Degenerate omplex Monge Ampère equations

Jean-Pierre Demailly

Description

Degenerate omplex Monge-Ampère equations over ompa t Kähler manifolds Jean-Pierre Demailly and Nefton Pali Abstra t We prove the existen e and uniqueness of the solutions of some very general type of degenerate omplex Monge-Ampère equations, and investigate their regularity. This type of equations are pre isely what is needed in order to onstru t Kähler-Einstein metri s over ir- redu ible singular Kähler spa es with ample or trivial anoni al sheaf and singular Kähler-Einstein metri s over varieties of general type. Contents 1 Introdu tion 1 2 General L∞-estimates for the solutions 5 3 Currents with Bedford-Taylor type singularities 23 4 Uniqueness of the solutions 35 5 Generalized Kodaira lemma 43 6 Existen e and higher order regularity of solutions 46 7 Appendix 60 1 Introdu tion In a elebrated paper [Yau? published in 1978, Yau solved the Calabi onje - ture. As is well known, the problem of pres ribing the Ri i urvature an be formulated in terms of non-degenerate omplex Monge-Ampère equations. Key words : Complex Monge-Ampère equations, Kähler-Einstein metri s, Closed posi- tive urrents, Plurisubharmoni fun tions, Capa ities, Orli z spa es. AMS Classi ation : 53C25, 53C55, 32J15.

• degenerate omplex


• ontinuous lo


• monge-ampère equations


• al potentials


• over ompa


• kähler-einstein metri


• general l∞

Sujets

• Demailly

Informations

• Publié par: pefav
• Nombre de lectures: 10
• Langue: English

Extrait

∞L
solutions
Monge-Amp
in
?re
[Y
equations
53C55,
o
of
v
ell
er
K?hler-Einstein

yp
K?hler
43
manifolds
In
Jean-Pierre
ed
Demailly

and
Key
Nefton
functions,
P
Bedford-T
ali
4

K
W
higher
e
App
pro
a
v
1978,
e
ture.
the
problem
existence
b
and

uniqueness
Complex
of
osi-
the
AMS
solutions
ts
of
ylor
some
singularities
v
of
ery
5
general
daira
t
Existence
yp
regularit
e
46
of
60
degenerate


pap
Monge-Amp
published
?re
au
equations,
Calabi
and
is
in
wn,
v

estigate
ature
their
form
regularit
of
y
?re
.
ords
This
?re
t
Closed
yp

e

of
:
equations
1
are
with
precisely
a
what
t
is
e
needed
23
in
Uniqueness
order
the
to
35

Generalized
K?hler-Einstein
o
metrics
lemma
o
6
v
and
er
order
ir-
y

solutions
singular
7
K?hler
endix
spaces
1
with
tro
ample
In
or

trivial
er

℄
sheaf
in
and
Y
singular
solv
K?hler-Einstein
the
metrics

o
As
v
w
er
kno
v
the
arieties
of
of
the
general

t

yp
e
e.
ulated
Con
terms
ten
non-degenerate
ts
Monge-Amp
1
equations.
In
w
tro
:

Monge-Amp
1
equations,
2
metrics,
General
p

tiv
e
Degenerate
ts,
Plurisubharmonic
-estimates

for
spaces.
the
Classication
solutions
53C25,
5
32J15.
3
CurrenX
n χ v > 0R R
nX v = χ ( )
X X
n n¯ ¯ω∈χ ( ω =ω +i∂∂ϕ ω ∈χ ) ω =(ω +i∂∂ϕ) =v0 0 0
n¯(ω+i∂∂ϕ) =v,
pω v≥ 0 L
ω
(1,1)
BT
Θ BT
k+1 k¯Θ = i∂∂(ϕΘ ),
kϕ Θ ϕΘ
(1,1) χ
BT := BT∩χ.χ
logBT ⊂ BT ,χχ
T ∈ BTχ
n 1T L
Z
n−log(T /Ω)Ω< +∞,
X
Another
ol1
ositiv
℄
Monge-Amp
pro

v
result
ed
s
the

existence
?re
of

solutions
unique
for
een
equations
the
of
?re
t
the
yp
b
e
p
o?o
for
K
e
results,

these
uniqueness
bining
un

of
y
In
b
for

mension
In

equations.
1.1
?re
p
Monge-Amp
as
where
as

where
a
the
K?hler
of
metric
eectiv
and

degenerate
monotone
ery
o
v
subset
a
ther
densit
of
y
a
in
oten
of
and
study
studies,
or
[Blo1
in
e
some
density
general
a

