Principles of Locally Conformally Kähler Geometry

• Autore: Ornea Liviu
• Pubblicazione: Cham : , : Springer International Publishing AG, , 2024; ©2024
• Edizione: 1st ed.
• Descrizione fisica: 1 online resource (729 pages)
• Altri autori: VerbitskyMisha
• Nota di contenuto: Intro -- 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.
• Titolo autorizzato: Principles of Locally Conformally Kähler Geometry
• ISBN: 3-031-58120-2
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910855368703321
• Lo trovi qui: Univ. Federico II
• Serie: Progress in Mathematics Series