For a complex manifold $M$, the complexified tangent space $\Omega^1(M)$ splits into a direct sum $\Omega^1(M) = \Omega^{(1,0)}(M) \oplus \Omega^{(0,1)}(M)$. As is well-known $\Omega^{(1,0)}(M)$ is canonically a holomorphic vector bundle over $M$.

Question: Is it true that $\Omega^{(0,1)}(M)$ can never be given a holomorphic vector bundle structure, and if so, then why?

For a complex manifold $M$, the complexified tangent space $\Omega^1(M)$ splits into a direct sum $\Omega^1(M) = \Omega^{(1,0)}(M) \oplus \Omega^{(0,1)}(M)$. As is well-known $\Omega^{(1,0)}(M)$ is canonically a holomorphic vector bundle over $M$.

Question: When can $\Omega^{(0,1)}(M)$ be given a holomorphic vector bundle structure? When will it be unique?