# Local System and Gauss-Manin connection

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

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Dear Pierre, Since $\mathcal O(D)$ and $\mathcal O(D')$ are different (though isomorphic) sheaves, and $F$ and $F'$ are subsheaves of these sheaves, how could they coincide exactly? Regards, – Emerton Jul 17 '13 at 14:23
Besides the "data-type" problem identified by Emerton, I think there's another: the Gauss-Manin connection is not a connection in some arbitrary line bundle, but rather, in the relative de Rham cohomology of some family of complex manifolds. – Tim Perutz Jul 17 '13 at 15:15
Dear Emerton, you are right! – Pierre MATSUMI Jul 18 '13 at 17:59
However, is it OK that F and F' will be isomorphic, i.e. the Gauss-Manin connections are in the similar manner defined. OK? – Pierre MATSUMI Jul 18 '13 at 18:02