Principles of Locally Conformally Kähler Geometry

Contents
Introduction
I Lectures in locally conformally Kähler geometry
Kähler manifolds
Complex manifolds
Holomorphic vector fields
Hermitian manifolds
Kähler manifolds
Examples of Kähler manifolds
Menagerie of complex geometry
Exercises
Kähler geometry and holomorphic vector fields
The Lie algebra of holomorphic Hamiltonian Killing fields
Connections in vector bundles and the Frobenius theorem
Introduction
Connections in vector bundles
Curvature of a connection
Ehresmann connections
Ehresmann connection on smooth fibrations
Linear Ehresmann connections on vector bundles
Frobenius form and Frobenius theorem
Basic forms
The curvature of an Ehresmann connection
The Riemann–Hilbert correspondence
Flat bundles and parallel sections
Local systems
Exercises
Locally conformally Kähler manifolds
Introduction
Locally conformally symplectic manifolds
Galois covers and the deck transform group
Locally conformally Kähler manifolds
LCK manifolds: the tensorial definition
The weight bundle and the homothety character
Automorphic forms related to the homothety character
Kähler covers of LCK manifolds: the second definition
LCK manifolds via an L-valued Kähler form: the third definition
Conformally equivalent Kähler forms
LCK manifolds via charts and atlases: the fourth definition
The LCK rank
A first example
Notes
Exercises
Hodge theory on complex manifolds and Vaisman's theorem
Introduction
Preliminaries
Hodge decomposition on complex manifolds
Holomorphic one-forms and first cohomology
Positive (1,1)-forms
Vaisman's theorem
Exercises
Holomorphic vector bundles
Introduction
Holomorphic vector bundles
Holomorphic structure operator
The -operator on vector bundles
Connections and holomorphic structure operators
Curvature of holomorphic line bundles
Kähler potentials and plurisubharmonic functions
Chern connection obtained from an Ehresmann connection
Calabi formula (2.6).
Positive line bundles
Exercises
CR, Contact and Sasakian manifolds
Introduction
CR-manifolds
Contact manifolds and pseudoconvex CR-manifolds
Contact manifolds and symplectic cones
Levi form and pseudoconvexity
Normal varieties
Stein completions and Rossi-Andreotti–Siu theorem
Sasakian manifolds
Notes
CR-structures and CR-manifolds
Sasakian manifolds: the tensorial definition by Sh. Sasaki
Exercises
Vaisman manifolds
Introduction
Many definitions of Vaisman manifolds
Riemannian cones
Basics of Vaisman geometry
Holonomy and the de Rham splitting theorem
Conical Riemannian metrics
Vaisman manifolds: local properties
Vaisman metrics obtained from holomorphic automorphisms
The canonical foliation on compact Vaisman manifolds
Exercises
The structure of compact Vaisman manifolds
Introduction
The Vaisman metric expressed through the Lee form
Decomposition for harmonic 1-forms on Vaisman manifolds
Rank 1 Vaisman structures
The structure theorem
Exercises
Orbifolds
Introduction
Groupoids and orbispaces
Real orbifolds
Complex orbifolds
Quotients by tori
Principal orbifold bundles
Exercises
Quasi-regular foliations
Introduction
Quasi-regular foliations and holonomy
Circle bundles over Riemannian orbifolds
Quasi-regular Sasakian manifolds
Notes
Exercises
Regular and quasi-regular Vaisman manifolds
Introduction
Quasi-regular Vaisman manifolds as cone quotients
Regular Vaisman manifolds
Quasi-regular Vaisman manifolds are orbifold elliptic fibrations
Density of quasi-regular Vaisman manifolds
Immersion theorem for Vaisman manifolds
Notes
Exercises
LCK manifolds with potential
Introduction
Deformations of LCK structures
LCK manifolds with potential
LCK manifolds with potential, proper and improper
