Principles of locally conformally Kähler geometry / Liviu Ornea, Misha Verbitsky.

Format:

Book

Author/Creator:

Ornea, Liviu, 1960- author.

Verbit︠s︡kiĭ, Misha, 1969- author.

Series:

Progress in mathematics (Boston, Mass.) ; 0743-1643 v. 354.

Progress in mathematics, 0743-1643 ; volume 354

Language:

English

Subjects (All):

Kählerian structures.

Physical Description:

xxi, 736 pages ; 25 cm

Place of Publication:

Cham, Switzerland : Birkhäuser, [2024]

Summary:

"This monograph introduces readers to locally conformally Kähler (LCK) geometry and provides an extensive overview of the most current results. A rapidly developing area in complex geometry dealing with non-Kähler manifolds, LCK geometry has strong links to many other areas of mathematics, including algebraic geometry, topology, and complex analysis. The authors emphasize these connections to create a unified and rigorous treatment of the subject suitable for both students and researchers. Part I builds the necessary foundations for those approaching LCK geometry for the first time with full, mostly self-contained proofs and also covers material often omitted from textbooks, such as contact and Sasakian geometry, orbifolds, Ehresmann connections, and foliation theory. More advanced topics are then treated in Part II, including non-Kähler elliptic surfaces, cohomology of holomorphic vector bundles on Hopf manifolds, Kuranishi and Teichmüller spaces for LCK manifolds with potential, and harmonic forms on Sasakian and Vaisman manifolds. Each chapter in Parts I and II begins with motivation and historic context for the topics explored and includes numerous exercises for further exploration of important topics. Part III surveys the current research on LCK geometry, describing advances on topics such as automorphism groups on LCK manifolds, twisted Hamiltonian actions and LCK reduction, Einstein-Weyl manifolds and the Futaki invariant, and LCK geometry on nilmanifolds and on solvmanifolds. New proofs of many results are given using the methods developed earlier in the text. The text then concludes with a chapter that gathers over 100 open problems, with context and remarks provided where possible, to inspire future research" -- Back cover.

Contents:

Introduction

Part I: Lectures in locally conformally Kähler geometry

Kähler manifolds

Connections in vector bundles and the Froebenius theorem

Locally conformally Kahler manifolds

Hodge theory on complex manifolds and Vaisman's theorem

Holomorphic vector bundles

CR, Contact and Sasakian manifolds

Vaisman manifolds

The structure of compact Vaisman manifolds

Orbifolds

Quasi-regular foliations

Regular and quasi-regular Vaisman manifolds

LCK manifolds with potential

Embedding LCK manifolds with potential in Hopf manifolds

Logarithms and algebraic cones

Pseudoconvex shells and LCK metrics on Hopf manifolds

Embedding theorem for Vaisman manifolds

Non-linear Hopf manifolds

Morse-Novikov and Bott-Chern cohomology of LCK manifolds

Existence of positive potentials

Holomorphic S^1 actions on LCK manifolds

Sasakian submanifolds in algebraic cones

Oeljeklaus-Toma manifolds

Appendices

Part II: Advanced LCK geometry

Non-Kähler elliptic surfaces

Kodaira classification for non-Kähler complex surfaces

Cohomology of holomorphic bundles on Hopf manifolds

Mall bundles and flat connections on Hopf manifolds

Kuranishi and Teichmüller spaces for LCK manifolds with potential

The set of Lee classes on LCK manifolds with potential

Harmonic forms on Sasakian and Vaisman manifolds

Dolbeault cohomology of LCK manifolds with potential

Calabi-Yau theorem for Vaisman manifolds

Holomorphic tensor fields on LCK manifolds with potential

Part III: Topics in locally conformally Kähler geometry

Twisted Hamiltonian actions and LCK reduction

Elliptic curves on Vaisman manifolds

Submersions and bimeromorphic maps of LCK manifolds

Bott-Chern cohomology of LCK manifolds with potential

Hopf surfaces in LCK manifolds with potential

Riemannian geometry of LCK manifolds

Einstein-Weyl manifolds and the Futaki invariant

LCK structures on homogeneous manifolds

LCK structures on nilmanifolds and solvmanifolds

Explicit LCK metrics on Inoue surfaces

More on Oeljeklaus-Toma manifolds

Locally conformally parallel and non-parallel structures

Open questions.

Notes:

Includes bibliographical references (pages 689-719) and indexes.

ISBN:

9783031581199

3031581199

OCLC:

1439006170