Atiyah sequence of a coherent sheaf

Question

I want to show that if $X$ is a smooth complex projective variety, then analytification induces an equivalence of categories between algebraic integrable connections on $X$ and analytic integrable connections on $X^\text{an}$. It does not follow immediately from GAGA that every analytic connection $(M, \nabla)$ comes from an algebraic connection, since $\nabla: M \to M \otimes \Omega^1_{X^\text{an}}$ is not $\mathcal{O}_{X^\text{an}}$-linear. A fix mentioned in this question is to view an integrable connection on $X^\text{an}$ as an $\mathcal{O}$-linear splitting of the Atiyah sequence

$$0 \mapsto \operatorname{End}(M) \to \operatorname{At}(M) \to \mathcal{T}_{X^\text{an}} \to 0$$

and apply GAGA.

I can't figure out what this exact sequence should be, and I couldn't find any references. What is $\operatorname{At}(\bullet)$, and what precisely is this exact sequence?

Answer 1

Here's how the argument goes, thanks to @abx. $\operatorname{At}(M)$ is given by $\mathbb{C}$-linear endomorphisms $D: M \to M$ for which the commutator $[D,f]$ given by $[D,f](m) := D(fm) - fD(m)$ acts via left multiplication by some element $h_{D,f} \in \mathcal{O}$. The map to the tangent sheaf $T$ is given by $$D \mapsto (f \mapsto h_{D,f})$$ and it's straightforward to check that this is a derivation. It's clear that the kernel is $\operatorname{End}_{\mathcal{O}}(M)$, and surjectivity follows by defining $$D_P\left( \sum_{i=1}^n f_i m_i\right) := \sum_{i=1}^n P(f_i) m_i$$ in some local basis $m_i$ of $M$, where $P \in T_X$. It is now immediate that $\mathcal{O}$-linear right splittings correspond to integrable connections on $M$.