ ∂Φ
∂ξNd
(z, ξ) ∈ Λ1+NdTX ,Φ(z,ξ)
of Φ∗Λ1+NdTX is not identically zero. Let us assume that JΦ(0) is nonzero
in the corresponding vector space. Remark that JΦk(0) = JΦ(0), for any
k > 1, where JΦk ∈ Φ∗kΛ1+NdTX is the section associated to the map Φk
as indicated in (1), is not identically zero, as a section. The positivity of
the vector bundle TX in the parameter space directions now comes into the
picture: thanks to Proposition 2, we can choose n− 2 vector fields
V1, . . . , Vn−2 ∈ TX ⊗ p∗OPn(1)
such that
JΦk(0) ∧ Φ∗k
(
V1 ∧ · · · ∧ Vn−2
)
6= 0
in K−1X ⊗ p
∗OPn(n− 2)Φk(0).
With the vector fields previously chosen, we consider the following section
σk = JΦk ∧ Φ∗k
(
V1 ∧ · · · ∧ Vn−2
)
of the bundle Φ∗k(K
−1
X ⊗ p∗OPn(n− 2))
)
over the polydisc. Its value at the
origin is independent of k, and of course nonzero. If q is the projection
of X on the parameter space PNd , then under the assumption d > 2n,
the restriction of KX ⊗ OPn(2 − n)) to q−1(U) is ample (eventually after
shrinking once again the open subset U), hence we can endow this bundle
with a metric h of positive curvature.
We now define a sequence of functions fk : B(δ0k)Nd+1 → R+ as follows
∀w ∈ B(δ0k)Nd+1 fk(w) = ‖σk(w)‖2/(Nd+1)Φ∗
k
h−1 .
Remark 3. — Notice that, by construction, there exists a positive num-
ber c such that for each k > 1, we have fk(0) = c.
We have the following lemma.
ANNALES DE L’INSTITUT FOURIER
NON-DEFORMABILITY OF ENTIRE CURVES 251
Lemma 4. — For each k > 1, there exists a positive constant C such
that we have ∆ log fk > Cfk pointwise over the polydisc B(δ0k)Nd+1.
Proof. — First, remark that by construction, the image of the map Φk
lies inside q−1(U), for each k > 1, so that
(2) i∂∂̄ log ‖σk‖2Φ∗
k
h−1 > Φ
∗
kΘh
(
KX ⊗ p∗OPn(2− n)
)
.
In the inequality (2), take the trace with respect to the flat metric on the
polydisc. We get
∆ log ‖σk‖2Φ∗
k
h−1 > C
(∥∥∥∂Φk
∂z
∥∥∥2
ω
+
Nd∑
j=1
∥∥∥∂Φk
∂ξj
∥∥∥2
ω
)
> C‖JΦk‖
2/(1+Nd)
Λ1+Ndω
> C‖σk‖2/(1+Nd)Φ∗
k
h−1
The constant C in the previous sequence of inequalities varies from one line
to another, but we still denote it by C as it is independent of k. The above
relations are obtained using the vector inequalities
‖W1 ∧ · · · ∧Ws‖ 6 ‖W1‖ . . . ‖Ws‖ 6 s−s(‖W1‖+ · · ·+ ‖Ws‖)s.
So the lemma is proved. 
Using the previous lemma we will prove a result whose proof is very close
to that of the Ahlfors-Schwarz lemma.
Proposition 5. — For each k > 1 we have fk(0) 6 Ck−2. In particular,
as k →∞, we have fk(0) → 0.
Proof. — Consider the volume form of the Poincaré metric on the poly-
disc
ψk =
1(
1− |z|
2
δ20k
2
)2 Nd∏
j=1
1(
1− |ξj |
2
δ20k
2
)2
A quick computation shows that
i∂∂̄ logψk = (δ0k)−2
(
idz ∧ dz(
1− |z|
2
δ20k
2
)2 + Nd∑
j=1
idξj ∧ dξj(
1− |ξj |
2
δ20k
2
)2
)
so if we take the trace of this equality with respect to the flat metric, we
get
(3) ∆ logψk 6 Ck−2ψk.
Remark that the previous inequality can be obtained precisely because we
have the same radius δ0k for the components of the polydisc which is the
TOME 56 (2006), FASCICULE 1
252 Olivier DEBARRE, Gianluca PACIENZA & Mihai PĂUN
domain of ψk. This is why we had to reparametrize our map Φ from the
very beginning.
Consider the function (z, ξ) 7→ fk(z, ξ)
ψk(z, ξ)
. Its maximum cannot be achieved
at a boundary point of the domain, since ψk goes to infinity as (z, ξ) goes
to the boundary. So at the maximum point (z0, ξ0), we have
(4) ∆ log fk/ψk 6 0.
This inequality, combined with Lemma 4 and (3), gives
(5) fk(z0, ξ0) 6 Ck−2ψk(z0, ξ0).
Since the relation (5) is verified at the maximum point of the quotient, it
follows that the same is true at an arbitrary point, so we get
(6) fk(z, ξ) 6 Ck−2ψk(z, ξ).
To finish the proof, it is sufficient to write the inequality (6) at the origin.

