Let $X$ be a smooth algebraic variety over $\mathbb{C}$ and $\Theta_X$ be its tangent sheaf. Giving an $\mathscr{O}_X$-module $M$ the structure of a left $\mathcal{D}_X$-module is equivalent to the data of a $\mathbb{C}$-linear morphism
$$\nabla:\Theta_X\to \underline{End}(M),$$
which is usually called a connection.
Well… for me, a connection on a locally free sheaf $M$ is a $\mathbb{C}$-linear morphism
$$\nabla':M\to M\otimes_{\mathscr{O}_X}\Omega^1_X.$$
What is the precise relation between those two notions?
Best Answer
If you have a morphism $M\to M\otimes \Omega_X$ and have a section of $\Theta_X=Hom(\Omega_X,O_X)$, you can combine these two to get a map $M\to M\otimes O_X=M$ so you get a connection in the first sense. conversely, if you have an action of the tangent bundle on M, you get an element of $Hom(\Theta_X,Hom(M,M))$ which by adjunction is equal to $Hom(\Theta_X\otimes M,M)$ which by duality is the same as $Hom(M,M\otimes\Omega_X)$ which is a connection in the second sense