On a compact locally conformal K & auml;hler manifold (which is also Gauduchon) satisfying an Einstein-type condition, an inequality between the integrations of Chern forms and the Bochner curvature tensor is proved. While on a compact Gauduchon surface, two similar inequalities are derived.
We revisit the second order estimate for solutions to the quaternionic Calabi-Yau problem on hyperkähler manifolds, originally established by Dinew and Sroka. In this note, we present a simplified argument to derive the estimate.
We study the stability at blow-ups and deformations of a class of Hermitian metrics whose fundamental two-form ω\omega satisfies the condition ∂∂¯ωk=0\partial \bar{\partial }{\omega }^{k}=0, for any kk between 1 and n−1n-1 (where nn is the complex dimension of the manifold). We are motivated by the existence of compact complex manifolds supporting such metrics.
In this article, we introduce certain arity-3 operations on complex manifolds arising from homotopy transfer theory. Such operations are related to the triple ABCABC-Massey products from Bott-Chern to Aeppli cohomology. We present a package for computing such products as well as other cohomological and homotopical invariants for complex nilmanifolds and give some examples.
We survey some results and open questions related to various algebraic invariants of compact complex manifolds that can be obtained from differential forms.
In this study, we extend the uniform L2{L}^{2}-estimate of ∂¯\bar{\partial }-equations for flat nontrivial line bundles, proved for compact Kähler manifolds by Hashimoto and Koike, to compact complex manifolds. In the proof, by tracing the Dolbeault isomorphism in detail, we derive the desired L2{L}^{2}-estimate directly from Ueda’s lemma.
We revisit Koiso’s original examples of rigid infinitesimally deformable Einstein metrics. We show how to compute Koiso’s obstruction to the integrability of the infinitesimal deformations on CP2n×CP1{{\mathbb{CP}}}^{2n}\times {{\mathbb{CP}}}^{1} using completely elementary complex differential geometry.
In this article, we briefly survey the classical problem of understanding which Lie algebras admit a complex structure, put in the broader perspective of almost complex structures with special properties. We focus on the different behavior of invariant and non-invariant structures, with special attention to their canonical bundle and Kodaira dimension. We provide new examples of computations of the Kodaira dimension of invariant and non-invariant structures.
Let Mi{M}_{i}, for i=1i=1, 2, be a Kähler manifold, and let GG be a compact Lie group acting on Mi{M}_{i} by Kähler isometries. Suppose that the action admits a momentum map μi{\mu }_{i}, and let Ni≔μi−1(0){N}_{i}:= {\mu }_{i}^{-1}\left(0) be a regular-level set. When the action of GG on Ni{N}_{i} is proper and free, the Meyer-Marsden-Weinstein quotient Pi≔Ni∕G{P}_{i}:= {N}_{i}/G is a Kähler manifold and πi:Ni→Pi{\pi }_{i}:{N}_{i}\to {P}_{i} is a principal fiber bundle with base Pi{P}_{i} and characteristic fiber GG. In this article, we define an almost-complex structure on the manifold N1×N2{N}_{1}\times {N}_{2} and give necessary and sufficient conditions for its integrability. In the integrable case, we find explicit holomorphic charts for N1×N2{N}_{1}\times {N}_{2}. As applications, we consider a nonintegrable almost-complex structure on the product of two complex Stiefel manifolds and the infinite Calabi-Eckmann manifolds S2n+1×S(ℋ){{\mathbb{S}}}^{2n+1}\times S\left({\mathcal{ {\mathcal H} }}), for n≥1n\ge 1, where S(ℋ)S\left({\mathcal{ {\mathcal H} }}) denotes the unit sphere of an infinite-dimensional complex Hilbert space ℋ{\mathcal{ {\mathcal H} }}.
We discuss complex quaternionic manifolds, i.e., those that have holonomy GL(n,H)U(1) $GL\left(n,\mathbb{H}\right)U\left(1\right)$ , which naturally arise via quaternionic Feix–Kaledin construction. We show that for a fixed c-projective class, any real analytic connection with type (1,1) curvature induces, via quaternionic Feix–Kaledin construction, an S 1-invariant connection with holonomy contained in GL(n,H)U(1) $GL\left(n,\mathbb{H}\right)U\left(1\right)$ . As an application, we characterize in this setting the distinguished U*(2n):=SL(n,H)U(1) ${U}^{{\ast}}\left(2n\right):=SL\left(n,\mathbb{H}\right)U\left(1\right)$ connection studied in Battaglia [Circle actions and Morse theory on quaternion-Kähler manifolds, J. Lond. Math. Soc. 59 (1999), 345–358] and Hitchin [Manifolds with holonomy U*(2m), Rev. Mat. Complut. 27 (2014), 351–368].