Since Proposition 5 and Remark 3 contradict each other, the theorem is
proved. 
BIBLIOGRAPHY
[1] R. Brody, “Compact manifolds in hyperbolicity”, Trans. Amer. Math. Soc. 235
(1978), p. 213-219.
[2] M. Brunella, “On entire curves tangent to a foliation”, preprint 2004.
[3] ——— , “Courbes entières dans les surfaces algébriques complexes (d’après McQuil-
lan, Demailly-El Goul,. . . )”, Astérisque (2002), no. 282, p. Exp. No. 881, vii, 39-61,
Séminaire Bourbaki, Vol. 2000/2001.
[4] H. Clemens, “Lower bounds on genera of subvarieties of generic hypersurfaces”,
Comm. Algebra (2003), no. 31, p. 3673-3711, Special issue in honor of Steven L.
Kleiman.
[5] J.-P. Demailly, “Algebraic criteria for Kobayashi hyperbolic projective varieties
and jet differentials”, in Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure
Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, p. 285-360.
[6] J. E. G. J.-P. Demailly, “Hyperbolicity of generic surfaces of high degree in pro-
jective 3-space”, Amer. J. Math. 122 (2000), p. 515-546.
[7] S. Kobayashi, Hyperbolic complex spaces, Grundlehren der Mathematischen Wis-
senschaften, no. 318, Springer-Verlag, Berlin, 1998.
[8] M. McQuillan, “Holomorphic curves on hyperplane sections of 3-folds”, Geom.
Funct. Anal. (1999), no. 9, p. 370-392.
[9] E. Rousseau, “Weak analytic hyperbolicity of generic hypersurfaces of high degree
in the complex projective space of dimension 4”, preprint math.AG/0510285.
[10] T. O. S. Kobayashi, “Meromorphic mappings onto compact complex spaces of
general type”, Invent. Math. (1975), no. 31, p. 7-16.
[11] Y. Siu, “Hyperbolicity in complex geometry”, in The legacy of Niels Henrik Abel,
Springer, Berlin, 2004, p. 543-566.
ANNALES DE L’INSTITUT FOURIER
NON-DEFORMABILITY OF ENTIRE CURVES 253
[12] C. Voisin, “On a conjecture of Clemens on rational curves on hypersurfaces”, J.
Diff. Geom. (1996), no. 44, p. 200-214.
[13] ——— , “A correction: “on a conjecture of Clemens on rational curves on hyper-
surfaces””, J. Diff. Geom. (1998), no. 49, p. 601-611.
Manuscrit reçu le 5 avril 2005,
accepté le 28 juillet 2005.
Olivier DEBARRE & Gianluca PACIENZA
Université L. Pasteur et CNRS
Institut de Recherche Mathématique Avancée
7, rue René Descartes
67084 Strasbourg Cedex (France)
debarre@math.u-strasbg.fr
pacienza@math.u-strasbg.fr
Mihai PĂUN
Université Henri Poincaré Nancy 1
Institut Élie Cartan
B.P. 239
54506 Vandœuvre-lès-Nancy Cedex (France)
Mihai.Paun@iecn.u-nancy.fr
TOME 56 (2006), FASCICULE 1


Non-deformability of entire curves in projective hypersurfaces of high degree

Stalder, Yves

Moyennabilité intérieure et extensions HNN

Nicolas, Jean-Philippe

On Lars Hörmander’s remark on the characteristic Cauchy problem

Ramsey, Nick

Geometric and $p$-adic Modular Forms of Half-Integral Weight

Brinon, Olivier

Représentations cristallines dans le cas d’un corps résiduel imparfait

Hajir, Farshid

Wong, Siman

Specializations of one-parameter families of polynomials

de Bartolomeis, Paolo

Tomassini, Adriano

On Solvable Generalized Calabi-Yau Manifolds

Not a member yet

Spectrum of the Laplace operator and periodic geodesics: thirty years after

Semi-classical formula beyond the Ehrenfest time in quantum chaos. (I) Trace formula

Homology and modular classes of Lie algebroids

Non-deformability of entire curves in projective hypersurfaces of high degree

Moyennabilité intérieure et extensions HNN

On Lars Hörmander’s remark on the characteristic Cauchy problem

Geometric and $p$ -adic Modular Forms of Half-Integral Weight

Représentations cristallines dans le cas d’un corps résiduel imparfait

Specializations of one-parameter families of polynomials

On Solvable Generalized Calabi-Yau Manifolds

3,503

3,732