ALE Ricci-flat K?hler metrics and deformations of quotient surface singularities

Let N₀=\mathbb C²/H{N_0={\mathbb C}^2/H} be an isolated quotient singularity with H ì U(2){H\subset U(2)} a finite subgroup. We show that for any \mathbb Q{\mathbb Q} -Gorenstein smoothings of N ₀ a nearby fiber admits ALE Ricci-flat K?hler metrics in any K?hler class. Moreover, we generalize Kronheimer’s results on hyperk?hler 4-manifolds (J Differ Geom 29(3):685–697, 1989), by giving an explicit classification of the ALE Ricci-flat K?hler surfaces. We construct ALF Ricci-flat K?hler metrics on the above non-simply connected manifolds. These provide new examples of ALF Ricci-flat K?hler 4-manifolds, with cubic volume growth and cyclic fundamental group at infinity.     

Uniqueness of extremal Kähler metrics

In the infinite dimensional space of Kähler potentials, the geodesic equation of disc type is a complex homogenous Monge–Ampère equation. The partial regularity theory established by Chen and Tian [C. R. Acad. Sci. Paris, Ser. I 340 (5) (2005)] amounts to an improvement of the regularity of the known $C^{1\text{,}1}$ solution to the geodesic of disc type to almost everywhere smooth. For such an almost smooth solution, we prove that the K-energy functional is sub-harmonic along such a solution. We use this to prove the uniqueness of extremal Kähler metrics and to establish a lower bound for the modified K-energy if the underlying Kähler class admits an extremal Kähler metric. To cite this article: X.X. Chen, G. Tian, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Ricci-flat Kähler metrics on crepant resolutions of Kähler cones

We prove that a crepant resolution π : Y → X of a Ricci-flat Kähler cone X admits a complete Ricci-flat Kähler metric asymptotic to the cone metric in every Kähler class in ${H^2_c(Y,\mathbb{R})}We prove that a crepant resolution π : Y → X of a Ricci-flat K?hler cone X admits a complete Ricci-flat K?hler metric asymptotic to the cone metric in every K?hler class in H²_c(Y,\mathbbR){H^2_c(Y,\mathbb{R})}. A K?hler cone (X,[`(g)]){(X,\bar{g})} is a metric cone over a Sasaki manifold (S, g), i.e. ${X=C(S):=S\times\mathbb{R}_{ >0 }}${X=C(S):=S\times\mathbb{R}_{ >0 }} with [`(g)]=dr² +r² g{\bar{g}=dr^2 +r^2 g}, and (X,[`(g)]){(X,\bar{g})} is Ricci-flat precisely when (S, g) Einstein of positive scalar curvature. This result contains as a subset the existence of ALE Ricci-flat K?hler metrics on crepant resolutions p:Y? X=\mathbbC<sup>n</sup> /G{\pi:Y\rightarrow X=\mathbb{C}^n /\Gamma}, with G ì SL(n,\mathbbC){\Gamma\subset SL(n,\mathbb{C})}, due to P. Kronheimer (n = 2) and D. Joyce (n > 2). We then consider the case when X = C(S) is toric. It is a result of A. Futaki, H. Ono, and G. Wang that any Gorenstein toric K?hler cone admits a Ricci-flat K?hler cone metric. It follows that if a toric K?hler cone X = C(S) admits a crepant resolution π : Y → X, then Y admits a T <sup>n</sup> -invariant Ricci-flat K?hler metric asymptotic to the cone metric (X,[`(g)]){(X,\bar{g})} in every K?hler class in H²_c(Y,\mathbbR){H^2_c(Y,\mathbb{R})}. A crepant resolution, in this context, is a simplicial fan refining the convex polyhedral cone defining X. We then list some examples which are easy to construct using toric geometry.     

K?hler–Einstein metrics on strictly pseudoconvex domains

Extending the results of Cheng and Yau it is shown that a strictly pseudoconvex domain ${M\subset X}$ in a complex manifold carries a complete K?hler–Einstein metric if and only if its canonical bundle is positive, i.e. admits an Hermitian connection with positive curvature. We consider the restricted case in which the CR structure on ${\partial M}$ is normal. In this case M must be a domain in a resolution of the Sasaki cone over ${\partial M}$ . We give a condition on a normal CR manifold which it cannot satisfy if it is a CR infinity of a K?hler–Einstein manifold. We are able to mostly determine those normal CR three-manifolds which can be CR infinities. We give many examples of K?hler–Einstein strictly pseudoconvex manifolds on bundles and resolutions. In particular, the tubular neighborhood of the zero section of every negative holomorphic vector bundle on a compact complex manifold whose total space satisfies c ₁?<?0 admits a complete K?hler–Einstein metric.     

