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Since Poonen's construction of a variety $X$ defined over a number field $k$for which $X(k)$ is empty and the \'etale Brauer--Manin set$X(\mathbf{A}_k)^\text{Br,et}$ is not, several other examples of smooth,projective varieties have been found for which the \'etale Brauer--Maninobstruction does not explain the failure of the Hasse principle. All knownexamples are constructed using "Poonen's trick", i.e. they have the distinctivefeature of being fibrations over a higher genus curve; in particular, theirAlbanese variety is non-trivial.  In this paper, we construct examples for which the Albanese variety istrivial. The new geometric ingredient in our construction is the appearance ofBeauville surfaces. Assuming the $abc$ conjecture and using geometric work ofCampana on orbifolds, we also prove the existence of an example which is simplyconnected.

Insufficiency of the \'etale Brauer-Manin obstruction: towards a simply connected example