Degenerate omplex Monge Ampère equations

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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∞

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Publié par

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Langue

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Demailly

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

yp
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43
manifolds
In
Jean-Pierre
ed
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P
Bedford-T
ali
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ture.
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ery
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general
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yp
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46
of
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estigate
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35

Generalized
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metrics
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o
6
v
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solutions
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7
K?hler
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tro
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Monge-Amp
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metrics,
General
p

tiv
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Degenerate
ts,
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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
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ositiv
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e
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mension
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equations.
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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
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y

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ent
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of
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6.1).
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ossessing
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this
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6.1,
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omplex
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to
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metrics
w
v
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equations

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(A)
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ect
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with
e
obtained
W
is
6.1).
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Theorem
maxim
(see
the
-density
degenerate.
singularities
of
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in

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-densit
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gularities
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existence
a
ne
unique
y

,
d
e
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an
ositive
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the
ent
of
the
os
is
p
this
of
example
.
or
t

e
that

F
?re

Voir

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