Kähler Einstein C urrents and Relative Pluricanonical Systems Hajime TSUJI Sophia Univesity Durhan July 2 2012 Main Result Scheme of the proof Canonical metrics ID: 428276
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Slide1
Twisted Kähler-Einstein Currents andRelative Pluricanonical Systems
Hajime TSUJI
Sophia
Univesity
Durhan
July 2 , 2012Slide2
Main ResultSlide3
Scheme of the proof Slide4
Canonical metricsConstruct a canonical singular hermitian metrics on the canonical bundle of the varieties.Requirement : The metrics varies in a plurisubharmonic way,i.e
. the metrics has
semipositive
curvature on projective families(hopefully also for
K
ähler families). The metrics defines the Monge
-Amp
è
re foliation on the family. Slide5
Kähler
-Einstein
Kähler
-Einstein
metrics
Theorem (
Aubin-Yau
)
Slide6
Canonical ring
We want to construct a (singular)
K
ä
hler
metric which reflects the
canonical ring. Slide7
Iitaka fibration
Iitaka
fibration
is the most naïve
geometric
realization of the positivity of the canonical ring.Slide8
Iiaka
fibration
2Slide9
Hodge Q-line bundleSlide10
Hodge metric
By the variation of Hodge structure we have : Slide11
F
ig.1Slide12
T
wisted
Kähler
-Einstein
currentsSlide13
Existence of Twisted Kähler
-Einstein currents
Theorem
Let
be a KLT pair with
And let
be the
Iitaka
fibration
of
. And let
be the Hodge line bundle with the Hodge metric.
Then there exists a unique twisted
Kähler
-Einstein current
on Slide14
Monge Ampère equation
Complex
Monge
-Amp
è
re equationSlide15
Monge-Ampère equations on compact Kähler manifoldsSlide16
Relative Iitaka fibrationsSlide17
Relative Twisted Kähler
-Einstein currentsSlide18
Relative Twisted Kähler-Einstein currents 2Slide19
Variation of Twisted Kä
hler
-Einstein currents
TheoremSlide20
Dynamical system of Bergman kernels
Approximate in terms of Bergman kernels.Slide21
Monge-Ampère equations and Bergman kernelsSlide22
Berndtsson’s theorem(with Pă
un)Slide23
Use of the Plurisubharmonicity of Bergman kernelsSlide24
Dirichlet problem for complex Monge-Ampère equations
We consider the
Dirichlet
problem: Slide25
Boudary regularitySlide26
Interior regularity Slide27
D
irichlet
construction of twisted
K
ä
hler
-Einstein currents
ISlide28
Dirichlet problem for complex Monge-Ampere equations IISlide29
SmoothnessSlide30
P
roof of the smoothness
(1) Construct the twisted
K
ä
hler-Eisntein
current as the limit of
Dirichlet
problems of complex
Monge
-Amp
è
re equations.
(2) Consider the family of exhaustion via strongly
pseudoconvex
domains and apply the implicit function theorem to the solution of complex
Monge
-Amp
è
re equations.
(3) Apply the weighted uniform estimates to the solution and taking the limit for the horizontal derivatives. Slide31
Monge-Ampère foliationsSlide32
Descent of leavesSlide33
Use of the weak semistabilitySlide34
Flatness of the relative canonical systems along leaves Slide35
IsometriesSlide36
C
losedness
of the
leavesSlide37
Decent of the positivitySlide38
Positivity of the determinant