Seminari del Dipartimento

 

Geometria

Stability conditions loathe phantomic summands

Domenico Fiorenza


10-06-2021 - 14:30
modalitĂ  telematica (telematic form)

 

Showing that a given exceptional collection in a triangulated category is full is generally a nontrivial problem. In the numerically finite case, an obvious necessary condition is that the exceptional collection is numerically full, i.e., it spans the numerical Grothendieck group of the triangulated category. This condition is however not sufficient, due to the possible presence of phantom subcategories, i.e., nontrivial subcategories that are invisible to numerical detection. A Bridgeland stability condition on the right orthogonal of the exceptional collection (when it exists), including a “positivity condition” for nonzero objects, notably forbids the presence of these phantoms thus implying fullness. As an explicit application I'll show how one can use this to rediscover Kapranov's full exceptional collection on a smooth quadric threefold. The same technique can be used to recover a few classical results on the commutativity of certain Kuznetsov components, e.g., Bondal-Orlov's proof of the commutativity of Ku(Y_4) and the Addington-Thomas theorem on cubic fourfolds. Joint work with Barbara Bolognese (arXiv:2103.15205)

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