On embeddings of almost complex manifolds in almost complex Euclidean spaces

We prove that any compact almost complex manifold $(M^{2m}, J)$ of real dimension $2m$ admits a pseudo-holomorphic embedding in $(R^{4m+2}, ilde{J})$ for a suitable positive almost complex structure $ ilde 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 $(R^{4m}, ilde{J})$. We also discuss the pseudo-holomorphic embeddings of an almost complex 4-manifold in $(R^6, ilde J)$.