Suppose $L$ is a line bundle over $\mathbb{C}P^1$, I wonder how to determine whether $H^{0,1}(\mathbb{C}P^1,L)$ is $0$ or not.
I want to use kodaira vanishing theorem at first, but in my case the assumption of kodaira vanishing theorem does not hold whatever $L$ is positive or anti-positive, I guess the answer depends on the sign of first Chern class, but I have no idea to calculate it, can anyone help me? Thanks!