Gluing local holomorphic connections

Question

On page $180$ of Complex Geometry by Daniel Huybrecths, he defines the so called Atiyah class of a holomorphic vector bundle by the Čech cocycle $$A(E)=\{U_{ij}, \psi^{-1}_j \circ (\psi^{-1}_{ij}d\psi_{ij}) \circ \psi_j\},$$ where $\psi_i: E|_{U_i}\cong U_i \times \Bbb C^r$ is a local trivialization and $\psi_{ij}$ the transition functions.

He then goes on to prove Proposition $4.2.19$ which states that a holomorphic vector bundle $E$ admits a holomorphic connection if and only if $A(E)=0$.

In the proof of this proposition its being stated that local holomorphic connections on $U_i \times \Bbb C^r$ are of the form $\partial + A_i$. This sounds all good to me, but then he states that on $U_{ij}$ these glue if and only if $$\psi_i^{-1}\circ (d+A_i) \circ \psi_i = \psi_j^{-1}\circ (d+A_j) \circ \psi_j.$$

This is an equation that does not really make sense to me. I've been taught that these local connections glue if and only if they satisfy $$A_j=\psi_{ij}^{-1}d\psi_{ij}+\psi^{-1}_{ij}A_i\psi_{ij}$$ which makes sense as this is only matrix multiplication. Is there some identification going on with the first equation I should be aware of or how is this derived (I was not able to go from the latter to this)? I tried to ask this on math.stackexchange before, but did not get any replies so apologies if this is not an appropriate question for this site.

Answer 1

(Updated, as I realized the formula stated by Huybrechts indeed also does make sense without making any identifications.)

One should interpret the two sides in the compatibility condition of Huybrechts as defining two morphisms taking sections of $E$ to sections of $E \otimes \Omega^1$, i.e., if $\xi$ is a section of $E$ (over $V \subseteq U_i \cap U_j$), then the morphism on the left-hand side is $$\xi \mapsto\psi_i^{-1} ((d + A_i) (\psi_{i} \xi)),$$ which does indeed make sense as $\psi_i \xi$ is a section of $V \times \mathbb{C}^r$. By a similar calculation as the one in Huybrechts after equation (4.4), one may show that this condition is equivalent to the compatibility condition that you write. I suppose Huybrechts has written it the way he did since he had not stated the compatibility condition that you write earlier in the text.

If one starts with your equation for compatibility, then one may compose this with $\psi_j^{-1}$ from the left and $\psi_j$ from the right, which yields $$ \psi_j^{-1} \psi_{ij}^{-1} d\psi_{ij} \psi_j = \psi_j^{-1} A_j \psi_j - \psi_i^{-1} A_i \psi_i $$ and then conclude the proof as in the last paragraph of Proposition 4.2.19. This would make the proof a bit shorter, but relies on the fact that one knows this equation for compatibility.