Volume 11, Issue 1

We study geodesics and magnetic trajectories in the model space F 4 {{\rm{F}}}^{4} . The space F 4 {{\rm{F}}}^{4} is isometric to the 4-dim simply connected Riemannian 3-symmetric space due to Kowalski. We describe the solvable Lie group model of F 4 {{\rm{F}}}^{4} and investigate its curvature properties. We introduce the symplectic pair of two Kähler forms on F 4 {{\rm{F}}}^{4} . Those symplectic forms induce invariant Kähler structure and invariant strictly almost Kähler structure on F 4 {{\rm{F}}}^{4} . We explore some typical submanifolds of F 4 {{\rm{F}}}^{4} . Next, we explore the general properties of magnetic trajectories in an almost Kähler 4-manifold and characterize Kähler magnetic curves with respect to the symplectic pair of Kähler forms. Finally, we study homogeneous geodesics and homogeneous magnetic curves in F 4 {{\rm{F}}}^{4} .

We consider complex structures with totally real zero section of the tangent bundle. We assume that the complex structure tensor is real-analytic along the fibers of the tangent bundle. This assumption is quite natural in view of a well-known result by Bruhat and Whitney. We provide explicit integrability equations for such complex structures in terms of the fiberwise Taylor expansion. In a particular geometric case considered in the literature, we explicit much further the fiberwise Taylor expansion of the complex structure as well as the integrability equations.

Beginning with the state of art around 1953, solutions of the Levi problem on complex manifolds will be recalled at first up to Takayama’s result in 1998. Then, the activity of extending the results by the L 2 {L}^{2} method in these decades will be reported. The method is by exploiting the finite dimensionality of certain L 2 {L}^{2} ∂ ¯ \bar{\partial } -cohomology groups to prove that a Hermitian holomorphic line bundle L L over a complex manifold M M is bimeromorphically equivalent to an ample bundle when it is restricted to a bounded locally pseudoconvex domain Ω ⋐ M \Omega \hspace{0.15em}\Subset \hspace{0.15em}M under the positivity of L ∣ ∂ Ω {L| }_{\partial \Omega } and the regularity of ∂ Ω \partial \Omega .

We can construct a real line bundle arising from the locally conformal Kähler (LCK) structure over an LCK manifold. We study the properties of this line bundle over an LCK solvmanifold whose complex structure is left-invariant. Mainly, we prove that this line bundle L L over an LCK solvmanifold Γ \ G \Gamma \backslash G with left-invariant complex structure is flat and G G has a global closed 2-form, which induces an Hermitian structure on the holomorphic tangent bundle twisted by the line bundle L C = L ⊗ C {L}^{{\mathbb{C}}}=L\otimes {\mathbb{C}} if the Lee form is cohomologous to a left-invariant 1-form on G G .

In their seminal work, Chen and Cheng proved a priori estimates for the constant scalar curvature metrics on compact Kähler manifolds. They also prove C 3 , α {C}^{3,\alpha } -estimate for the potential of the Kähler metrics under boundedness assumption on the scalar curvature and the entropy. The goal of this article is to replace the uniform boundedness of the scalar curvature to the L p {L}^{p} -boundedness of the scalar curvature.

On Riemann surfaces M M , there exists a canonical correspondence between a possibly multivalued function Ψ X {\Psi }_{X} whose differential is single-valued (i.e. an additively automorphic singular complex analytic function) and a vector field X X . From the point of view of vector fields, the singularities that we consider are zeros, poles, isolated essential singularities, and accumulation points of the above. The theory of singularities of the inverse function Ψ X ‒ 1 {\Psi }_{X}^{‒1} is extended from meromorphic functions to additively automorphic singular complex analytic functions. The main contribution is a complete characterization of when a singularity of Ψ X − 1 {\Psi }_{X}^{-1} is algebraic, is logarithmic, or arises from a zero with non-zero residue of X X . Relationships between analytical properties of Ψ X {\Psi }_{X} , singularities of Ψ X − 1 {\Psi }_{X}^{-1} and singularities of X X are presented. Families and sporadic examples showing the geometrical richness of vector fields on the neighbourhoods of the singularities of Ψ X − 1 {\Psi }_{X}^{-1} are studied. As applications, we have; a description of the maximal univalence regions for complex trajectory solutions of a vector field X X , a geometric characterization of the incomplete real trajectories of a vector field X X , and a description of the singularities of the vector field associated with the Riemann ξ {\rm{\xi }} -function.

The first goal of this article is to give a complete classification (up to Real biholomorphisms) of Real primary Hopf surfaces ( H , s ) \left(H,s) , and, for any such pair, to describe in detail the following naturally associated objects : the group Aut h ( H , s ) {{\rm{Aut}}}_{h}\left(H,s) of Real automorphisms, the Real Picard group ( Pic ( H ) , s ˆ * ) \left({\rm{Pic}}\left(H),{\hat{s}}^{* }) , and the Picard group of Real holomorphic line bundles Pic R ( H ) {{\rm{Pic}}}_{{\mathbb{R}}}\left(H) . Our second goal is the classification of Real primary Hopf surfaces up to equivariant diffeomorphisms, which will allow us to describe explicitly in each case the real locus H ( R ) = H s H\left({\mathbb{R}})={H}^{s} and the quotient H ⁄ ⟨ s ⟩ H/\langle s\rangle .