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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 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]

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 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.

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 define the Monge-Ampère operator $${(i\partial {\bar{\partial }}u)^{2}}$$ for continuous J-plurisubharmonic functions on four dimensional almost complex manifolds. [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]

We prove that any compact almost complex manifold ( M 2 m , J ) of real dimension 2 m admits a pseudo-holomorphic embedding in ( R 4 m + 2 , J ̃ ) for a suitable positive almost complex structure J ̃ . Moreover, we give a necessary and sufficient condition, expressed in terms of the Segre class s m ( M , J ) , for the existence of an embedding or an immersion in ( R 4 m , J ̃ ) . We also discuss the pseudo-holomorphic embeddings of an almost complex 4-manifold in ( R 6 , J ̃ ) . [ABSTRACT FROM AUTHOR]

We study the Dirichlet problem for the Monge-Ampère equation on almost complex manifolds. We obtain the existence of the unique smooth solution in strictly pseudoconvex domains. [ABSTRACT FROM AUTHOR]

Given an integer 1 ≤ p ≤ n − 1, we consider a plurisubharmonic positive currentTof bidimension (p, p) on an open subset of ℂncontaining the origin. The tangent cone toTat the origin, if it exists, is the weak limit of dilations ofTby homotheties. Even when the tangent cone fails to exist, every weak limit at 0 of a subsequence of dilations ofTis a positive, plurisubharmonic conical current on ℂnwith the same Lelong number at the origin asT. We give a sufficient condition, in terms of estimates of the mass of the trace measure, for the existence of the tangent cone. [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]