Let $E\to M$ be a holomorphic vector bundle ($M$ is compact) and assume $s\colon M\to E$ is a non-trivial section transverse to the zero section.

Is it be possible that $s^{-1}(0)\neq \emptyset$, yet $c_{top}(E)=0$?

I exect the answer to be NO. If $M$ is Kahler, integrating the Kahler form on $s^{-1}(0)$ shows that it is homologically non-trivial. But how do you prove it for general complex manifold or without using the Kahler form.

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