Winner of the 2016 EMS Monograph Award! Complex Monge–Ampère equations have been one of the most powerful tools in Kähler geometry since Aubin and Yau’s classical works, culminating in Yau’s solution ...to the Calabi conjecture. A notable application is the construction of Kähler-Einstein metrics on some compact Kähler manifolds. In recent years degenerate complex Monge–Ampère equations have been intensively studied, requiring more advanced tools. The main goal of this book is to give a self-contained presentation of the recent developments of pluripotential theory on compact Kähler manifolds and its application to Kähler–Einstein metrics on mildly singular varieties. After reviewing basic properties of plurisubharmonic functions, Bedford–Taylor’s local theory of complex Monge–Ampère measures is developed. In order to solve degenerate complex Monge–Ampère equations on compact Kähler manifolds, fine properties of quasi-plurisubharmonic functions are explored, classes of finite energies defined and various maximum principles established. After proving Yau’s celebrated theorem as well as its recent generalizations, the results are then used to solve the (singular) Calabi conjecture and to construct (singular) Kähler–Einstein metrics on some varieties with mild singularities. The book is accessible to advanced students and researchers of complex analysis and differential geometry.
We prove the existence of the first eigenvalue and an associated eigenfunction with Dirichlet condition for the complex Monge–Ampère operator on a bounded strongly pseudoconvex domain in
C
n
. We ...show that the eigenfunction is plurisubharmonic, smooth with bounded Laplacian in
Ω
and boundary values 0. Moreover it is unique up to a positive multiplicative constant. To this end, we follow the strategy used by P. L. Lions in the real case. However, we have to prove a new theorem on the existence of solutions for some special complex degenerate Monge–Ampère equations. This requires establishing new a priori estimates of the gradient and Laplacian of such solutions using methods and results of Caffarelli et al. (Commun Pure Appl Math 38(2):209–252, 1985) and Guan (Commun Anal Geom 6(4):687–703, 1998). Finally we provide a Pluripotential variational approach to the problem and using our new existence theorem, we prove a Rayleigh quotient type formula for the first eigenvalue of the complex Monge–Ampère operator.
A variational approach to complex Monge-Ampère equations Berman, Robert J.; Boucksom, Sébastien; Guedj, Vincent ...
Publications mathématiques. Institut des hautes études scientifiques,
06/2013, Letnik:
117, Številka:
1
Journal Article
Recenzirano
We show that degenerate complex Monge-Ampère equations in a big cohomology class of a compact Kähler manifold can be solved using a variational method, without relying on Yau’s theorem. Our ...formulation yields in particular a natural pluricomplex analogue of the classical logarithmic energy of a measure. We also investigate Kähler-Einstein equations on Fano manifolds. Using continuous geodesics in the closure of the space of Kähler metrics and Berndtsson’s positivity of direct images, we extend Ding-Tian’s variational characterization and Bando-Mabuchi’s uniqueness result to singular Kähler-Einstein metrics. Finally, using our variational characterization we prove the existence, uniqueness and convergence as
k
→∞ of
k
-balanced metrics in the sense of Donaldson both in the (anti)canonical case and with respect to a measure of finite pluricomplex energy.
We study fine properties of quasiplurisubharmonic functions on compact Kähler manifolds. We define and study several intrinsic capacities which characterize pluripolar sets and show that locally ...pluripolar sets are globally “quasi-pluripolar.”
Kähler–Einstein currents, also known as singular Kähler–Einstein metrics, have been introduced and constructed a little over a decade ago. These currents live on mildly singular compact Kähler spaces ...X and their two defining properties are the following: They are genuine Kähler–Einstein metrics on
$X_{\mathrm {reg}}$
, and they admit local bounded potentials near the singularities of X. In this note, we show that these currents dominate a Kähler form near the singular locus, when either X admits a global smoothing, or when X has isolated smoothable singularities. Our results apply to klt pairs and allow us to show that if X is any compact Kähler space of dimension three with log terminal singularities, then any singular Kähler–Einstein metric of nonpositive curvature dominates a Kähler form.
We compare various notions of weak subsolutions to degenerate complex Monge-Amp\`ere equations, showing that they all coincide. This allows us to give an alternative proof of mixed Monge-Amp\`ere ...inequalities due to Kolodziej and Dinew.
We study degenerate complex Monge–Ampère equations on a compact Kähler manifold
(
X
,
ω
)
. We show that the complex Monge–Ampère operator
(
ω
+
d
d
c
⋅
)
n
is well defined on the class
E
(
X
,
ω
)
...of
ω-plurisubharmonic functions with finite weighted Monge–Ampère energy. The class
E
(
X
,
ω
)
is the largest class of
ω-psh functions on which the Monge–Ampère operator is well defined and the comparison principle is valid. It contains several functions whose gradient is not square integrable. We give a complete description of the range of the operator
(
ω
+
d
d
c
⋅
)
n
on
E
(
X
,
ω
)
, as well as on some of its subclasses. We also study uniqueness properties, extending Calabi's result to this unbounded and degenerate situation, and we give applications to complex dynamics and to the existence of singular Kähler–Einstein metrics.
Let
be a compact
Kähler manifold. We show that the Kähler–Ricci flow (as well as its twisted versions) can be run from an arbitrary positive closed current with zero Lelong numbers and immediately ...smoothes it.
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