A remark on generators of $\mathtt D(\hbox {X})$ and flags

Abstract

We give a simple proof of the following fact. Let $\hbox {X}$ be an n-dimensional, smooth, projective variety with ample or anti-ample canonical bundle, over an algebraically closed base field. Let $\hbox {Y}_0 \subset \hbox {Y}_{1} \subset \cdots \subset \hbox {Y}_n = \hbox {X}$ be a complete flag of closed smooth subvarieties, where $\hbox {Y}_{j+1} {\setminus } \hbox {Y}_{j}$ is affine. Then $\hbox {G} = \bigoplus _{j=0}^n \mathcal O_{\mathrm{Y}_{j}}$ is a generator of the (bounded coherent) derived category $\mathtt D(\hbox {X})$. Moreover, from the endomorphism dg-algebra ${{\mathrm{REnd}}}_{\mathrm{X}}(\hbox {G})$ one can recover not only $\hbox {X}$ but also the flag $\hbox {Y}_0 \subset \hbox {Y}_{1} \subset \cdots \subset \hbox {Y}_n$.