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Let (M, J) be a 2n-dimensional almost complex manifold and let x ∈ M . We define the notion of almost complex blow-up of (M, J) at x. We prove the existence of almost complex blow-ups at x under suitable assumptions on the almost complex structure J and we provide explicit examples of such a construction. We note that almost complex blow-ups are unique if they exist. When (M, J) is a 4-dimensional almost complex manifold, we give an obstruction on J to the existence of almost complex blow-ups at a point and prove that the almost complex blow-up at a point of a compact almost Kähler manifold is almost Kähler. [ABSTRACT FROM AUTHOR]

We study Chern flat, Chern–Ricci flat twisted product almost Hermitian manifolds. We extend the twisted product to almost Hermitian manifolds, and give some formulae of the Chern curvature, the Chern–Ricci curvature and the Chern scalar curvature on the twisted product almost Hermitian manifold under the quasi-Kähler condition. [ABSTRACT FROM AUTHOR]

Given a formally integrable almost complex structure J defined on the closure of a bounded domain D ⊂ C n , and provided that J is sufficiently close to the standard complex structure, the global Newlander–Nirenberg problem asks whether there exists a global diffeomorphism defined on D ¯ that transforms J into the standard complex structure, under certain geometric and regularity assumptions on D. In this paper we prove a quantitative result of this problem. Assuming D is a strictly pseudoconvex domain in C n with C 2 boundary, and that the almost complex structure J belongs to the Hölder–Zygmund class Λ r (D ¯) for r > 3 2 , we show the existence of a global diffeomorphism (independent of r) in the class Λ r + 1 2 - ε (D ¯) , for any ε > 0 . [ABSTRACT FROM AUTHOR]

We prove two results on geometric consequences of the representation of restricted holonomy group of a Hermitian connection. The first result concerns when such a Hermitian manifold is Kähler in terms of the torsion and the irreducibility of the holonomy action. As a consequence we obtain a criterion of when a Hermitian manifold (and connection) is a generalized Calabi–Yau (in the sense that the first Ricci vanishes or equivalently that the restricted holonomy is inside SU (m) ). The second result concerns when a compact Kähler manifold with a generic restricted holonomy group is projective, under some nonnegativity assumptions in terms of the Ricci and other curvatures. [ABSTRACT FROM AUTHOR]

In this paper, we compare two types of complex non-Kähler manifolds: LVM and lck manifolds. First, lck manifolds (for locally conformally Kähler manifolds) admit a metric which is locally conformal to a Kähler metric. On the other side, LVM manifolds (for López de Medrano, Verjovsky and Meersseman) are quotients of an open subset of C n by an action of C ∗ × C m . LVM and lck manifolds have a fundamental common point: Hopf manifolds which are a specific case of LVM manifolds and which admit also lck metric. Therefore, the question of this paper is: Are LVM manifolds lck ? We provide some answers to this question. The results obtained are as follows. In the set of all LVM manifolds, there is a dense subset of LVM manifolds which are not lck. And if we consider lck manifolds with potential (whose metric derives from a potential), the diagonal Hopf manifolds are the only LVM manifolds which admit an lck metric with potential. However, we show that there exists an lck covering with potential (non-compact) of a certain subclass of LVM manifolds. Finally, we present some examples. [ABSTRACT FROM AUTHOR]

We develop computational techniques which allow us to calculate the Kodaira dimension as well as the dimension of spaces of Dolbeault harmonic forms for left-invariant almost complex structures on the generalised Kodaira–Thurston manifolds. [ABSTRACT FROM AUTHOR]

We show that extended graph 4-manifolds with positive Euler characteristic cannot support a complex structure. This result stems from a new proof of the fact that there is no compact complex surface which is homotopy equivalent to a closed real-hyperbolic 4-manifold. Finally, we construct infinitely many extended graph 4-manifolds with positive Euler characteristic which support almost complex structures. [ABSTRACT FROM AUTHOR]

