I'm reading a vanishing theorem from Kobayashis' differential geometry of complex bundles and I'm stuck with an argument.
Let $E\rightarrow M$ a holomorphic vector bundle over a compact Kahler manifold.
At some point of the proof he says that a nonzero holomorphic section $\xi$ that is parallel (wrt Chern connection) never vanishes. Why is this true?
This is not related to the holomorphic setup but holds for parallel sections of any linear connection. It needs the assumption that $M$ is connected, though. Suppose that $\nabla s=0$ and that $x_0\in M$ is such that $s(x_0)=0$. Then for any point $x\in M$, connectedness implies that there is a smooth curve in $M$ that connects $x_0$ to $x$ and since $\nabla s=0$, $s(x)$ has to be obtained by parallely transporting $s(x_0)$ to $x$ along $c$. But parallel transport is a linear map, so the parallel transport of $0$ is $0$.