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Let $X$ be a nice variety resp. a manifold over the complex numbers. One defines a connection on a vector bundle $V$ on over $X$ as a $\mathbb C-$linear sheaf homomorphism

$\nabla : V\rightarrow V\otimes \Omega^1$

which satisfies the Leibniz rule.

I have read that this is equivalent to giving for each local vector field $Y\in Der_{\mathbb C}(\mathcal O_X)$ a $\mathbb C-$ linear sheaf homomorphism

$\nabla_Y : V \rightarrow V$

with

(1) Leibniz rule

(2) $\nabla_{fY+gZ}=f\nabla_Y+g\nabla_Z$ for $f,g \in \mathcal O_X$ and $Y,Z$ local vector fields.

I can prove that each connection in the first sense implies a connection in the second sense. But I don't see how you get from the datum of thhe $\nabla_Y$ a connection in the first sense.

Remark: By a vector field I understand a linear derivation of the structure sheaf into itself.

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Using the fact that $\Omega^1$ is $\mathcal{O}_X$-locally free and dual to the sheaf of vector fields you should be able to prove $Hom_{\mathcal{O}_X}(\Theta_X,\mathcal{E}nd_\mathbb{C}(V)) = Hom_\mathbb{C}(V,V\otimes_{\mathcal{O}_X} \Omega^1)$. In local complex coordinates $(x_1,\ldots,x_n)$, $\{\nabla_Y\}$ is mapped to $\nabla: V \to V\otimes \Omega^1$ defined by $$ \nabla(v) = \sum_{i=1}^n \nabla_{\partial_i}(v) \otimes dx_i $$ (This the analogue of $df = \sum_i \frac{\partial f}{\partial x_i} dx_i$). Condition (2) ensures that this formula does not depend on the choice of coordinates.

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  • $\begingroup$ Hi YBL, thanks for your enlightening answer. But I think you mean "this formula does not depend of..."? $\endgroup$ – Veen Oct 23 '11 at 14:07
  • $\begingroup$ Of course. I edited it. $\endgroup$ – AFK Oct 23 '11 at 15:44

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