Let $X$ be a complex manifold of complex dimension $n$ and let $(E,h)$ be a smooth Hermitian vector bundle over $X$. Then there is a correspondence between unitary connections on $(E,h)$ and almost complex structures on $E$, these identified with Dolbeault operators $\bar{\partial}_E:\Omega^0(E)\rightarrow \Omega^{0,1}(E)$.
This correspondence goes even further: integrable almost complex structures correspond to unitary connections such that the curvature is of type (1,1), $F_A^{2,0}=F_A^{0,2}=0$.
I want to verify the following: in Riemann surfaces $n=1$, every almost complex structure on $E$ is integrable (i.e. a holomorphic structure), since there are no non-trivial (0,2) or (2,0) forms over a Riemann surface. Is this correct or am I making a horrible mistake somewhere? I feel that should be explicitly stated in the books that I am working with but they seem to overlook this (or maybe it is completely false!)