A report on locally conformally Kaehler manifolds

Ornea L.; M. Verbitsky

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A report on locally conformally Kaehler manifolds

Ch. 10. P. 135–151.

Ornea L., Verbitsky M.

We present an overview of recent results in locally conformally K¨ahler geometry, with focus on the topological properties which obstruct the existence of such structures on compact manifolds.

Language: English

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Keywords: locally conformally Kaehler manifolds локально конформно кэлерово многообразие

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Embedding of LCK manifolds with potential into Hopf manifolds using Riesz-Schauder theorem

Ornea L., Verbitsky M., / Series arXiv "math". 2017.

An locally conformally Kahler (LCK) manifold with potential is a complex manifold with a cover which admits an automorphic Kahler potential. An LCK manifold with potential can be embedded to a Hopf manifold, if its dimension is at least 3. We give a functional-analytic proof of this result based on Riesz-Schauder theorem and Montel theorem. ...

Added: February 6, 2017

LCK rank of locally conformally Kähler manifolds with potential

Ornea L., Verbitsky M., Journal of Geometry and Physics 2016 Vol. 107 P. 92–98

An LCK manifold with potential is a compact quotient of a Kähler manifold X equipped with a positive Kähler potential f, such that the monodromy group acts on X by holomorphic homotheties and multiplies f by a character. The LCK rank is the rank of the image of this character, considered as a function from ...

Added: June 5, 2016

Locally conformally Kähler metrics obtained from pseudoconvex shells

Ornea L., Verbitsky M., Proceedings of the American Mathematical Society 2016 No. 144 P. 325–335

A locally conformally Kähler (LCK) manifold is a complex manifold M admitting a Kähler covering \tilde{M}, such that its monodromy acts on this covering by homotheties. A compact LCK manifold is called LCK with potential if its covering admits an automorphic Kähler potential. It is known that in this case \tilde{M} is an algebraic cone, ...

Added: January 28, 2016

Blow-ups of locally conformally Kahler manifolds

Ornea L., Verbitsky M., Vuletescu V., International Mathematics Research Notices 2013 No. 12 P. 2809–2821

A locally conformally Khler (LCK) manifold is a complex manifold which admits a covering endowed with a Kähler metric with respect to which the covering group acts through homotheties. We show that the blow-up of a compact LCK manifold along a complex submanifold admits an LCK structure if and only if this submanifold is globally ...

Added: October 10, 2013

Locally conformally Kahler manifolds admitting a holomorphic conformal flow

Ornea L., Verbitsky M., Mathematische Zeitschrift 2013 Vol. 273 No. 3-4 P. 605–611

A manifold M is locally conformally Kähler (LCK) if it admits a Kähler covering M˜ with monodromy acting by holomorphic homotheties. Let M be an LCK manifold admitting a holomorphic conformal flow of diffeomorphisms, lifted to a non-isometric homothetic flow on M˜ . We show that M admits an automorphic potential, and the monodromy group ...

Added: March 19, 2013