LCK manifolds with potential and preferred gauge
The monodromy of LCK manifolds with proper potential
ddc-potential
Deforming an LCK potential to a proper potential
Stein manifolds and normal families
Stein manifolds
Normal families of functions
The C0 - topology on spaces of functions
The C1 - topology on spaces of sections
Montel theorem for normal families
The Stein completion of the Kähler cover
Appendix 1: another construction of the Stein completion
Appendix 2: the proof of the Kodaira–Spencer stability theorem
Notes
Exercises
Embedding LCK manifolds with potential in Hopf manifolds
Introduction
Preliminaries on functional analysis
The Banach space of holomorphic functions
Compact operators
Holomorphic contractions
The Riesz–Schauder theorem
The embedding theorem
Density implies the embedding theorem
Density of *-finite functions on the minimal Kähler cover
Notes
Exercises
Logarithms and algebraic cones
Introduction
The logarithm of an automorphism
Logarithms of an automorphism of a Banach ring
Logarithms of the homothety of the cone
Algebraic structures on Stein completions
Ideals of the embedding to a Hopf manifold
Algebraic structures on Stein completions: the existence
Algebraic structures on Stein completions: the uniqueness
Algebraic cones
Algebraic cones defined in terms of C*-action
Algebraic cones and Hopf manifolds
Exercises
Pseudoconvex shells and LCK metrics on Hopf manifolds
Introduction
LCK metrics on Hopf manifolds
Affine cones of projective varieties
Pseudoconvex shells
Pseudoconvex shells in algebraic cones
All linear Hopf manifolds are LCK with potential
Pseudoconvex shells in algebraic cones
LCK manifolds admitting an S1-action
Existence of S1-action on an LCK manifold with potential
Quotients of algebraic cones are LCK
Holomorphic isometries of LCK manifolds with potential
Algebraic cones as total spaces of C*-bundles
Algebraic cones: an alternative definition
Closed algebraic cones and normal varieties
Exercises
Embedding theorem for Vaisman manifolds
Introduction
Embedding Vaisman manifolds to Hopf manifolds
Semisimple Hopf manifolds are Vaisman
Algebraic groups and Jordan–Chevalley decomposition
The algebraic cone of an LCK manifold with potential
Deforming an LCK manifold with proper potential to Vaisman manifolds
Exercises
Non-linear Hopf manifolds
Introduction
Hopf manifolds and holomorphic contractions
Holomorphic contractions on Stein varieties
Non-linear Hopf manifolds are LCK
Minimal Hopf embeddings
Poincaré–Dulac normal forms
Exercises
Morse–Novikov and Bott–Chern cohomology of LCK manifolds
Introduction
Preliminaries on differential operators
Differential operators
Elliptic complexes
Bott–Chern cohomology
Morse–Novikov cohomology
Morse–Novikov class of an LCK manifold
Twisted Dolbeault cohomology
Twisted Bott–Chern cohomology
Bott–Chern classes and Morse–Novikov cohomology
Exercises
Existence of positive potentials
Introduction
A counterexample to the positivity of the potential
A ddc-potential on a compact LCK manifold is positive somewhere
Stein manifolds with negative ddc-potential
Remmert theorem and 1-jets on Stein manifolds
Negative sets for ddc-potentials are Stein
Stein LCK manifolds admit a positive ddc-potential
Gluing the LCK forms
Regularized maximum of plurisubharmonic functions
Gluing of LCK potentials
Exercises
Holomorphic S1 actions on LCK manifolds
Introduction
S1-actions on compact LCK manifolds
The averaging procedure
Holomorphic homotheties on a Kähler manifold
Vanishing of the twisted Bott–Chern class on manifolds endowed with an S1-action
Exercises
Sasakian submanifolds in algebraic cones
Introduction
Sasakian structures on CR-manifolds
