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Let $X$ and $Y$ be smooth projective varieties over $\mathbb C$. Suppose I have given a line bundle $L$ on $X\times Y$ with a connection relative to $Y$, i.e.

$\nabla: L \rightarrow \Omega^1_{X\times Y/Y} \otimes L$.

One can push it down via the first projection $p$ and so gets a connection on $p_{*}L$ on $X$.

Furthermore let $V$ be any quasicoherent sheaf on $X \times Y$; then there is no canonical way to associate a connection to $L\otimes V$ relative to $Y$ from the given data, simply because $V$ doesn't have a connection. But please correct me if I am wrong here.

Question: But is there nevertheless a natural way to get a connection on $p_{*}(L\otimes V)$ from the data?

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Perhaps better suited for MO. – Martin Brandenburg Oct 31 '11 at 15:17
up vote 1 down vote accepted

The quasi-coherent sheaf $L\otimes V$ does have a connection, take $\nabla\otimes \mathrm{Id}_V$.

If $F\to \Omega^1_{X\times Y/Y} \otimes F$ is a (relative) connection on some quasi-coherent sheaf $F$ on $X\times Y$, it induces canonically a map $$ p_*F \to p_*(\Omega^1_{X\times Y/Y} \otimes F)=p_*(p^*\Omega_{X/\mathbb C}\otimes F)\simeq \Omega^1_{X/\mathbb C}\otimes p_*F. $$ The second isomorphism comes from the projection formula and because $\Omega^1_{X/\mathbb C}$ is locally free. And it is easy to check that this defines a connection on $p_*F$.

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