Entire curves and algebraic differential equations
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Documents
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
- constant holomorphic map
- simply connected
- open subset
- has no
- dimensional manifold
- liouville's theorem
- entire curves
JaenP-irereDmeiallyG(reEntirecurvesand
algebraicdifferentialequations
nboleJean-PierreDemailly
InstitutFourier,Universite´deGrenobleI,France
April162009,Saint-Martind’He`res
IF-IMPAConference
)I,610//40290Enitreucrvsenadlaegrbiacdiffreneitlaqeutainos/FI-MIAP
JeEntirecurves
naP-irereDmeDefinition.
Byan
entirecurve
wemeananonconstant
holomorphicmap
f
:
C
→
X
intoacomplex
n
-dimensionalmanifold.
iallyG(rnebole)I,610//40290Enitreucrvsenadlaegrbiacdiffreneitlaqeutainos/FI-MIAP
JeEna-niPretreirDeemcruvesDefinition.
Byan
entirecurve
wemeananonconstant
holomorphicmap
f
:
C
→
X
intoacomplex
n
-dimensionalmanifold.
If
X
isa
bounded
opensubsetΩ
⊂
C
n
,thenthereareno
entirecurves
f
:
C
→
Ω(
Liouville’stheorem
)
iallyG(rnebole)I,610//40290Enitreucrvsenadlaegrbiacdiffreneitlaqeutainos/FI-MIAP
JeEna-niPretreirDeemcruvesDefinition.
Byan
entirecurve
wemeananonconstant
holomorphicmap
f
:
C
→
X
intoacomplex
n
-dimensionalmanifold.
If
X
isa
bounded
opensubsetΩ
⊂
C
n
,thenthereareno
entirecurves
f
:
C
→
Ω(
Liouville’stheorem
)
X
=
Cr
{
0
,
1
,
∞}
=
Cr
{
0
,
1
}
hasnoentirecurves
(
Picard’stheorem
)
iallyG(rnebole)I,610//40290Enitreucrvsenadlaegrbiacdiffreneitlaqeutainos/FI-MIAP
JeEna-niPretreirDeemcruvesDefinition.
Byan
entirecurve
wemeananonconstant
holomorphicmap
f
:
C
→
X
intoacomplex
n
-dimensionalmanifold.
If
X
isa
bounded
opensubsetΩ
⊂
C
n
,thenthereareno
entirecurves
f
:
C
→
Ω(
Liouville’stheorem
)
X
=
Cr
{
0
,
1
,
∞}
=
Cr
{
0
,
1
}
hasnoentirecurves
(
Picard’stheorem
)
Acomplextorus
X
=
C
n
/
Λ(Λlattice)hasalotofentire
curves.As
C
simplyconnected,every
f
:
C
→
X
=
C
n
/
Λ
liftsas
f
˜:
C
→
C
n
,
f
˜(
t
)=(
f
˜
1
(
t
)
,...,
f
˜
n
(
t
))
and
f
˜
j
:
C
→
C
canbearbitraryentirefunctions.
iallyG(rnebole)I,610//40290Enitreucrvsenadlaegrbiacdiffreneitlaqeutainos/FI-MIAP
JePnaP-rieorrejeDecmtivealegrbaicvraietiesConsidernowthecomplexprojective
n
-space
P
n
=
P
n
C
=(
C
n
+1
r
{
0
}
)
/
C
∗
,
[
z
]=[
z
0
:
z
1
:
...
:
z
n
]
.
iallyG(rnebole)I,610//40290Enitreucrvsenadlaegrbiacdiffreneitlaqeutainos/FI-MIAP
JePnaP-rieorrejeDecmtivealegrbaicvraietiesConsidernowthecomplexprojective
n
-space
P
n
=
P
n
C
=(
C
n
+1
r
{
0
}
)
/
C
∗
,
[
z
]=[
z
0
:
z
1
:
...
:
z
n
]
.
Anentirecurve
f
:
C
→
P
n
isgivenbyamap
t
7−→
[
f
0
(
t
):
f
1
(
t
):
...
:
f
n
(
t
)]
where
f
j
:
C
→
C
areholomorphicfunctionswithout
commonzeroes(sotherearealotofthem).
iallyG(rnebole)I,610//40290Enitreucrvsenadlaegrbiacdiffreneitlaqeutainos/FI-MIAP
JePnaP-rieorrejeDecmtivealegrbaicvraietiesConsidernowthecomplexprojective
n
-space
P
n
=
P
n
C
=(
C
n
+1
r
{
0
}
)
/
C
∗
,
[
z
]=[
z
0
:
z
1
:
...
:
z
n
]
.
Anentirecurve
f
:
C
→
P
n
isgivenbyamap
t
7−→
[
f
0
(
t
):
f
1
(
t
):
...
:
f
n
(
t
)]
where
f
j
:
C
→
C
areholomorphicfunctionswithout
commonzeroes(sotherearealotofthem).
Moregenerally,lookata(complex)
projectivemanifold
,
.e.i
X
n
⊂
P
N
,
X
=
{
[
z
];
P
1
(
z
)=
...
=
P
k
(
z
)=0
}
where
P
j
(
z
)=
P
j
(
z
0
,
z
1
,...,
z
N
)arehomogeneous
polynomials(ofsomedegree
d
j
),suchthat
X
is
nonsingular
.
iallyG(rnebole)I,610//40290Enitreucrvsenadlaegrbiacdiffreneitlaqeutainos/FI-MIAP
JeKobayashimetric/hyperbolicmanifolds
naP-irereDmeForacomplexmanifold,
n
=dim
C
X
,onedefines
the
Kobayashipseudo-metric
:
x
∈
X
,
ξ
∈
T
X
κ
x
(
ξ
)=inf
{
λ>
0;
∃
f
:
D
→
X
,
f
(0)=
x
,λ
f
∗
(0)=
ξ
}
On
C
n
,
P
n
orcomplextori
X
=
C
n
/
Λ,onehas
κ
X
≡
0
.
nO
iallyG(rneo,
ble)I,1otxelpmocroir
/6402/009Ent=
ireucrvsesaheno,Λ/
nadlaegrbiacdiffreneitlaqeutainos/FI-MIAP
JeKobayashimetric/hyperbolicmanifolds
naP-irereDmeForacomplexmanifold,
n
=dim
C
X
,onedefines
the
Kobayashipseudo-metric
:
x
∈
X
,
ξ
∈
T
X
κ
x
(
ξ
)=inf
{
λ>
0;
∃
f
:
D
→
X
,
f
(0)=
x
,λ
f
∗
(0)=
ξ
}
On
C
n
,
P
n
orcomplextori
X
=
C
n
/
Λ,onehas
κ
X
≡
0
.
X
issaidtobe
hyperbolic(inthesenseofKobayashi)
if
theassociatedintegratedpseudo-distanceisadistance
(i.e.itseparatespoints–Hausdorfftopology),
iallyG(rnebole)I,610//40290Enitreucrvsenadlaegrbiacdiffreneitlaqeutainos/FI-MIAP