Isometric embeddings of Kähler and Vaisman manifolds
Embedding Sasakian manifolds in spheres
Kodaira-like embedding for Sasakian manifolds
Optimality of the embedding result
Notes
Exercises
Oeljeklaus–Toma manifolds
Introduction
Many species of Inoue surfaces
Class VII0 surfaces with b2=0
Oeljeklaus–Toma manifolds and LCK geometry
Subvarieties in the OT-manifolds
Number theory: local and global fields
Normed fields
Local fields
Valuations and extensions of global fields
Dirichlet's unit theorem
Oeljeklaus–Toma manifolds
The solvmanifold structure
The LCK metric
Non-existence of complex subvarieties in OT-manifolds
Non-existence of curves on OT-manifolds
Exercises
Idempotents in tensor products
OT-manifolds
Appendices
Appendix A. Gauduchon metrics
Appendix B. An explicit formula of the Weyl connection
II Advanced LCK geometry
Non-Kähler elliptic surfaces
Introduction
Gauss–Manin local systems and variations of Hodge structure
The Gauss–Manin connection
Variations of Hodge structures
Gromov's compactness theorem
Barlet spaces
Elliptic fibrations with multiple fibres
Multiple fibres of elliptic fibrations and the relative Albanese map
Structure of a neighbourhood of a multiple fiber
Non-Kähler elliptic surfaces
Structure of elliptic fibrations on non-Kähler surfaces
Isotrivial elliptic fibrations
The Blanchard theorem
Exercises
Group structure on a curve of genus 1
Elliptic fibrations
Kodaira classification for non-Kähler complex surfaces
Introduction
An overview of this chapter
The Buchdahl–Lamari theorem
Locally conformally Kähler surfaces
Cohomology of non-Kähler surfaces
Bott–Chern cohomology of a surface
First cohomology of non-Kähler surfaces
Second cohomology of non-Kähler surfaces
Vanishing in multiplication of holomorphic 1-forms
Structure of multiplication in de Rham cohomology of non-Kähler surfaces without curves
Elliptic fibrations on non-Kähler surfaces
Class VII surfaces
The Riemann–Roch formula for embedded curves
(-1)-curves
Non-Kähler surfaces are either class VII or elliptic
Brunella's theorem: all Kato surfaces are LCK
The embedding theorem in complex dimension 2
Inoue surfaces
Exercises
Riemann–Roch formula for a curve
Complex surfaces
Cohomology of holomorphic bundles on Hopf manifolds
Introduction
Derived functors and the Grothendieck spectral sequence
Equivariant sheaves, local systems and cohomology
Equivariant sheaves and equivariant objects
Group cohomology, local systems and Ext groups
Group cohomology of Z
Directed sheaves and cohomology of Cn0
Directed sheaves: definition and examples
Serre duality with compact supports
Cohomology of Cn0
Contractions define compact operators on holomorphic functions
Mall's theorem on cohomology of vector bundles
Exercises
Mall bundles and flat connections on Hopf manifolds
Introduction
Mall bundles and coherent sheaves
Flat affine structures and the development map
Coherent sheaves
Normal sheaves and reflexive sheaves
Extension of coherent sheaves on complex varieties
Dolbeault cohomology of Hopf manifolds
Degree of a line bundle
Computation of H0,p(H) for a Hopf manifold
Holomorphic differential forms on Hopf manifolds
Mall bundles on Hopf manifolds
Mall bundles: definition and examples
The Euler exact sequence and an example of a non-Mall bundle on a classical Hopf manifold
Resonance in Mall bundles
Resonant matrices
Resonant equivariant bundles
Holomorphic connections on vector bundles
The flat connection on a non-resonant Mall bundle
Flat connections on Hopf manifolds
Developing map for flat affine manifolds
Flat affine connections on a Hopf manifold
A new proof of Poincaré theorem about linearization of non-resonant contractions
