Why $\Lambda^{p,q}M$ are not holomorphic bundles for $q\neq 0$?

Question

I just read from a lecture note that $\Lambda^{p,q}M$ are not holomorphic bundles for $q\neq 0$. But, take $\Lambda^{0,1}M$ for example, locally we have $$ d\bar{z} = \frac{\partial \bar{z}}{\partial w}dw + \frac{\partial\bar{z}}{\partial \bar{w}}d\bar{w} $$ where $z,w$ are two holomorphic coordinates. Since $z(w)$ is holomorphic, we have $\partial\bar{z}/\partial w = \partial{z}/\partial \bar{w}= 0$, so $$ d\bar{z} = \frac{\partial\bar{z}}{\partial \bar{w}}d\bar{w} $$ and the transition function $\partial\bar{z}/\partial \bar{w} = \partial{z}/\partial {w}$ is holomorphic. So what's wrong here?

Answer 1

No, it's anti-holomorphic, as $\dfrac{\partial\bar z}{\partial\bar w} = \overline{\dfrac{\partial z}{\partial w}}$.

And with $p,q\ne 0$, the transition functions are products of both holomorphic and anti-holomorphic functions.