We prove a formula for the first few terms of the asymptotic expansion of the holomorphic analytic torsion of the Dirac operator modified by the Clifford action of a real and closed three-form.
We explore submersions introduced by reducible holonomy representations of connections with parallel skew torsion. A submersion theorem extending previous, less general, results is given. As our main application, we show that parallel 3-(α,δ)\left(\alpha ,\delta )-Sasaki manifolds admit one-dimensional submersions onto nearly Kähler orbifolds. As a secondary application, we reprove that a given class of nearly Kähler manifolds submerges onto quaternionic Kähler manifolds. This new proof gives a direct expression for the quaternionic structure on the base.
We study infinite superelliptic curves as translation surfaces and explore their Veech groups. These objects are branched covering of the complex plane, branching over infinitely many points. We provide a criterion for isomorphism between a special family of infinite superelliptic curves. We describe the geometry of saddle connections and holonomy vectors on these infinite superelliptic curves. In addition, we prove that the Veech group of an infinite superelliptic curve consists of matrices arising from the differentials of the affine mappings from C{\mathbb{C}} to itself, which permutes the branched points. We obtain necessary and sufficient conditions to guarantee that the Veech group of an infinite superelliptic curve is uncountable. Furthermore, we establish a trichotomy on the holonomy vector set and precisely characterize certain countable groups that can appear as Veech groups of an infinite superelliptic curve. Finally, we also construct and study several examples of interesting infinite superelliptic curves illustrating our results.
In their seminal work, Chen and Cheng proved a priori estimates for the constant scalar curvature metrics on compact Kähler manifolds. They also prove C3,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 Lp{L}^{p}-boundedness of the scalar curvature.
On Riemann surfaces MM, there exists a canonical correspondence between a possibly multivalued function ΨX{\Psi }_{X} whose differential is single-valued (i.e. an additively automorphic singular complex analytic function) and a vector field XX. 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 ΨX‒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 ΨX−1{\Psi }_{X}^{-1} is algebraic, is logarithmic, or arises from a zero with non-zero residue of XX. Relationships between analytical properties of ΨX{\Psi }_{X}, singularities of ΨX−1{\Psi }_{X}^{-1} and singularities of XX are presented. Families and sporadic examples showing the geometrical richness of vector fields on the neighbourhoods of the singularities of ΨX−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 XX, a geometric characterization of the incomplete real trajectories of a vector field XX, and a description of the singularities of the vector field associated with the Riemann ξ{\rm{\xi }}-function.
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 L2{L}^{2} method in these decades will be reported. The method is by exploiting the finite dimensionality of certain L2{L}^{2} ∂¯\bar{\partial }-cohomology groups to prove that a Hermitian holomorphic line bundle LL over a complex manifold MM 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 LL over an LCK solvmanifold Γ\G\Gamma \backslash G with left-invariant complex structure is flat and GG has a global closed 2-form, which induces an Hermitian structure on the holomorphic tangent bundle twisted by the line bundle LC=L⊗C{L}^{{\mathbb{C}}}=L\otimes {\mathbb{C}} if the Lee form is cohomologous to a left-invariant 1-form on GG.
We study geodesics and magnetic trajectories in the model space F4{{\rm{F}}}^{4}. The space F4{{\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 F4{{\rm{F}}}^{4} and investigate its curvature properties. We introduce the symplectic pair of two Kähler forms on F4{{\rm{F}}}^{4}. Those symplectic forms induce invariant Kähler structure and invariant strictly almost Kähler structure on F4{{\rm{F}}}^{4}. We explore some typical submanifolds of F4{{\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 F4{{\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.
The first goal of this article is to give a complete classification (up to Real biholomorphisms) of Real primary Hopf surfaces $(H,s)$, and, for any such pair, to describe in detail the following naturally associated objects : the group $\mathrm{Aut}_h(H,s)$ of Real automorphisms, the Real Picard group $(\mathrm{Pic}(H),\hat s^*)$, and the Picard group of Real holomorphic line bundles $\mathrm{Pic}_{\mathbb{R}}(H)$. Our second goal: the classification of Real primary Hopf surfaces up to equivariant diffeomorphisms, which will allow us to describe explicitly in each case the real locus $H(\mathbb{R})=H^s$ and the quotient $H/\langle s\rangle$.