Harmonic forms on Hopf manifolds with coefficients in a bundle
The Hodge * operator and cohomology of holomorphic bundles
Multiplication in cohomology of holomorphic bundles on Vaisman-type Hopf manifolds
Appendix: cohomology of local systems on S1 and the multiplication in cohomology of holomorphic bundles on Hopf manifolds
Exercises
Kuranishi and Teichmüller spaces for LCK manifolds
Introduction
Deformation spaces
Deformations of Hopf surfaces: a short survey
Teichmüller space of Hopf manifolds and applications to LCK geometry
The Kuranishi space
Nijenhuis–Schouten and Frölicher–Nijenhuis brackets
Kuranishi space: the definition
Kuranishi to Teichmüller map
The Kuranishi space for Hopf manifolds
Vanishing of H2(TH) for a Hopf manifold
Kuranishi space and linear vector fields
Kuranishi to Teichmüller map for Hopf manifolds
Hilbert schemes
The space of complex structures on LCK manifolds with potential
The conjugation orbit of a linear operator
Diffeomorphism orbits of LCK structures with potential have Vaisman limit points
The Teichmüller space of LCK manifolds with potential
Notes
Exercises
Hilbert polynomials
Deformation theory
The set of Lee classes on LCK manifolds with potential
Introduction
LCK metrics on Vaisman manifolds
Opposite Lee forms on LCK manifolds with potential
Hodge decomposition of H1(M) on LCK manifolds with potential
The set of Lee classes on Vaisman manifolds
The set of Lee classes on LCK manifolds with potential
Notes
Exercises
Harmonic forms on Sasakian and Vaisman manifolds
Introduction
Basic cohomology and taut foliations
Hattori spectral sequence and Hattori differentials
Supersymmetry and geometric structures on manifolds
Lie superalgebras acting on the de Rham algebra
Lie superalgebras and superderivations
Differential operators on graded commutative algebras
Supersymmetry on Kähler manifolds
Hattori differentials on Sasakian manifolds
Hattori spectral sequence and associated differentials
Hattori differentials on Sasakian manifolds
Transversally Kähler manifolds
Basic cohomology and Hodge theory on Sasakian manifolds
The cone of a morphism of complexes and cohomology of Sasakian manifolds
Harmonic form decomposition on Sasakian manifolds
Hodge theory on Vaisman manifolds
Basic cohomology of Vaisman manifolds
Harmonic forms on Vaisman manifolds
The supersymmetry algebra of a Sasakian manifold
Notes
Exercises
Dolbeault cohomology of LCK manifolds with potential
Introduction
Weights of a torus action on the de Rham algebra
Dolbeault cohomology on manifolds with a group action
Dolbeault cohomology of Vaisman manifolds
Basic and Dolbeault cohomologies of Vaisman manifolds
Harmonic decomposition for the Dolbeault cohomology
Dolbeault cohomology of LCK manifolds with potential
Exercises
Isometry groups
Aeppli and Dolbeault cohomologies of Vaisman manifolds
Aeppli cohomology and strongly Gauduchon metrics
Calabi–Yau theorem for Vaisman manifolds
Introduction
The Lee field on a compact Vaisman manifold
The complex Monge-Ampère equation
Exercises
Holomorphic tensor fields on LCK manifolds with potential
Introduction
Holomorphic tensors on LCK manifolds with potential
Zariski closures and the Chevalley theorem
Holomorphic tensors on Vaisman manifolds
Exercises
III Topics in locally conformally Kähler geometry
Automorphism groups of LCK manifolds
Infinitesimal automorphisms
Lifting a transformation group to a Kähler cover
Affine vector fields
Conformal vector fields on compact LCK manifolds
Holomorphic Lee field
Twisted Hamiltonian actions and LCK reduction
Twisted Hamiltonian actions
The LCK momentum map
LCK reduction at 0
Complex Lie group acting by holomorphic isometries