e
spaces.
v
Ho
omplex
w

[K
ev
et
er,
if
in
e
v
w
arious
e

Bedford-T
applications,
hiev
it
tions
is


is
to
tial


the
lo

mass.
where
ery
the
.
is
i.e.
merely
,
semip
e
ositiv
v
e.
exterior
This
ers
more
in
dicult
oth
situation
exists
has
that
b
the
een
degenerate
examined
equations
rst

b
ounded
y
has
T
big
suji
ob
[T
tensiv
s],
e.g.
and
[Ti-Zha
his
[E-G-Z1


hnique
tro
has

b
smo
een


(closed
in

the
let

h
t
a
w

orks
ossessing
[Ca-La
-densit
℄
that
[Ti-Zha
act
℄
e
[E-G-Z1
au).
℄
ylor
and
the
[P

au
exterior
℄
o
In
ers
this
b
pap

er
a
w
y
e
ed
push
ac
further
w
the
equa-

Monge-Amp
hniques
of
dev
study
elop
a
ed
oten
so
of
far
and
and
breakthrough
w
is
e

obtain
nite
some
Then
v
ev
ery
pseudo
general
e
and
that
sharp
with
results
-cohomology
on
metric
for
w
dziej
pro
v
uniqueness
a
and

regularit
ergence
y
for
of
p
the
w
solutions
of
of
ts
degenerate
the

K?hler
Monge-Amp
smo
?re
a
equations.
e
In
,
order
The
to
of
dene
solutions
the
the
relev

an
?re
t
in

reasonable
of
of
uniqueness
b
of
p
the
tials
solutions,
b
w
a
e
issue
in
the
tro


in
a
e
suitable
see
subset
[T
of
℄
the
℄
space
℄
of
℄

this
d
w
metho
in
new

a
subset
initiated
on
-curren
oth
ts,
any
namely
Then
the
K?hler
domain
of
of
p
denition
e)
They
ts
℄
b
of
and
the
whic

ha
Monge-Amp
e
?re
Monge-Amp
op
pro
erator
di-
in
p
the
an
sense
of
of
y
Bedford-T
h
a
manifold
ylor:
K?hler
a
omp

a
t
b
a
L
is
(Y
in
Theorem
[Be-T
the
2
existence,Ω> 0
n 1T L
logBTχ
n+εω≥ 0 Llog L
X n
χ (1,1) R
n(1,1) ω ω >0
X
R R
n+ε nLlog L v ≥ 0 ε > 0 v = χ
X X
nT ∈ BT T =vχ
X
Σ ⊂Xχ
χ χ
v ≥ 0
T
Σ ∪Z(v)⊂X Z(v) vχ
Ric(ω) = −λω+ρ, λ≥ 0.
∞L
∞L
zer
y


of

ent
analytic
K?hler-Einstein
singularities
relev
(see
of
Theorems
and
6.2
equation
and
It
6.1).
righ
As

a
Laplacian

ossessing
of

this
erties
results,
The
w

e
K?hler
deriv
singular
e
e
the
erate
follo
the
wing
bining
generalization
metric
of
obtained
Y
℄
au's
sin-
theorem.
smo
Theorem
y
1.2
prop
L
is
et
oles
v
of
b
Theorem
e
in
a
er

or
omp
also
act
v
K?hler
e
manifold
the
of
of

a
omplex
with
dimension
Theorem
pro
metrics)
and
one
let

e
pro
b
6.1,
e
bination
a
[T
w
particular

omplex
last
the
the
is
In
outside
-c
analytic
ohomolo
e
gy
regularit

wher
admitting
ossesses
a
set
smo
and
oth
side

yp
d
Monge-Amp
semip
ed
ositive
is
.
is

to
the
o
in

dened
with
-form

ell
b

to
that
metrics
w
v
is
t
erator
to
op
equations

hand
the
when
.
solutions
(A)
en
F
ect
or
t
any
needed
that
of
e
(in
observ
with
e
obtained
W
is
6.1).
not
Theorem
maxim
(see
the
-density
degenerate.
singularities
of
analytic
in

h
,
a
with
the
y
[Y
-densit
℄

vides
that
p
an

ossesses
analytic
p
gularities
t


or
the
also
when
oth

the
,
omplex
ther
subset
e
-densit
exists
existence
a
ne
unique
y

,
d
e
p
an
ositive
p

the
ent
of
the
os
is
p
this
of
example
.
or
t

e
that

F
?re