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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]