Minimal Lefschetz decompositions of the derived categories for Grassmannians

Abstract

We construct two Lefschetz decompositions of the derived category of coherent sheaves on the Grassmannian of k-dimensional subspaces in a vector space of dimension n. Both of them admit a Lefschetz basis consisting of equivariant vector bundles. We prove fullness of the first decomposition and conjecture it for the second one. In the case when n and k are coprime these decompositions coincide and are minimal. In general, we conjecture minimality of the second decomposition.Exceptional collections can be considered as a kind of basis for the triangulated category: in this situation every object admits a unique filtration with the i-th quotient isomorphic to a direct sum of shifts of the i-th object in the collection.The simplest example of a variety with a full exceptional collection is a projective space. A. Beilinson showed in [Beȋ78] that the collection (O P n , O P n (1), . . . O P n (n)) is full and exceptional in D b (P n ).One can slightly generalize the notion of an exceptional collection and consider subcategories instead of single objects. For a full triangulated subcategory A of a triangulated category T the right orthogonal (resp. left orthogonal) to A in T is the full triangulated subcategory A ⊥ (resp. ⊥ A) consisting of all the objects T ∈ T such that Hom T (A, T ) = 0 (resp. Hom T (T, A) = 0) for all A ∈ A.

On residual categories for Grassmannians

“…In , an exceptional collection in the derived category generalizing was suggested. To describe it, we introduce some combinatorics of Young diagrams.…”

“…We define an action of the group on the set by letting the generator act as See [, Section 3.1] for a useful geometric description of this action.…”

“…A Young diagram is called if it lies above the diagonal of the rectangle that goes from the lower‐left to the upper‐right corner, that is, we have for all . By [, Lemma 3.2], every orbit of the cyclic group action on contains an upper triangular representative. Moreover, if , such representative is unique.…”

“…In this paper, we discuss Conjecture for Grassmannians . In this case, a nice Lefschetz collection in is known after Fonarev's work . In general, this collection is not known but expected to be full (Conjecture ).…”

On the Kuznetsov-Polishchuk conjecture

“…A class of exact complexes of equivariant vector bundles on Grassmannians was constructed in [3]. These complexes, called staircase, are a natural generalization of the twist by O(1) of the Koszul complex on P(V )…”