We investigate the exterior differential operator on quasi-Kähler manifolds and show some relations of its components for smooth functions. [ABSTRACT FROM AUTHOR]

We study the Kodaira dimension of a real parallelizable manifold M, with an almost complex structure J in standard form with respect to a given parallelism. For X = (M , J) we give conditions under which kod (X) = 0 . We provide examples in the case M = G × G , where G is a compact connected real Lie group. Finally we describe geometrical properties of real parallelizable manifolds in the framework of statistical geometry. [ABSTRACT FROM AUTHOR]

Let (X , J , ω) be a compact 2n-dimensional almost Kähler manifold. We prove primitive decompositions for Bott–Chern and Aeppli harmonic forms in special bidegrees and show that such bidegrees are optimal. We also show how the spaces of primitive Bott–Chern, Aeppli, Dolbeault and ∂ -harmonic forms on (X , J , ω) are related. [ABSTRACT FROM AUTHOR]

We study the deformed Hermitian–Yang–Mills equation (dHYM equation, for short) with gradient terms on a compact almost Hermitian manifold and show the existence of solutions for the dHYM equation in the case of the hypercritical phase. [ABSTRACT FROM AUTHOR]

In this note, we determine the structure of the associative algebra generated by the differential operators μ¯,∂¯,∂\overline{\mu },\overline{\partial },\partial , and μ\mu that act on complex-valued differential forms of almost complex manifolds. This is done by showing that it is the universal enveloping algebra of the graded Lie algebra generated by these operators and determining the structure of the corresponding graded Lie algebra. We then determine the cohomology of this graded Lie algebra with respect to its canonical inner differential [d,−]\left[d,-], as well as its cohomology with respect to all its inner differentials.

On an almost complex manifold, a quasi-Kähler metric, with canonical connection in the c-projective class of a given minimal complex connection, is equivalent to a nondegenerate solution of the c-projectively invariant metrizability equation. For this overdetermined equation, replacing this maximal rank condition on solutions with a nondegeneracy condition on the prolonged system yields a strictly wider class of solutions with non-vanishing (generalized) scalar curvature. We study the geometries induced by this class of solutions. For each solution, the strict point-wise signature partitions the underlying manifold into strata, in a manner that generalizes the model, a certain Lie group orbit decomposition of C P m . We describe the smooth nature and geometric structure of each strata component, generalizing the geometries of the embedded orbits in the model. This includes a quasi-Kähler metric on the open strata components that becomes singular at the strata boundary. The closed strata inherit almost CR-structures and can be viewed as a c-projective infinity for the given quasi-Kähler metric. [ABSTRACT FROM AUTHOR]

In this paper we consider the Monge–Ampère type equations on compact almost Hermitian manifolds. We derive C∞ a priori estimates under the existence of an admissible C -subsolution. Finally, we obtain an existence result if there exists an admissible supersolution. [ABSTRACT FROM AUTHOR]

In this paper, we establish a priori estimates for solutions of a general class of fully non-linear equations on compact almost Hermitian manifolds. As an application, we solve the complex Hessian equation and the Monge–Ampère equation for (n - 1) -plurisubharmonic functions in the almost Hermitian setting. [ABSTRACT FROM AUTHOR]

We give an exposition of Sullivan’s theorem on realizing rational homotopy types by closed smooth manifolds, including a discussion of the necessary rational homotopy and surgery theory, adapted to the realization problem for almost complex manifolds: namely, we give a characterization of the possible simply connected rational homotopy types, along with a choice of rational Chern classes and fundamental class, realized by simply connected closed almost complex manifolds in real dimensions six and greater. As a consequence, beyond demonstrating that rational homotopy types of closed almost complex manifolds are plenty, we observe that the realizability of a simply connected rational homotopy type by a simply connected closed almost complex manifold depends only on its cohomology ring. We conclude with some computations and examples.

