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Journal ArticleDOI

A variational approach to complex Monge-Ampère equations

TL;DR: In this article, the degenerate complex Monge-Ampere equations in a big cohomology class of a compact Kahler manifold can be solved using a variational method, without relying on Yau's theorem.
Abstract: We show that degenerate complex Monge-Ampere equations in a big cohomology class of a compact Kahler 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 Kahler-Einstein equations on Fano manifolds. Using continuous geodesics in the closure of the space of Kahler metrics and Berndtsson’s positivity of direct images, we extend Ding-Tian’s variational characterization and Bando-Mabuchi’s uniqueness result to singular Kahler-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.
Citations
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Journal ArticleDOI
TL;DR: In this paper, the existence and regularity of a compact Kahler manifold M with edge singularities with cone angle 2 along a smooth divisor D was studied and it was shown that solutions of this problem are polyhomogeneous, i.e., have a complete asymptotic expansion with smooth coecients along D for all 2 < 2.
Abstract: This article considers the existence and regularity of Kahler{Einstein metrics on a compact Kahler manifold M with edge singularities with cone angle 2 along a smooth divisor D. We prove existence of such metrics with negative, zero and some positive cases for all cone angles 2 2 . The results in the positive case parallel those in the smooth case. We also establish that solutions of this problem are polyhomogeneous, i.e., have a complete asymptotic expansion with smooth coecients along D for all 2 < 2 .

239 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that any singular Fano variety X admitting a Kahler-Einstein metric is K-polystable, thus confirming one direction of the Yau-Tian-Donaldson conjecture in the setting of Fano varieties equipped with their anti-canonical polarization.
Abstract: It is shown that any, possibly singular, Fano variety X admitting a Kahler-Einstein metric is K-polystable, thus confirming one direction of the Yau-Tian-Donaldson conjecture in the setting of $$\mathbb {Q}$$ -Fano varieties equipped with their anti-canonical polarization. The proof is based on a new formula expressing the Donaldson-Futaki invariants in terms of the slope of the Ding functional along a geodesic ray in the space of all bounded positively curved metrics on the anti-canonical line bundle of X. One consequence is that a toric Fano variety X is K-polystable iff it is K-polystable along toric degenerations iff 0 is the barycenter of the canonical weight polytope P associated to X. The results also extend to the logarithmic setting and in particular to the setting of Kahler-Einstein metrics with edge-cone singularities. Applications to geodesic stability, bounds on the Ricci potential and Perelman’s $$\lambda $$ -entropy functional on K-unstable Fano manifolds are also given.

210 citations

Journal ArticleDOI
TL;DR: In this paper, the existence and uniqueness of the normalized Kahler-Ricci flow on non-singular Fano manifolds with log terminal singularities has been proved.
Abstract: We prove the existence and uniqueness of Kahler-Einstein metrics on Q-Fano varieties with log terminal singularities (and more generally on log Fano pairs) whose Mabuchi functional is proper. We study analogues of the works of Perelman on the convergence of the normalized Kahler-Ricci flow, and of Keller, Rubinstein on its discrete version, Ricci iteration. In the special case of (non-singular) Fano manifolds, our results on Ricci iteration yield smooth convergence without any additional condition, improving on previous results. Our result for the Kahler-Ricci flow provides weak convergence independently of Perelman's celebrated estimates.

209 citations

Journal ArticleDOI
TL;DR: In this article, the energy at equilibrium of a big line bundle on a compact complex manifold X is defined as the Monge-AmpSre energy of the extremal psh weight associated to (K,phi).
Abstract: Let L be a big line bundle on a compact complex manifold X. Given a non-pluripolar compact subset K of X and a continuous Hermitian metric e (-phi) on L, we define the energy at equilibrium of (K,phi) as the Monge-AmpSre energy of the extremal psh weight associated to (K,phi). We prove the differentiability of the energy at equilibrium with respect to phi, and we show that this energy describes the asymptotic behaviour as k -> a of the volume of the sup-norm unit ball induced by (K,k phi) on the space of global holomorphic sections H (0)(X,kL). As a consequence of these results, we recover and extend Rumely's Robin-type formula for the transfinite diameter. We also obtain an asymptotic description of the analytic torsion, and extend Yuan's equidistribution theorem for algebraic points of small height to the case of a big line bundle.

188 citations

Posted Content
TL;DR: In this article, it was shown that any singular, possibly singular, Fano variety X admitting a Kahler-Einstein metric is K-polystable, thus confirming one direction of the Yau-Tian-Donaldson conjecture in the setting of Q-Fano varieties equipped with their anti-canonical polarization.
Abstract: It is shown that any, possibly singular, Fano variety X admitting a Kahler-Einstein metric is K-polystable, thus confirming one direction of the Yau-Tian-Donaldson conjecture in the setting of Q-Fano varieties equipped with their anti-canonical polarization. The proof exploits convexity properties of the Ding functional along weak geodesic rays in the space of all bounded positively curved metrics on the anti-canonical line bundle of X and also gives a new proof in the non-singular case. One consequence is that a toric Fano variety X is K-polystable iff it is K-polystable along toric degenerations iff 0 is the barycenter of the canonical weight polytope P associated to X. The results also extend to the logarithmic setting and in particular to the setting of Kahler-Einstein metrics with edge-cone singularities. Furthermore, applications to geodesic stability, bounds on the Ricci potential and Perelman's entropy functional on K-unstable Fano manifolds are given.

183 citations

References
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Journal ArticleDOI
TL;DR: In this paper, the Ricci form of some Kahler metric is shown to be closed and its cohomology class must represent the first Chern class of M. This conjecture of Calabi can be reduced to a problem in non-linear partial differential equation.
Abstract: Therefore a necessary condition for a (1,l) form ( G I a ' r r ) I,,, Rlr dz' A d? to be the Ricci form of some Kahler metric is that it must be closed and its cohomology class must represent the first Chern class of M. More than twenty years ago, E. Calabi [3] conjectured that the above necessary condition is in fact sufficient. This conjecture of Calabi can be reduced to a problem in non-linear partial differential equation.

2,903 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that pl-flips exist in dimension n − 1, assuming finite generation in dimension N − 1 and assuming that pl flips exist in all dimensions.
Abstract: Assuming finite generation in dimension n − 1, we prove that pl-flips exist in dimension n.

1,612 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that the existence of Kahler-Einstein metrics implies the stability of the underlying Kahler manifold in a suitable sense, which disproves a long-standing conjecture that a compact KG admits KG metrics if it has positive first Chern class and no nontrivial holomorphic vector fields.
Abstract: In this paper, we prove that the existence of Kahler-Einstein metrics implies the stability of the underlying Kahler manifold in a suitable sense. In particular, this disproves a long-standing conjecture that a compact Kahler manifold admits Kahler-Einstein metrics if it has positive first Chern class and no nontrivial holomorphic vector fields. We will also establish an analytic criterion for the existence of Kahler-Einstein metrics. Our arguments also yield that the analytic criterion is satisfied on stable Kahler manifolds, provided that the partial C 0-estimate posed in [T6] is true.

1,038 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that a metric of constant scalar curvature on a polarised Kahler manifold is the limit of metrics induced from a specific sequence of projective embeddings.
Abstract: We prove that a metric of constant scalar curvature on a polarised Kahler manifold is the limit of metrics induced from a specific sequence of projective embeddings; satisfying a condition introduced by H. Luo. This gives, as a Corollary, the uniqueness of constant scalar curvature Kahler metrics in a given rational cohomology class. The proof uses results in the literature on the asymptotics of the Bergman kernel. The arguments are presented in a general framework involving moment maps for two different group actions.

734 citations