Minimal Lefschetz decompositions of the derived categories for Grassmannians (Q2866966)

The author uses Kapranov's exceptional collection on the derived category of bounded coherent sheaves on the Grassmannian \(\mathrm{Gr}(k,n)\) of \(k\) dimensional subspaces in an \(n\) dimensional vector space, to construct two different Lefschetz decompositions of the respective derived category.NEWLINENEWLINELefschetz decompositions are important in the context of homological projective duality, as introduced by Kuznetsov, and are special semiorthogonal decompositions that behave well (they induce a Lefschetz type property) with respect to taking linear sections (and, in particular, hyperplane sections). Such decompositions were already known for \(\mathrm{Gr}(2,n)\), the Grassmannian of 2 dimensional subpaces in an \(n\) dimensional vector space, and in other sporadic examples.NEWLINENEWLINEThe author proves that one of the collections is full and, for \(k\) and \(n\) coprime, that it is also minimal. For \(k\) and \(n\) not coprime, the other collection is smaller and the conjecture is that it is both minimal and full. If \(k=2\) then both collections coincide to the one constructed in an earlier work by Kuznetsov.NEWLINENEWLINEWhile proving the statements in the paper, the author constructs a new class of exact Schur-Weyl type complexes of vector bundles on the Grassmannian which are interesting by themselves.