Volume 7, Issue 1

We classify and investigate locally conformally Kähler structures on four-dimensional solvable Lie algebras up to linear equivalence. As an application we can produce many examples in higher dimension, here including lcK structures on Oeljeklaus-Toma manifolds, and we also give a geometric interpretation of some of the 4-dimensional structures in our classification.

Given a semi-Riemannian 4-manifold ( M , g ) with two distinguished vector fields satisfying properties determined by their shear, twist and various Lie bracket relations, a family of Kähler metrics g K is constructed, defined on an open set in M , which coincides with M in many typical examples. Under certain conditions g and g K share various properties, such as a Killing vector field or a vector field with a geodesic flow. In some cases the Kähler metrics are complete. The Ricci and scalar curvatures of g K are computed under certain assumptions in terms of data associated to g . Many examples are described, including classical spacetimes in warped products, for instance de Sitter spacetime, as well as gravitational plane waves, metrics of Petrov type D such as Kerr and NUT metrics, and metrics for which g K is an SKR metric. For the latter an inverse ansatz is described, constructing g from the SKR metric.

We study the behavior of the degeneration at the second step of the Frölicher spectral sequence of a 𝒞 ∞ family of compact complex manifolds. Using techniques from deformation theory and adapting them to pseudo-differential operators we prove a result à la Kodaira-Spencer for the dimension of the second step of the Frölicher spectral sequence and we prove that, under a certain hypothesis, the degeneration at the second step is an open property under small deformations of the complex structure.

In this paper, for a non-compact Riemman surface S homeomorphic to either: the Infinite Loch Ness monster, the Cantor tree and the Blooming Cantor tree, we give a precise description of an infinite set of generators of a Fuchsian group Γ < PSL (2, ℝ), such that the quotient space ℍ/Γ is a hyperbolic Riemann surface homeomorphic to S . For each one of these constructions, we exhibit a hyperbolic polygon with an infinite number of sides and give a collection of Mobius transformations identifying the sides in pairs.

We show that a 2 k -current T on a complex manifold is a real holomorphic k -chain if and only if T is locally real rectifiable, d -closed and has ℋ 2 k -locally finite support. This result is applied to study homology classes represented by algebraic cycles.

We consider several differential operators on compact almost-complex, almost-Hermitian and almost-Kähler manifolds. We discuss Hodge Theory for these operators and a possible cohomological interpretation. We compare the associated spaces of harmonic forms and cohomologies with the classical de Rham, Dolbeault, Bott-Chern and Aeppli cohomologies.

In this paper we study an analog of minimal surfaces called Weyl-minimal surfaces in conformal manifolds with a Weyl connection ( M 4 , c , D ). We show that there is an Eells-Salamon type correspondence between nonvertical 𝒥-holomorphic curves in the weightless twistor space and branched Weyl-minimal surfaces. When ( M , c , J ) is conformally almost-Hermitian, there is a canonical Weyl connection. We show that for the canonical Weyl connection, branched Weyl-minimal surfaces satisfy the adjunction inequality χ(Tf∑)+χ(Nf∑)≤±c1(f*T(1,0)M).\chi \left( {{T_f}\sum } \right) + \chi \left( {{N_f}\sum } \right) \le \pm {c_1}\left( {f*{T^{\left( {1,0} \right)}}M} \right). The ± J -holomorphic curves are automatically Weyl-minimal and satisfy the corresponding equality. These results generalize results of Eells-Salamon and Webster for minimal surfaces in Kähler 4-manifolds as well as their extension to almost-Kähler 4-manifolds by Chen-Tian, Ville, and Ma.

Serre’s duality theorem implies a symmetry between the Hodge numbers, h p , q = h n − p , n − q , on a compact complex n –manifold. Equivalently, the first page of the associated Frölicher spectral sequence satisfies dimE1p,q=dimE1n−p,n−q\dim E_1^{p,q} = \dim E_1^{n - p,n - q} for all p , q . Adapting an argument of Chern, Hirzebruch, and Serre [3] in an obvious way, in this short note we observe that this “Serre symmetry” dimEkp,q=dimEkn−p,n−q\dim E_k^{p,q} = \dim E_k^{n - p,n - q} holds on all subsequent pages of the spectral sequence as well. The argument shows that an analogous statement holds for the Frölicher spectral sequence of an almost complex structure on a nilpotent real Lie group as considered by Cirici and Wilson in [4].

We introduce Kähler-like, G-Kähler-like metrics on almost Hermitian manifolds. We prove that a compact Kähler-like and G-Kähler-like almost Hermitian manifold equipped with an almost balanced metric is Kähler. We also show that if a Kähler-like and G-Kähler-like almost Hermitian manifold satisfies Bi¯j¯λBλji≥0B_{\bar i\bar j}^\lambda B_{\lambda j}^i \ge 0, then the metric is almost balanced and the almost complex structure is integrable, which means that the metric is balanced. We investigate a G-Kähler-like almost Hermitian manifold under some assumptions.

We present a binary code for spinors and Clifford multiplication using non-negative integers and their binary expressions, which can be easily implemented in computer programs for explicit calculations. As applications, we present explicit descriptions of the triality automorphism of Spin (8), explicit representations of the Lie algebras 𝔰𝔭𝔦𝔶 (8), 𝔰𝔭𝔦𝔶 (7) and 𝔤 2 , etc.

We construct a simply-connected compact complex non-Kähler manifold satisfying the ∂ ̅∂ -Lemma, and endowed with a balanced metric. To this aim, we were initially aimed at investigating the stability of the property of satisfying the ∂ ̅∂ -Lemma under modifications of compact complex manifolds and orbifolds. This question has been recently addressed and answered in [34, 39, 40, 50] with different techniques. Here, we provide a different approach using Čech cohomology theory to study the Dolbeault cohomology of the blowup ̃ X Z of a compact complex manifold X along a submanifold Z admitting a holomorphically contractible neighbourhood.

This is a review article on pseudo-holomorphic curves which attempts at touching all the main analytical results. The goal is to make a user friendly introduction which is accessible to those without an analytical background. Indeed, the major accomplishment of this review is probably its short length. Nothing in here is original and can be found in more detailed accounts such as [6] and [8]. The exposition of the compactness theorem is somewhat different from that in the standard references and parts of it are imported from harmonic map theory [7], [5]. The references used are listed, but of course any mistake is my own fault.

Let M be the moduli space of complex Lagrangian submanifolds of a hyperKähler manifold X , and let ω̄ : 𝒜̂ → M be the relative Albanese over M . We prove that 𝒜̂ has a natural holomorphic symplectic structure. The projection ω̄ defines a completely integrable structure on the symplectic manifold 𝒜̂. In particular, the fibers of ω̄ are complex Lagrangians with respect to the symplectic form on 𝒜̂. We also prove analogous results for the relative Picard over M .

We describe natural deformation classes of generalized Kähler structures using the Courant symmetry group, which determine natural extensions of the notions of Kähler class and Kähler cone to generalized Kähler geometry. We show that the generalized Kähler-Ricci flow preserves this generalized Kähler cone, and the underlying real Poisson tensor.