We show the short-time existence of the second Ricci flow on a complete noncompact almost Hermitian manifold, and prove that along the second Ricci flow, the non-positivity of the first Chern–Ricci curvature can be preserved if the initial almost Hermitian metric has non-positive bisectional curvature. If additionally the first Chern–Ricci curvature of the initial metric is negative at least at one point, then we show that the almost complex structure of a complete noncompact non-quasi-Kähler almost Hermitian manifold equipped with such a metric cannot be integrable. [ABSTRACT FROM AUTHOR]

We study almost complex structures with lower bounds on the rank of the Nijenhuis tensor. Namely, we show that they satisfy an h-principle. As a consequence, all parallelizable manifolds and all manifolds of dimension 2 n ≥ 10 (respectively ≥ 6 ) admit a almost complex structure whose Nijenhuis tensor has maximal rank everywhere (resp. is nowhere trivial). For closed 4-manifolds, the existence of such structures is characterized in terms of topological invariants. Moreover, we show that the Dolbeault cohomology of non-integrable almost complex structures is often infinite dimensional (even on compact manifolds). [ABSTRACT FROM AUTHOR]

We introduce the conformally balanced condition on almost Hermitian manifolds. We show that a compact Kähler-like and G-Kähler-like almost Hermitian manifold with the conformally balanced condition becomes Kähler. [ABSTRACT FROM AUTHOR]

We give several explicit examples of compact manifolds with a 1-parameter family of almost complex structures having arbitrarily small Nijenhuis tensor in the C0-norm. The 4-dimensional examples possess no complex structure, whereas the 6-dimensional example does not possess a left invariant complex structure, and whether it possesses a complex structure appears to be unknown.

The Donaldson–Fujiki Kähler reduction of the space of compatible almost complex structures, leading to the interpretation of the scalar curvature of Kähler metrics as a moment map, can be lifted canonically to a hyperkähler reduction. Donaldson proposed to consider the corresponding vanishing moment map conditions as (fully nonlinear) analogues of Hitchin's equations, for which the underlying bundle is replaced by a polarised manifold. However this construction is well understood only in the case of complex curves. In this paper we study Donaldson's hyperkähler reduction on abelian varieties and toric manifolds. We obtain a decoupling result, a variational characterisation, a relation to K-stability in the toric case, and prove existence and uniqueness under suitable assumptions on the "Higgs tensor". We also discuss some aspects of the analogy with Higgs bundles. [ABSTRACT FROM AUTHOR]

We investigate the geometry of a twisting non-shearing congruence of null geodesics on a conformal manifold of even dimension greater than four and Lorentzian signature. We give a necessary and sufficient condition on the Weyl tensor for the twist to induce an almost Robinson structure, that is, the screen bundle of the congruence is equipped with a bundle complex structure. In this case, the (local) leaf space of the congruence acquires a partially integrable contact almost CR structure of positive definite signature. We give further curvature conditions for the integrability of the almost Robinson structure and the almost CR structure and for the flatness of the latter. We show that under a mild natural assumption on the Weyl tensor, any metric in the conformal class that is a solution to the Einstein field equations determines an almost CR–Einstein structure on the leaf space of the congruence. These metrics depend on three parameters and include the Fefferman–Einstein metric and Taub–NUT–(A)dS metric in the integrable case. In the non-integrable case, we obtain new solutions to the Einstein field equations, which, we show, can be constructed from strictly almost Kähler–Einstein manifolds. [ABSTRACT FROM AUTHOR]

We prove two results related to the Schwarz lemma in complex geometry. First, we show that if the inequality in the Schwarz lemmata of Yau, Royden and Tosatti becomes equality at one point, then the equality holds on the whole manifold. In particular, the holomorphic map is totally geodesic and has constant rank. In the second part, we study the holomorphic sectional curvature on an almost Hermitian manifold and establish a Schwarz lemma in terms of holomorphic sectional curvatures in almost Hermitian setting. [ABSTRACT FROM AUTHOR]

We investigate Monge–Ampère type fully nonlinear equations on compact almost Hermitian manifolds with boundary and show a priori gradient estimates for a smooth solution of these equations. [ABSTRACT FROM AUTHOR]