The Calabi metric for the space of Kähler metrics

Given any closed Kähler manifold we define, following an idea by Calabi (Bull. Am. Math. Soc. 60:167–168, 1954), a Riemannian metric on the space of Kähler metrics regarded as an infinite dimensional manifold. We prove several geometrical features of the resulting space, some of which we think were already known to Calabi. In particular, the space is a portion of an infinite dimensional sphere and explicit unique smooth solutions for the Cauchy and the Dirichlet problems for the geodesic equation are given.     

Projectively induced rotation invariant Kähler metrics

We classify Kähler–Einstein manifolds admitting a Kähler immersion into a finite dimensional complex projective space endowed with the Fubini–Study metric, whose codimension is less than or equal to 3 and whose metric is rotation invariant.     

Some new domains with complete Kähler metrics of negative curvature

We add to the known examples of complete Kähler manifolds with negative sectional curvature by showing that the following three classes of domains in euclidean spaces also belong: perturbations of ellipsoidal domains in ?ⁿ, intersections of complex-ellipsoidal domains in ?², and intersections of fractional linear transforms of the unit ball in ?². In the process, we prove the following theorem in differential geometry: in the intersection of two complex-ellipsoidal domains in ?², the sum of the Bergman metrics is a Kähler metric with negative curvature operator.     

Almost Kähler metrics of negative scalar curvature on symplectic manifolds

We show that every symplectic manifold of dimension ≥ 4 admits a complete compatible almost Kähler metric of negative scalar curvature. And we discuss the C ⁰-closure of the set of almost Kähler metrics of negative scalar curvature. Some local versions are also proved.     

On asymptotics of complete Ricci-flat Kähler metrics on open manifolds

Tian and Yau constructed in J. Am. Math. Soc., 3(3):579–609, 1990, a complete Ricci-flat Kähler metric on the complement of an ample and smooth anticanonical divisor. For the explicitly constructed referential metric ω of Tian and Yau (J. Am. Math. Soc., 3(3):579–609, 1990) we prove a property that ${\|\partial\overline\partial u\|_\omega}$ has the same decay rate as Δ _ω u provided u satisfies some decay conditions on higher Laplacians. As an application we describe the behaviour of this metric towards the boundary divisor and prove the best possible decay rate of the difference to ω.     

Extremal Kähler metrics and complex deformation theory

Let (M, J, g) be a compact Kähler manifold of constant scalar curvature. Then the Kähler class [] has an open neighborhood inH ^1,1 (M, ) consisting of classes which are represented by Kähler forms of extremal Kähler metrics; a class in this neighborhood is represented by the Kähler form of a metric of constant scalar curvature iff the Futaki invariant of the class vanishes. If, moreover, the derivative of the Futaki invariant at [] is nondegenerate, every small deformation of the complex manifold (M, J) also carries Kähler metrics of constant scalar curvature. We then apply these results to prove new existence theorems for extremal Kähler metrics on certain compact complex surfaces.The first author is supported in part by NSF grant DMS 92-04093.     

Some remarks on the Kähler geometry of the Taub-NUT metrics

In this article, we investigate the balanced condition and the existence of an Engliš expansion for the Taub-NUT metrics on \mathbbC²{\mathbb{C}^2} . Our first result shows that a Taub-NUT metric on \mathbbC²{\mathbb{C}^2} is never balanced unless it is the flat metric. The second one shows that an Engliš expansion of the Rawnsley’s function associated to a Taub-NUT metric always exists, while the coefficient a ₃ of the expansion vanishes if and only if the Taub-NUT metric is indeed the flat one.     

Constructions of Kähler–Einstein metrics with negative scalar curvature

We show that on Kähler manifolds with negative first Chern class, the sequence of algebraic metrics introduced by Tsuji converges uniformly to the Kähler–Einstein metric. For algebraic surfaces of general type and orbifolds with isolated singularities, we prove a convergence result for a modified version of Tsuji’s iterative construction.