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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]

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]

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]

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]

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.

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 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.

The well-known Kähler identities naturally extend to the non-integrable setting. This paper deduces several geometric and topological consequences of these extended identities for compact almost Kähler manifolds. Among these are identities of various Laplacians, generalized Hodge and Serre dualities, a generalized hard Lefschetz duality, and a Lefschetz decomposition, all on the space of d-harmonic forms of pure bidegree. There is also a generalization of Hodge Index Theorem for compact almost Kähler 4-manifolds. In particular, these provide topological bounds on the dimension of the space of d-harmonic forms of pure bidegree, as well as several new obstructions to the existence of a symplectic form compatible with a given almost complex structure. [ABSTRACT FROM AUTHOR]

We consider the evolution of an almost Hermitian metric by the (1, 1) part of its Chern-Ricci form on almost complex manifolds. This is an evolution equation first studied by Chu and coincides with the Chern-Ricci flow if the complex structure is integrable and with the Kähler-Ricci flow if moreover the initial metric is Kähler. We find the maximal existence time for the flow in term of the initial data and also give some convergence results. As an example, we study this flow on the (locally) homogeneous manifolds in more detail. [ABSTRACT FROM AUTHOR]

The almost complex Lie algebroids over smooth manifolds are considered in the paper. In the first part, we give some examples and we extend some basic results from almost complex manifolds to almost complex Lie algebroids. Next the almost Hermitian Lie algebroids and some related structures on the associated complex Lie algebroid are studied. [ABSTRACT FROM AUTHOR]

Let $$(X,J)$$ be an almost-complex manifold. The almost-complex structure $$J$$ acts on the space of $$2$$ -forms on $$X$$ as an involution. A $$2$$ -form $$\alpha $$ is $$J$$ - anti-invariant if $$J\alpha =-\alpha $$ . We investigate the anti-invariant forms and their relation to taming and compatible symplectic forms. For every closed almost-complex manifold, in contrast to invariant forms, we show that the space of closed anti-invariant forms has finite dimension. If $$X$$ is a closed almost-complex manifold with a taming symplectic form, then we show that there are no non-trivial exact anti-invariant forms. On the other hand, we construct many examples of almost-complex manifolds with exact anti-invariant forms, which are therefore not tamed by any symplectic form. In particular, we use our analysis to give an explicit example of an almost-complex structure which is locally almost-Kähler but not globally tamed. The non-existence of exact anti-invariant forms, however, does not in itself imply that there exists a taming symplectic form. We show how to construct examples in all dimensions. [ABSTRACT FROM AUTHOR]

The goal of this paper is to generalize the notion of almost analyticity in the almost (para)complex Lie algebroids framework. We use a global formalism which yields to generalizations of the main results of previous known almost (para)complex manifolds case. [ABSTRACT FROM AUTHOR]