Here the existence of a complete Ricci-flat K¨ahler metric is encoded in terms of Sasakian geometry. At the basis of this technique lies the fact that the metric cone C(S) =S×R+with ˜g:=dr2+r2gover a Sasaki manifold(S,g)– called “K¨ahler cone” in this context – is Ricci-flat and K¨ahler if and only ifSis Sasaki-Einstein with positive scalar curvature. Now the aim is to resolve the singularity at the apex in such a way that a Ricci-flat K¨ahler metric asymptotic to the cone metric can be obtained on the resolution. Necessarily, the resolutionπ:Y −→C(S)has to be crepant, i.e.π∗KC(S)=KY. For this class, van Coevering found a beautiful result.
Theorem 4.1 ([C10]) Letπ:Y −→C(S)be a crepant resolution of the Ricci-flat K¨ahler cone C(S). Then, for each K ¨ahler class α∈Hc2(Y,R)there is a complete Ricci-flat K¨ahler metric g with[g] =α and g is asymptotic to the cone metric for any order of derivatives.
In particular, the volume growth of geodesic balls is of orderr2n. Many examples for this Theorem have been constructed before: Joyce [J01] and Kronheimer [K89]
proved the existence of an almost locally euclidean Ricci-flat K¨ahler metric in every K¨ahler class for crepant resolutions ofCn\ {0}/Γ withΓ⊂SL(n,C)a finite group.
The conditions of Theorem 4.1 need some discussion. The existence of a crepant resolution in dimension 3 implies that the singularity ofX is Gorenstein. For toric varieties the situation clarifies further: Futaki, Ono and Wang [FOW06] proved that a toric Gorenstein metric cone over a Sasaki manifold admits a Ricci-flat K¨ahler cone metric. In dimension 3, any toric Gorenstein singularity allows for a crepant resolution, so that we obtain
Example 3LetX be a 3-dimensional toric Gorenstein K¨ahler cone andπ:Y−→X a crepant resolution. In every class ofHc2(Y,R)there is a complete, Ricci-flatTn -invariant K¨ahler metric onXasymptotic to the cone metric.
There are cases of crepant resolutions not covered by Theorem 4.1.
Example 4Whenever a crepant resolution is small, i.e. the exceptional locus has codimension>1, then there cannot be a K¨ahler class inHc2(Y,R). For instance, the total space of the line bundleY :=O(−1)⊕O(−1)−→CP1can be obtained by a small resolution fromX={z21+z22+z23+z24=0} ⊂C4, the cone overS2×S3. The latter allows for a Sasaki-Einstein metric andY is known to support a complete, Ricci-flat K¨ahler metric [CO90].
This example is also instructive from the viewpoint of open manifolds.Y is an open manifold inY:=P(O(−1)⊕O(−1)⊕O)−→P1withD=O(1). In particu-lar,−KY is ample and not a multiple ofD.
The examples show that as well for open manifolds as for crepant resolutions a classification of the Ricci-flat case is still missing. Apart from the ambitious goal to achieve such a classification, one may also ask for a unifying definition of a class of non-compact manifolds allowing for a classification as systematic as the one for compact manifolds. But there are still too many questions unanswered to raise such an issue.
References
[A76] Aubin, T.,Equations du type Monge-Amp`ere sur les variet´es k¨ahl´eriennes compactes, C.R.A.S. Paris283A, (1976) 119
[BK88] Bando, S., Kobayashi, R.,Ricci-flat K¨ahler metrics on affine algebraic manifolds, Ge-ometry and analysis on manifolds (Katata/Kyoto, 1987), 20–31, Lecture Notes in Math., 1339, Springer, Berlin, 1988
[BK90] Bando, S., Kobayashi, R.,Ricci-flat K¨ahler metrics on affine algebraic manifolds. II, Math. Ann.287(1990), no. 1, 175–180
[C54] Calabi, E.,The space of K¨ahler metrics, Proc. Int. Cong. Math. Amsterdem2(1954) 206–
207
[CO90] Candelas, P., de la Ossa, X.C.,Comments on conifolds, Nucl. Phys. B342(1) (1990) 246–
268
[CY80] Cheng, S.-Y., Yau, S.-T.,On the existence of a complete K ¨ahler metric on non-compact complex manifolds and the regularity of Fefferman’s equation, Commun. Pure Appl. Math.