We prove some Liouville type results for generalized holomorphic maps in three classes: maps from pseudo-Hermitian manifolds to almost Hermitian manifolds, maps from almost Hermitian manifolds to pseudo-Hermitian manifolds and maps from pseudo-Hermitian manifolds to pseudo-Hermitian manifolds, assuming that the domains are compact. For instance, we show that any (J , J N) holomorphic map from a compact pseudo-Hermitian manifold M with nonnegative (resp. positive) pseudo-Hermitian sectional curvature to an almost Hermitian manifold N with negative (resp. nonpositive) holomorphic sectional curvature is constant. We also construct explicit almost CR structures on a complex vector bundle over an almost CR manifold. [ABSTRACT FROM AUTHOR]

We investigate when the Chevalley-Eilenberg differential of a complex Lie algebroid on a manifold with boundary admits a Hodge decomposition. We introduce the concepts of Cauchy-Riemann structures, elliptic and non-elliptic boundary points and Levi-forms, which we use to define the notion of q-convexity. We show that the Chevalley-Eilenberg complex of an elliptic, q-convex Lie algebroid admits a Hodge decomposition in degree q. This generalizes the well-known Hodge decompositions for the exterior derivative on real manifolds and the delbar-operator on q-convex complex manifolds. We establish the results in a more general setting, where the differential does not necessarily square to zero and moreover varies in a family, including an analysis of the behaviour on the deformation parameter. As application we give a proof of a classical holomorphic tubular neighbourhood theorem (which implies the Newlander-Nirenberg theorem) based on the Moser trick, and we provide a finite-dimensionality result for certain holomorphic Poisson cohomology groups. [ABSTRACT FROM AUTHOR]

In this paper, we consider the deformed Hermitian-Yang-Mills equation on closed almost Hermitian manifolds. In the case of the hypercritical phase, we derive a priori estimates under the existence of an admissible C -subsolution. As an application, we prove the existence of solutions for the deformed Hermitian-Yang-Mills equation under the condition of existence of a supersolution. [ABSTRACT FROM AUTHOR]

We introduce the k-th almost Gauduchon condition on almost Hermitian manifolds. We show that if there is a k-th almost Gauduchon metric for some k, then it is k-th almost Gauduchon for any other k. [ABSTRACT FROM AUTHOR]

In this paper we relate the cohomology of J-invariant forms to the Dolbeault cohomology of an almost complex manifold. We find necessary and sufficient condition for the inclusion of the former into the latter to be true up to isomorphism. We also extend some results obtained by J. Cirici and S. O. Wilson about the computation of the left-invariant cohomology of nilmanifolds to the setting of solvmanifolds. Several examples are given. [ABSTRACT FROM AUTHOR]

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 (M4, 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

We study the moduli space of J-holomorphic subvarieties in a 4-dimensional symplectic manifold. For an arbitrary tamed almost complex structure, we show that the moduli space of a sphere class is formed by a family of linear system structures as in algebraic geometry. Among the applications, we show various uniqueness results of J-holomorphic subvarieties, e.g. for the fiber and exceptional classes in irrational ruled surfaces. On the other hand, non-uniqueness and other exotic phenomena of subvarieties in complex rational surfaces are explored. In particular, connected subvarieties in an exceptional class with higher genus components are constructed. The moduli space of tori is also discussed, and leads to an extension of the elliptic curve theory. [ABSTRACT FROM AUTHOR]

We discuss the complex geometry of two complex five-dimensional Kähler manifolds which are homogeneous under the exceptional Lie group G 2 . For one of these manifolds, rigidity of the complex structure among all Kählerian complex structures was proved by Brieskorn; for the other one, we prove it here. We relate the Kähler assumption in Brieskorn's theorem to the question of existence of a complex structure on the six-dimensional sphere, and we compute the Chern numbers of all G 2 -invariant almost complex structures on these manifolds. [ABSTRACT FROM AUTHOR]