On embeddings of almost complex manifolds in almost complex Euclidean spaces

We prove that any compact almost complex manifold (M2m,J)(M2m,J) of real dimension 2m2m admits a pseudo-holomorphic embedding in (R4m+2,J̃) for a suitable positive almost complex structure J̃. Moreover, we give a necessary and sufficient condition, expressed in terms of the Segre class sm(M,J)sm(M,J), for the existence of an embedding or an immersion in (R4m,J̃). We also discuss the pseudo-holomorphic embeddings of an almost complex 4-manifold in (R6,J̃).