Strongly not relatives Kähler manifolds

Zedda, Michela

doi:10.1515/coma-2017-0001

Cited by 8 publications

(6 citation statements)

References 18 publications

(35 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Dealing with the symmetric case, observe that since a Hermitian symmetric space of compact type with integral Kähler form admits a Kähler immersion into some finite dimensional complex projective space, then due to theorems 6.1.1 and This result follows from Theorem 6.2.1 and the following general lemma, which proves that when the Kähler manifold considered is regular, in the sense that it is projectively induced when rescaled by a great enough constant, the property of not being relative to any projective Kähler manifold is invariant by the multiplication of the metric by a positive constant. Lemma 6.2.3 (M. Zedda [82]). Assume that (M, βg) is infinite projectively induced for any β > β 0 ≥ 0.…”

Section: Homogeneous Kähler Manifolds Are Not Relative To Projective mentioning

confidence: 99%

See 1 more Smart Citation

Kähler Immersions of Kähler Manifolds into Complex Space Forms

Loi

Zedda

2018

Lecture Notes of the Unione Matematica Italiana

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Homogeneous Kähler Manifolds Are Not Relative To Projective mentioning

confidence: 99%

“…Cartan-Hartogs domains has been considered by many authors (see e.g. [30,31,50,51,53,76,77,79,80,81,82]) under different points of view. Their importance relies on being examples of nonhomogeneous domains which for a particular value of the parameter µ are Kähler-Einstein.…”

Section: Cartan-hartogs Domainsmentioning

confidence: 99%

Kähler Immersions of Kähler Manifolds into Complex Space Forms

Loi

Zedda

2018

Lecture Notes of the Unione Matematica Italiana

Self Cite

View full text Add to dashboard Cite

show abstract

“…Lemma 3.7. [19] Let (M, g) be a Kähler manifold. If (M, ag) is full Kähler immersion submanifold of CP n for any a > a 0 > 0 and if (M, g) and CP n are not relatives for any n < +∞, then (M, g) and CP n are not strongly relatives for any n < +∞.…”

Section: Almost Kähler-einstein Metricmentioning

confidence: 99%

“…Loi and Mossa [12] showed that a bounded homogeneous domains with a homogeneous Kähler metric and any projective Kähler manifold are not relatives in 2015. Zedda [19] gave a sufficient condition for a Kähler manifold are strongly not relative to any projective Kähler manifold in 2017. As an application, they got that the Bergman-Hartogs domain and Fock-Bargmann-Hartogs domain are strongly not relative to any projective Kähler manifold.…”

Section: Introductionmentioning

confidence: 99%

Non-relativity of Kähler manifold and complex space forms

Cheng¹,

Hao²

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Inspired by the work of Umehara, Di Scala and Loi [10] introduced the concept of "relatives" between two Kähler manifolds (i.e., they are said to be relatives if they share a common Kähler submanifold, otherwise, we say that they are not relatives) in 2010, and they proved that a bounded domain with its Bergman metric and a projective Kähler manifold with the restriction of the Fubini-Study metric are not relatives. For related problems, see Cheng, Di Scala and Yuan [6], Di Scala and Loi [9], Mossa [17] and Zedda [23].…”

Section: Introductionmentioning

confidence: 99%