Does there exist a smooth nonrational 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).
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3$\begingroup$ Perhaps this question should count as an "open problem". $\endgroup$– Leo AlonsoCommented Aug 31, 2018 at 14:28

$\begingroup$ @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. $\endgroup$– user74900Commented Aug 31, 2018 at 14:36

$\begingroup$ 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). $\endgroup$– Leo AlonsoCommented Aug 31, 2018 at 14:39

2$\begingroup$ 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. $\endgroup$– Ennio Mori coneCommented Aug 31, 2018 at 22:41
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