EXISTENCE AND COMPACTNESS THEORY FOR ALE SCALAR-FLAT KÄHLER SURFACES

Our main result in this article is a compactness result which states that a noncollapsed sequence of asymptotically locally Euclidean (ALE) scalar-flat Kähler metrics on a minimal Kähler surface whose Kähler classes stay in a compact subset of the interior of the Kähler cone must have a convergent subsequence. As an application, we prove the existence of global moduli spaces of scalar-flat Kähler ALE metrics for several infinite families of Kähler ALE spaces.

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PROJECTIVE STRUCTURES, FLAT BUNDLES, AND KÄHLER METRICS ON MODULI SPACES

Mathematics of the USSR-Sbornik ◽

10.1070/sm1988v061n01abeh003203 ◽

1988 ◽

Vol 61 (1) ◽

pp. 211-224 ◽

Cited By ~ 11

Author(s):

N V Ivanov

Keyword(s):

Moduli Spaces ◽

Projective Structures ◽

Kähler Metrics ◽

Kahler Metrics

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A SPLITTING THEOREM FOR KÄHLER MANIFOLDS WHOSE RICCI TENSORS HAVE CONSTANT EIGENVALUES

International Journal of Mathematics ◽

10.1142/s0129167x01001052 ◽

2001 ◽

Vol 12 (07) ◽

pp. 769-789 ◽

Cited By ~ 17

Author(s):

VESTISLAV APOSTOLOV ◽

TEDI DRĂGHICI ◽

ANDREI MOROIANU

Keyword(s):

Ricci Tensor ◽

Kähler Manifolds ◽

Kahler Manifolds ◽

Kähler Metrics ◽

Kahler Metrics ◽

Complex Dimension ◽

Negative Eigenvalues ◽

Kähler Surfaces ◽

Almost Kähler ◽

Compact Kähler Manifold

It is proved that a compact Kähler manifold whose Ricci tensor has two distinct constant non-negative eigenvalues is locally the product of two Kähler–Einstein manifolds. A stronger result is established for the case of Kähler surfaces. Without the compactness assumption, irreducible Kähler manifolds with Ricci tensor having two distinct constant eigenvalues are shown to exist in various situations: there are homogeneous examples of any complex dimension n ≥ 2 with one eigenvalue negative and the other one positive or zero; there are homogeneous examples of any complex dimension n ≥ 3 with two negative eigenvalues; there are non-homogeneous examples of complex dimension 2 with one of the eigenvalues zero. The problem of existence of Kähler metrics whose Ricci tensor has two distinct constant eigenvalues is related to the celebrated (still open) conjecture of Goldberg [24]. Consequently, the irreducible homogeneous examples with negative eigenvalues give rise to complete Einstein strictly almost Kähler metrics of any even real dimension greater than 4.

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Moser Vector Fields and Geometry of the Mabuchi Moduli Space of Kähler Metrics

Geometry ◽

10.1155/2014/968064 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6

Author(s):

Daniel Guan

Keyword(s):

Moduli Space ◽

Moduli Spaces ◽

Symmetric Spaces ◽

Vector Fields ◽

Constant Scalar Curvature ◽

Explicit Construction ◽

Kähler Metrics ◽

Cohomogeneity One ◽

Kahler Metrics ◽

Kähler Einstein Metrics

There is a natural Moser type transformation along any curve in the moduli spaces of Kähler metrics. In this paper we apply this transformation to give an explicit construction of the parallel transformation along a curve in the Mabuchi moduli space of Kähler metrics. This is crucial in the proof of the equivalence between the existence of the Kähler metrics with constant scalar curvature and the geodesic stability for the type II compact almost homogeneous manifolds of cohomogeneity one mentioned in (Guan 2013). We also explain a new description of the geodesics and prove a curvature property of the moduli space, called curvature symmetric, which makes it similar to some special symmetric spaces with nonpositive curvatures, although the spaces are usually not complete. Finally, we generalize our geodesic stability conjectures in (Guan 2003) and give several results on the Lie algebra structures related to the parallel transformations. In the last section, we generalize the Futaki obstruction of the Kähler-Einstein metrics to the parallel vector fields of the invariant Mabuchi moduli space. We call the related stability the parallel stability. This includes the toric and cohomogeneity one cases as well as the spherical manifolds.

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Geometry and topology of the space of Kähler metrics on singular varieties

Compositio Mathematica ◽

10.1112/s0010437x18007170 ◽

2018 ◽

Vol 154 (8) ◽

pp. 1593-1632 ◽

Cited By ~ 3

Author(s):

Eleonora Di Nezza ◽

Vincent Guedj

Keyword(s):

Einstein Metrics ◽

Normal Space ◽

Fano Varieties ◽

Kähler Metrics ◽

Kahler Metrics ◽

Metric Properties ◽

Singular Varieties ◽

Underlying Space ◽

And Topology ◽

Kähler Einstein Metrics

Let $Y$ be a compact Kähler normal space and let $\unicode[STIX]{x1D6FC}\in H_{\mathit{BC}}^{1,1}(Y)$ be a Kähler class. We study metric properties of the space ${\mathcal{H}}_{\unicode[STIX]{x1D6FC}}$ of Kähler metrics in $\unicode[STIX]{x1D6FC}$ using Mabuchi geodesics. We extend several results of Calabi, Chen, and Darvas, previously established when the underlying space is smooth. As an application, we analytically characterize the existence of Kähler–Einstein metrics on $\mathbb{Q}$-Fano varieties, generalizing a result of Tian, and illustrate these concepts in the case of toric varieties.

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