Timeline for Topological information in sheaf cohomology?

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Jun 10, 2020 at 13:27	comment	added	Donu Arapura		Even if $X$ is simply connected, a Zarsiki open $U\subset X$ need not be. Suppose additionally that $D=X-U$ is a divisor with normal crossings. If one starts with a unitary representation of $U$ forms the associated vector bundle, and extends to bundle $E$ $X$ a la Deligne. Then this gives something nontrivial example along the lines of your question. I have no idea, if this is the sort of thing you want however.
Jun 10, 2020 at 10:34	comment	added	Alex Gavrilov		@Piotr Achinger, If a connection is integrable and the base is simply connected, doesn't it mean that the bundle is trivial?
Jun 10, 2020 at 10:00	comment	added	Alex Gavrilov		I edited it so hopefully now it makes some sense.
Jun 10, 2020 at 9:59	history	edited	Alex Gavrilov	CC BY-SA 4.0	There was a stupid mistake in the original question.
Jun 10, 2020 at 9:53	history	undeleted	Alex Gavrilov
Jun 10, 2020 at 7:42	history	deleted	Alex Gavrilov		via Vote
Jun 10, 2020 at 7:15	comment	added	Piotr Achinger		P.S. Maybe the following is of interest to you: if $E$ carries a holomorphic integrable connection $\nabla$, then the horizontal sections of $E$ form a local system of $\mathbb{C}$-vector spaces $\mathcal{E}$ on $X$ and there is an isomorphism $H^_{\rm dR}(X, E) \simeq H^(X, \mathcal{E})$, where the first group is the de Rham cohomology of $E$ i.e. hypercohomology of the complex $(\Omega^\bullet_X \otimes E, \nabla)$.
Jun 10, 2020 at 7:12	comment	added	Piotr Achinger		Where did you get that isomorphism from? For $X$ an elliptic curve, $H^1(X, \mathcal{O}_X)$ is one-dimensional while $H^1(X, \mathbb{C})$ is two-dimensional. I do not understand the second paragraph.
Jun 10, 2020 at 6:55	history	asked	Alex Gavrilov	CC BY-SA 4.0