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