The question is pretty much as in the title: are there $\mathbb{G}_\mathrm{a}$-torsors $\pi:P\to X$ for some Grothendieck topology finer than the Zariski one, over a complex algebraic variety $X$ , that are not Zariski-locally trivial? Here $\mathbb{G}_\mathrm{a}$ denotes the additive group scheme over $\mathbb{C}$, a.k.a $\mathbb{A}^1$ with additive law.
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10$\begingroup$ In most topologies one could be interested in (e.g. etale or fppf), we have $H^1(X, \mathbf{G}_a) = H^1(X, \mathcal{O}_X) = H^1(X_{\rm Zar}, \mathcal{O}_X)$, which suggests that the answer should be no. $\endgroup$– Piotr AchingerJan 15, 2018 at 21:40
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