33, 507–544 (1980)
[C09] van Coevering, C.,A construction of complete Ricci-flat K¨ahler manifolds, preprint, arXiv:math.DG/0803.0112
[C10] van Coevering, C.,Ricci-flat K¨ahler metrics on crepant resolutions of K ¨ahler cones, Math.
Ann., DOI: 10.1007/s00208-009-0446-1, published online 18 November 2009
[F83] Futaki, A.,An obstruction to the existence of Einstein-K¨ahler metrics, Invent. Math.73 (1983) 437–443
[FOW06] Futaki, A., Ono, H., Wang, G.,Transverse K¨ahler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds, preprint, arXiv:math.DG/0607586
[J01] Joyce, D.,Asymptotically locally euclidean metrics with holonomy SU(m), Ann. Glob.
Anal. Geom.19(1) (2001) 55–73
[K84] Kobayashi, R.,K¨ahler-Einstein metric on an open algebraic manifold, Osaka J. Math.21 (1984) 399–418
[KK06] Koehler, B., K¨uhnel, M.,On invariance and Ricci-flatness of Hermitian metrics on open manifolds, Comment. Math. Helv.81(3), 543–567 (2006)
[KK10] Koehler, B., K¨uhnel, M.,On asymptotics of complete Ricci-flat K¨ahler metrics on open manifolds, to appear in Manuscripta Mathematica
[K89] Kronheimer, P.B.,The construction of ALE spaces as hyper-K¨ahler quotients, J. Diff.
Geom.29(3) (1989) 665-683
[K10] K¨uhnel, M.,Ricci-flat Complex Geometry and Its Applications, Habilitationsschrift, Otto-von-Guericke Universit¨at Magdeburg, 2010
[M57] Matsushima, Y.,Sur la structure du groupe d’hom´eomorpismes analytiques d’une certaine variet´e k¨ahl´erienne, Nagoya Math. J.11(1957) 145–150
[MY83] Mok, N., Yau, S.-T.,Completeness of the K ¨ahler-Einstein metric on bounded domains and the characterization of domains of holomorphy by curvature conditions, The mathematical heritage of Henri Poincar´e, Part 1 (Bloomington, Ind., 1980), 41–59, Proc. Sympos. Pure Math., 39, Amer. Math. Soc., Providence, RI, 1983
[M41] Myers, S. B.,Riemannian manifolds with positive mean curvature, Duke Math. J.8No.2 (1941) 401?404
[S08] Santoro, B.,On the asymptotic expansion of complete ricciflat k¨ahler metrics on quasi-projective manifolds, J. Reine Angew. Math. 615, 59-91 (2008)
[Sch98] Schumacher, G.,Asymptotics of K¨ahler-Einstein metrics on quasi-projective manifolds and an extension theorem on holomorphic maps, Math. Ann. 311, No.4, 631-645 (1998) [TY86] Tian, G., Yau, S.-T.,Existence of K¨ahler-Einstein metrics on complete K¨ahler manifolds
and their applications to algebrqic geometry, In: Mathematical Aspects of String Theory (San Diego, 1986), Adv. Ser. Math. Phys., 1, World Sci. Publ. 1987, 574–628
[TY90] Tian, G., Yau, S.-T.,Complete K¨ahler manifolds with zero Ricci curvature. I; J. Am. Math.
Soc. 3, No.3, 579–609 (1990)
[TY91] Tian, G., Yau, S.-T.,Complete K¨ahler manifolds with zero Ricci curvature. II; Invent.
Math. 106, No.1, 27–60 (1991)
[W05] Wu, D.,Higher Canonical Asymptotics of K¨ahler-Einstein Metrics on Quasi-Projective Manifolds, doctoral thesis, MIT, 2005
[Y78] Yau, S.-T.,On the Ricci curvature of a compact K¨ahler manifold and the complex Monge-Amp`ere equation. IComm. Pure Appl. Math. 31, No. 3, 339–411 (1978)
[Y78b] Yau, S.-T., A general Schwarz lemma for K¨ahler manifolds, Amer. J. Math100No. 1 (1978) 197–203
[Y93] Yau, S.-T.,A splitting theorem and an algebraic geometric characterization of locally sym-metric Hermitian spaces, Comm. Anal. Geom.1(3-4) (1993) 473–486
[Yu08] Yu, C.,Calabi-Yau metrics on compact K¨ahler manifolds with some divisors deleted, Front.
Math. China3(4) (2008) 589–598