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

Loi, Andrea; Zedda, Michela; Zuddas, Fabio

doi:10.1007/s10455-011-9297-6

Cited by 20 publications

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References 41 publications

(70 reference statements)

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“…) and u and v are implicitly defined by |z 1 | = e m(u 2 +v 2 ) u, |z 2 | = e m(v 2 −u 2 ) v (notice that for m = 0 one gets the flat metric on C 2 ). Then one can prove [30] that for m > 1 2 there does not exist a Kähler immersion of (C 2 , ω m ) into CP ∞ .…”

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“…In this regard Z. Lu and G. Tian [32] (see also [16] and [4] for the symmetric and homogenous case respectively) prove that the PDEs a j = f (j ≥ 2 and f a smooth function on M ) are elliptic and that if the logterm of the Bergman and Szegö kernel of the unit disk bundle over M vanishes then a k = 0, for k > n (n being the complex dimension of M ). The study of these PDEs makes sense regardless of the existence of a TYZ expansion and so given any Kähler manifold (M, g) it makes sense to call the a j 's the coefficients associated to metric g. In the noncompact case in [30] one can find a characterization of the flat metric as a Taub-Nut metric with a 3 = 0 while Feng and Tu [17] solve a conjecture formulated in [41] by showing that the complex hyperbolic space is the only Cartan-Hartogs domain where the coefficient a 2 is constant. In a recent paper [28] the first author together with M. Zedda prove that a locally hermitian symmetric space with vanishing a 1 and a 2 is flat.…”

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Two conjectures on Ricci-flat Kähler metrics

2018

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We propose two conjectures about Ricci-flat Kähler metrics:Conjecture 1: A Ricci-flat projectively induced metric is flat.Conjecture 2: A Ricci-flat metric on an n-dimensional complex manifold such that the a n+1 coefficient of the TYZ expansion vanishes is flat.We verify Conjecture 1 (see Theorem 1.1) under the assumptions that the metric is radial and stable-projectively induced and Conjecture 2 (see Theorem 1.2) for complex surfaces whose metric is either radial or complete and ALE. We end the paper by showing, by means of the Simanca metric, that the assumption of Ricci-flatness in Conjecture 1 and in Theorem 1.2 cannot be weakened to scalar-flatness (see Theorem 1.3).2010 Mathematics Subject Classification. 53C55; 58C25; 58F06.

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Two conjectures on Ricci-flat Kähler metrics

2018

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“…. The interested reader is referred to [5,15,23,24,28,32,52,53,55] for more details on these metrics.…”

Section: Chapter 3 Homogeneous Kähler Manifoldsmentioning

confidence: 99%

Kähler Immersions of Kähler Manifolds into Complex Space Forms

Loi

Zedda

2018

Lecture Notes of the Unione Matematica Italiana

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The study of Kähler immersions of a given real analytic Kähler manifold into a finite or infinite dimensional complex space form originates from the pioneering work of Eugenio Calabi [10]. With a stroke of genius Calabi defines a powerful tool, a special (local) potential called diastasis function, which allows him to obtain necessary and sufficient conditions for a neighbourhood of a point to be locally Kähler immersed into a finite or infinite dimensional complex space form. As application of its criterion, he also provides a classification of (finite dimensional) complex space forms admitting a Kähler immersion into another. Although, a complete classification of Kähler manifolds admitting a Kähler immersion into complex space forms is not known, not even when the Kähler manifolds involved are of great interest, e.g. when they are Kähler-Einstein or homogeneous spaces.In fact, the diastasis function is not always explicitely given and Calabi's criterion, although theoretically impeccable, most of the time is of difficult application.Nevertheless, throughout the last 60 years many mathematicians have worked on the subject and many interesting results have been obtained.The aim of this book is to describe Calabi's original work, to provide a detailed account of what is known today on the subject and to point out some open problems.Each chapter begins with a brief summary of the topics discussed and ends with a list of exercises which help the reader to test his understanding.Apart from the topics discussed in Section 3.1 of Chapter 3, which could be skipped without compromising the understanding of the rest of the book, the requirements to read this book are a basic knowledge of complex and Kähler iii geometry (treated, e.g. in Moroianu's book [59]).The authors are grateful to Claudio Arezzo and Fabio Zuddas for a careful reading of the text and for valuable comments that have improved the book's exposure.

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“…[21], [2]). Furthermore, in the noncompact case, one can find in [19] a characterization of the flat metric as a Taub-NUT metric with a 3 = 0, while Z.…”

Section: Introductionmentioning

confidence: 99%