Fix a complex manifold $X$. Then if we have a line bundle $L=\mathcal{O}(D)$ together with Gauss-Manin connection $\nabla: L \rightarrow L \otimes \Omega^{1}_X$, we get the locally constant sheaf $F$ by $F:= \ker \nabla$.

Assume that $D$ is linearly equivalent to $D'$, namely $D = D' +(f)$ for some $f$. Then, if $F'$ is in the similarly defined local system on $X$, is it true that F$ = F'$ exactly on $X$?

In particular I would like to know $F$ and $F'$ coincide with each other exactly as a sheaf (not isomorphism). Please teach me.

Pierre