# A non-rational variety with a full exceptional collection?

Does there exist a smooth non-rational projective variety whose bounded derived category of coherent sheaves admits a full exceptional collection? I could not find any examples in the literature (for instance projective spaces and intersections of quadrics, which admit full exceptional collections, are rational).

• Perhaps this question should count as an "open problem". – Leo Alonso Aug 31 '18 at 14:28
• @LeoAlonso is it? I did not know it. If you could post a link to a paper where it is stated to be an open problem, I would accept it as an answer. – user74900 Aug 31 '18 at 14:36
• Unfortunately, I'm not aware of such a reference. In fact I can't figure out any example. On the other hand, you have a full description of the derived category of Abelian varieties, though not through an exceptional collection (if I'm not mistaken). – Leo Alonso Aug 31 '18 at 14:39
• Bondal and Polishchuk's "Homological properties of associative algebras: the method of helices." shows that a variety with a full exceptional collection of length $\dim X+1$ must be a fano variety. – wnx Aug 31 '18 at 22:41