共查询到20条相似文献,搜索用时 15 毫秒
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Craig van Coevering 《Mathematische Annalen》2010,347(3):581-611
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
H2c(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=\mathbbCn /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
n
-invariant Ricci-flat K?hler metric asymptotic to the cone metric (X,[`(g)]){(X,\bar{g})} in every K?hler class in
H2c(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. 相似文献
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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 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). 相似文献
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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. 相似文献
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Filippo Salis 《Archiv der Mathematik》2017,109(3):285-292
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. 相似文献
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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. 相似文献
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Jean Varouchas 《Mathematische Annalen》1989,283(1):13-52
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Daniele Grandini 《Annals of Global Analysis and Geometry》2007,32(3):225-252
The main purpose of the following article is to introduce a Lie theoretical approach to the problem of classifying pseudo quaternionic-Kähler (QK) reductions of the pseudo QK symmetric spaces, otherwise called generalized Wolf spaces. 相似文献
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Julien Keller Christina W. Tønnesen-Friedman 《Central European Journal of Mathematics》2012,10(5):1673-1687
We provide nontrivial examples of solutions to the system of coupled equations introduced by M. García-Fernández for the uniformization problem of a triple (M; L; E), where E is a holomorphic vector bundle over a polarized complex manifold (M, L), generalizing the notions of both constant scalar curvature Kähler metric and Hermitian-Einstein metric. 相似文献
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Mancho Manev 《Journal of Geometry》2012,103(3):519-530
Almost hypercomplex manifolds with Hermitian and Norden metrics and more specially the corresponding quaternionic Kähler manifolds are considered. Some necessary and sufficient conditions for the investigated manifolds to be isotropic hyper-Kählerian and flat are found. It is proved that the quaternionic Kähler manifolds with the considered metric structure are Einstein for dimension at least 8. The class of the non-hyper-Kähler quaternionic Kähler manifolds of the considered type is determined. 相似文献
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Marco Brunella 《Mathematische Annalen》2010,346(3):629-639
We show that every Enoki surface, i.e. a non-Kählerian compactification of an affine line bundle over an elliptic curve, admits a locally conformally Kähler metric. 相似文献
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We show that if a Fano manifold M is K-stable with respect to special degenerations equivariant under a compact group of automorphisms, then M admits a Kähler–Einstein metric. This is a strengthening of the solution of the Yau–Tian–Donaldson conjecture for Fano manifolds by Chen–Donaldson–Sun (Int Math Res Not (8):2119–2125, 2014), and can be used to obtain new examples of Kähler–Einstein manifolds. We also give analogous results for twisted Kähler–Einstein metrics and Kahler–Ricci solitons. 相似文献
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Haozhao Li 《Annals of Global Analysis and Geometry》2012,41(4):423-445
In this article, we prove the equivalence of the existence of extremal Kähler metrics and the properness of the modified K-energy on projective bundles. Moreover, we discuss the relations of the lower boundedness of the K-energy, the infimum of the Calabi energy and the extremal polynomials. In particular, the author gives an example where the modified K-energy is bounded from below but not proper. 相似文献
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Yujen Shu 《Geometriae Dedicata》2009,138(1):151-172
In this article, I prove the following statement: Every compact complex surface with even first Betti number is deformation equivalent to one which admits an extremal Kähler metric. In fact, this extremal Kähler metric can even be taken to have constant scalar curvature in all but two cases: the deformation equivalence classes of the blow-up of \({\mathbb {P}_2}\) at one or two points. The explicit construction of compact complex surfaces with constant scalar curvature Kähler metrics in different deformation equivalence classes is given. The main tool repeatedly applied here is the gluing theorem of C. Arezzo and F. Pacard which states that the blow-up/resolution of a compact manifold/orbifold of discrete type, which admits cscK metrics, still admits cscK metrics. 相似文献