LCK manifolds admitting a torus action with an open orbit
Toric LCK manifolds
Elliptic curves on Vaisman manifolds
Counting elliptic curves
Application to Sasaki manifolds: closed Reeb orbits
Boothby–Wang theorem for Besse contact manifolds
Weinstein conjecture for Sasakian manifolds
Submersions and bimeromorphic maps of LCK manifolds
A topological criterion
Holomorphic submersions
LCK metrics on fibrations
LCK metrics on products
Blow-up at points
Blow-up along submanifolds
Weak LCK structures
Moishezon manifolds are not LCK
LCK currents and Fujiki LCK class
LCK manifolds in terms of currents
An analogue of Fujiki class C
Bott–Chern cohomology of LCK manifolds with potential
Bott–Chern versus Dolbeault cohomology
Generic vanishing of Bott–Chern cohomology
Hopf surfaces in LCK manifolds with potential
Diagonal and non-diagonal Hopf surfaces
Complex curves in non-diagonal Hopf surfaces
Gauduchon metrics on LCK manifolds with potential
Complex surfaces of Kähler rank 1
Surfaces in compact LCK manifolds with potential
Algebraic groups
Orbits of algebraic groups in Hopf manifolds
Hopf surfaces in LCK manifolds with potential
The pluricanonical condition
Riemannian geometry of LCK manifolds
Existence of parallel vector fields
LCK metrics are not Einstein
The Vaisman condition in terms of the Bismut connection
Notes
Bismut connections
Curvature properties
Harmonic maps and distributions
Einstein–Weyl manifolds and the Futaki invariant
The Einstein–Weyl condition
The Futaki invariant of Hermitian manifolds
The Futaki invariant on LCK manifolds
LCK structures on homogeneous manifolds
Introduction
Homogeneous LCK manifolds
Homogeneous Vaisman manifolds
Notes
LCK structures on nilmanifolds and solvmanifolds
Invariant geometric structures on Lie groups
Twisted Dolbeault cohomology on nilpotent Lie algebras
LCK nilmanifolds
LCK solvmanifolds
Explicit LCK metrics on Inoue surfaces
Inoue surfaces of class S0
Inoue surfaces of class S+
The solvable group Sol41
The structure of complex Lie groups
The group (Sol41, I0)
The group (Sol'41, I1)
Non-existence of LCK metrics on sol'41
Cocompact lattices in Sol41 and Sol'41
Equivalence with the Inoue's description of the surfaces of class S+.
The LCK metric on S+N,p,q,r,0
Non-existence of LCK metrics on S+N,p,q,r,t, t=0
Inoue surfaces of class S-.
More on Oeljeklaus–Toma manifolds
Cohomology of OT-manifolds
LCK structures on general OT manifolds
Cohomology of LCK OT-manifolds
LCK rank of OT manifolds
Locally conformally parallel and non-parallel structures
Locally conformally hyperkähler structures
Locally conformally balanced structures
Locally conformally parallel G2, Spin(7) and Spin(9) structures
Notes
Open questions
Existence of LCK structures
LCK structures on complex manifolds
Existence of LCK potential and Vaisman structures
Complex geometry of LCK manifolds
Hodge theory on LCK manifolds
Bimeromorphic geometry of LCK manifolds
Complex subvarieties in LCK manifolds
Sasakian and Vaisman manifolds
Vaisman manifolds
Sasakian manifolds
LCK manifolds with potential
Extremal metrics on LCK manifolds
LCHK and holomorphic symplectic structures
LCHK structures
Locally conformally holomorphic symplectic structures
Holomorphic Poisson structures
Riemannian geometry of LCK manifolds
Curvature of Vaisman manifolds
Special Hermitian metrics on LCK manifolds
LCK reduction and LCS geometry
The Lee cone of taming LCS structures
Twisted Hamiltonian action, LCS reduction and toric Vaisman geometry
Hopf manifolds
Foliations on LCK manifolds
Logarithmic foliations on LCK manifolds with potential
Flat affine structures on LCK manifolds
Bibliography
Subject Index
Name Index