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The purpose of this paper is to study holomorphic curves f from C to C³ avoiding four complex hyperplanes and a real subspace of real dimension four in C³ . We show that the projection of f into the complex projective space CP2 does not remain constant as in the complex case studied by Green, which indicates that the complex structure of the avoided hyperplanes is a necessary condition in the Green theorem. [ABSTRACT FROM AUTHOR]

We study the stability of compact pseudo-Kähler manifolds, i.e. compact complex manifolds X endowed with a symplectic form compatible with the complex structure of X. When the corresponding metric is positive-definite, X is Kähler and any sufficiently small deformation of X admits a Kähler metric by a well-known result of Kodaira and Spencer. We prove that compact pseudo-Kähler surfaces are also stable, but we show that stability fails in every complex dimension n ≥ 3. Similar results are obtained for compact neutral Kähler and neutral Calabi-Yau manifolds. Finally, motivated by a question of Streets and Tian in the positive-definite case, we construct compact complex manifolds with pseudo-Hermitian-symplectic structures that do not admit any pseudo-Kähler metric. [ABSTRACT FROM AUTHOR]

In this paper, we study several complex manifolds by using the following idea. First, we construct a certain moduli space and study the fundamental group of this space. This fundamental group is naturally mapped to the groups G n k and Γ n k . This is the step towards "complexification" of the G n k and Γ n k approach first developed in [V. O. Manturov, D. A. Fedoseev, S. Kim and I. M. Nikonov, On groups G n k and Γ n k : A study of manifolds, dynamics, and invariants, preprint (2019), arXiv:1905.08049]. [ABSTRACT FROM AUTHOR]

Let M be a complete complex Hermitian manifold with metric E M. A holomorphic mapping f : C m → M is called p-Yosida mapping if ∥ z ∥ 2 − p E M (f (z) , d f (z) (ξ)) is bounded above for z , ξ ∈ C m with ∥ ξ ∥ = 1 , where d f (z) is the mapping from T z ( C m) to T f (z) (M) induced by f. We formulate and prove a criterion of holomorphic mappings belonging to the class of all p-Yosida mappings and its applications to establish two different analogues of Lappan's five-point theorem. [ABSTRACT FROM AUTHOR]

Complex manifolds with compatible metric have a naturally defined subspace of harmonic differential forms that satisfy Serre, Hodge, and conjugation duality, as well as hard Lefschetz duality. This last property follows from a representation of sl(2,C), generalizing the well-known structure on the harmonic forms of compact Kähler manifolds. Some topological implications are deduced. [ABSTRACT FROM AUTHOR]

In this note we show that on any compact subdomain of a K¨ahler manifold that admits sufficiently many global holomorphic functions, the products of harmonic functions form a complete set. This gives a positive answer to the linearized anisotropic Calder´on problem on a class of complex manifolds that includes compact subdomains of Stein manifolds and sufficiently small subdomains of K¨ahler manifolds. Some of these manifolds do not admit limiting Carleman weights, and thus cannot be treated by standard methods for the Calder´on problem in higher dimensions. The argument is based on constructing Morse holomorphic functions with approximately prescribed critical points. This extends earlier results from the case of Riemann surfaces to higher dimensional complex manifolds. [ABSTRACT FROM AUTHOR]

Abstract The symmetrized bidisc G = def { (z + w , z w) : | z | < 1 , | w | < 1 } has interesting geometric properties. While it has a plentiful supply of complex geodesics and of automorphisms, there is nevertheless a unique complex geodesic R in G that is invariant under all automorphisms of G. Moreover, G is foliated by those complex geodesics that meet R in one point and have nontrivial stabilizer. We prove that these properties, together with two further geometric hypotheses on the action of the automorphism group of G , characterize the symmetrized bidisc in the class of complex manifolds. [ABSTRACT FROM AUTHOR]

We prove an equivalence of categories from formal complex structures with formal holomorphic maps to homotopy algebras over a simple operad with its associated homotopy morphisms. We extend this equivalence to complex manifolds. A complex structure on a smooth manifold corresponds in this way to a family of algebras indexed by the points of the manifold. [ABSTRACT FROM AUTHOR]

We investigate degenerate special-Hermitian metrics on compact complex manifolds; in particular, degenerate Kahler and locally conformally Kahler metrics on special classes of non-Kahler manifolds. [ABSTRACT FROM AUTHOR]

We introduce a property of compact complex manifolds under which the existence of balanced metric is stable by small deformations of the complex structure. This property, which is weaker than the ∂ ∂ ‾ -Lemma, is characterized in terms of the strongly Gauduchon cone and of the first ∂ ∂ ‾ -degree measuring the difference of Aeppli and Bott–Chern cohomologies with respect to the Betti number b 1 . [ABSTRACT FROM AUTHOR]

We introduce a property of compact complex manifolds under which the existence of balanced metric is stable by small deformations of the complex structure. This property, which is weaker than the ∂ ∂ ‾ -Lemma, is characterized in terms of the strongly Gauduchon cone and of the first ∂ ∂ ‾ -degree measuring the difference of Aeppli and Bott–Chern cohomologies with respect to the Betti number b 1 . [ABSTRACT FROM AUTHOR]

In this paper, we show the convexity of the image of a moment map on a transverse symplectic manifold equipped with a torus action under a certain condition. We also study properties of moment maps in the case of transverse Kähler manifolds. As an application, we give a positive answer to the conjecture posed by Cupit-Foutou and Zaffran. [ABSTRACT FROM AUTHOR]

