Semiorthogonal decompositions and components of derived categories of orthogonal Grassmannian fibrations

TL;DR

This paper explores semiorthogonal decompositions of derived categories related to orthogonal Grassmannian fibrations, extending Kuznetsov's work on quadric fibrations and analyzing residual categories in specific cases.

Contribution

It generalizes Kuznetsov's semiorthogonal decomposition to orthogonal Grassmannian fibrations with minimal conditions, introducing new embeddings and residual category computations.

Findings

The category D^b(S, Cl_0) embeds fully faithfully into D^b(OGr(k, Q)).

For k=2, a semiorthogonal decomposition of D^b(OGr(2, Q)) is constructed.

Residual categories are computed in the smooth case and conjectured for pencils of quadrics.

Abstract

Kuznetsov showed that for a flat quadric fibration $\mathcal{Q}$ over a smooth base $S$, $\mathrm{D}^b(\mathcal{Q})$ admits a semiorthogonal decomposition where one of the components is the derived category of the sheaf of even parts of a Clifford algebra $\mathrm{D}^b(S,\mathcal{C}l_0)$. \par As progress towards a generalization, we show that for a quadric fibration with a fairly minor condition on the rank of the quadric fibers, the category $\mathrm{D}^b(S,\mathcal{C}l_0)$ embeds fully faithfully into the derived category of the relative orthogonal Grassmannian $\mathrm{D}^b(\mathrm{OGr}(k,\mathcal{Q}))$. When $k = 2$, we use this to produce a semiorthogonal decomposition of $\mathrm{D}^b(\mathrm{OGr}(2,\mathcal{Q}))$ up to a residual category; we compute this residual category in the smooth case and produce a conjecture for in the case of a pencil of quadrics with smooth base locus.