On residual categories for Grassmannians

TL;DR

This paper explores the structure of residual categories in derived categories of Grassmannians, conjecturing their generation by orthogonal exceptional collections and proving this for prime dimensions.

Contribution

It introduces a conjecture about residual categories in Grassmannians and proves it for prime dimensions, advancing understanding of their categorical decompositions.

Findings

Conjecture that residual categories are generated by orthogonal exceptional collections.

Proof of the conjecture for Grassmannians with prime dimension k=p.

Verification of completeness for Fonarev's collection when p=3.

Abstract

We define and discuss some general properties of residual categories of Lefschetz decompositions in triangulated categories. In the case of the derived category of coherent sheaves on the Grassmannian $\text{G}(k,n)$ we conjecture that the residual category associated with Fonarev's Lefschetz exceptional collection is generated by a completely orthogonal exceptional collection. We prove this conjecture for $k = p$, a prime number, modulo completeness of Fonarev's collection (and for $p = 3$ we check this completeness).