We introduce a "qualitative property for Bott-Chern cohomology of complex non-K¨ahler manifolds, which is motivated in view of the study of the algebraic structure of Bott-Chern cohomology. We prove that such a property characterizes the validity of the ∂∂-Lemma. This follows from a quantitative study of Bott-Chern cohomology. In this context, we also prove a new bound on the dimension of the BottChern cohomology in terms of the Hodge numbers. We also give a generalization of this upper bound, with applications to symplectic cohomologies. [ABSTRACT FROM AUTHOR]

The aim of this work is to study some properties of the Lie derivative of forms of bidegree on a n-dimensional complex manifold M. More precisely, we give suffcient conditions such that the Lie derivative LXω of bidegree (p, p) form ω along the vector field X is also the bidegree (p, p) form. [ABSTRACT FROM AUTHOR]

We study a geometric notion related to formality for Bott–Chern cohomology on complex manifolds. [ABSTRACT FROM AUTHOR]

The Oeljeklaus-Toma manifolds are complex non-Kähler manifolds constructed by Oeljeklaus and Toma from certain number fields and generalizing the Inoue surfaces S. We prove that the Oeljeklaus-Toma manifolds contain no compact complex curves. [ABSTRACT FROM AUTHOR]

Abstract: A hypercomplex manifold is a manifold with a triple of complex structure operators satisfying quaternionic relations. For each quaternion , , is also a complex structure operator on , called an induced complex structure. We study compact complex subvarieties of , for a generic induced complex structure. Under additional assumptions (Obata holonomy contained in , the existence of an HKT-metric), we prove that contains no divisors, and all complex subvarieties of codimension 2 are trianalytic (that is, also hypercomplex). [Copyright &y& Elsevier]

Considering countably connected domains of the plane, we solve the problem of existence of finitely valent locally biholomorphic mappings of a bounded domain onto a disk. We obtain sufficient conditions for the existence of the corresponding mappings in higher dimensions. [ABSTRACT FROM AUTHOR]

Two examples of 4-dimensional complex manifolds with Norden metric are constructed by means of Lie groups and Lie algebras. Both manifolds are characterized geometrically. The form of the curvature tensor for each of the examples is obtained. Conditions these manifolds to be isotropic-Kählerian are given. [ABSTRACT FROM AUTHOR]

Abstract: Let be a complex manifold endowed with a strongly pseudoconvex Finsler metric , and be a complex submanifold of endowed with the induced complex Finsler metric . In this paper, the Gauss, Codazzi and Ricci equations are obtained with respect to the Chern–Finsler connection on , the relationship between the torsion of the induced Chern–Finsler connection and the torsion of the Chern–Finsler connection on the ambient Finsler manifold are obtained. As applications of the fundamental formulas, we first prove that the holomorphic curvature of the induced complex Finsler metric does not exceed the holomorphic curvature of , and then give a characterization of the totally geodesic complex Finsler submanifold in terms of the horizontal components of the second fundamental form of . [Copyright &y& Elsevier]

Abstract: This paper shows that an arbitrary generic submanifold in a complex manifold can be deformed into a 1-parameter family of generic submanifolds satisfying strong nondegeneracy conditions. The proofs use a careful analysis of the jet spaces of embeddings satisfying certain nondegeneracy properties, and also make use of Thom''s transversality theorem, as well as the stratification of real-algebraic sets. Optimal results on the order of nondegeneracy are given. [Copyright &y& Elsevier]

Abstract: In this note we study the vortex equation in holomorphic line bundle over non-Kähler complex manifolds. We prove a existence theorem to that equation by means of the upper and lower solution method to some Kazdan–Warner type equation. [Copyright &y& Elsevier]

Let D be a complete hyperbolic domain in Cn, n>1, and N a compact Hermitian manifold. We prove a criterion for the existence of the K-limit of an arbitrary holomorphic map f: D ? N at an arbitrary boundary point D under the condition that f has the corresponding radial limit at this point. [ABSTRACT FROM AUTHOR]

We introduce a new notion of complex oriented matroid and develop some basic properties of this object. Our definition of complex oriented matroids bears the same relationship to classical oriented matroids that the stratification of the complex plane into nine components corresponding to the signs of the complex and real parts has with the three-component sign stratification of the real line. We then use these complex oriented matroids to set up the foundations of a combinatorial version of complex geometry analogous to MacPherson''s combinatorial differential manifolds; in this world, the representing object for the functor of (combinatorial) complex vector bundles is the nerve of a poset of complex oriented matroids. We conclude by showing that this space is homotopy equivalent to the complex Grassmannian, thus deducing that our combinatorial world is able to completely capture the notion of complex vector bundles. [Copyright &y& Elsevier]

The authors prove a version, in utmost generality, of the Bun Wong-Rosay theorem on a complex manifold M . The essence of the result is that a domain $ \Omega \subseteq M $ with non-compact automorphism group and boundary orbit accumulation point that is strongly pseudoconvex must be biholomorphic to the unit ball in $ {\shadC}^n $ . [ABSTRACT FROM AUTHOR]

Abstract: Let G be a connected complex Lie group, and let Γ be a cocompact discrete subgroup of G. If admits an astheno-Kähler metric, then we show that G is abelian and is a compact complex torus. [Copyright &y& Elsevier]