Maps into complex space

If the dimension of $M$ is denoted by $2k-1$ or $2k$, then a generic map $F:M\to C^k$ satisfies $dF_1\wedge \ldots\wedge dF_k \neq 0$, while in certain cases there is no map $F: M\to C^{k+1}$ that satisfies $dF_1\wedge \ldots\wedge dF_{k+1} \neq 0$.