Almost complex blow-ups and positive closed $(1,1)$-forms on $4$-dimensional almost complex manifolds

Abstract: Let $(M,J)$ be a $2n$-dimensional almost complex manifold and let $x\in M$. 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. 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 K\"ahler manifold is almost K\"ahler.