A derived category analogue of the Nakai--Moishezon criterion

TL;DR

This paper characterizes line bundles on proper varieties whose tensor powers generate the derived category, extending classical positivity notions and providing new examples of reconstructible varieties.

Contribution

It offers a Nakai--Moishezon-like criterion for derived category generation by line bundles, answering a question from 2010 and generalizing to all Noetherian schemes.

Findings

Characterization of line bundles generating derived categories.

Extension of the criterion to all Noetherian schemes.

New examples of varieties reconstructible from their derived categories.

Abstract

We give a complete characterization of the line bundles on a proper variety whose tensor powers generate the derived category, answering a 2010 question of Chris Brav. The condition is analogous to the Nakai--Moishezon criterion and can be stated purely in terms of classical notions of positivity of line bundles. There is also a generalization which works for all Noetherian schemes. We use our criterion to prove basic properties of such line bundles and provide non-trivial examples of them. As an application, we give new examples of varieties which can be reconstructed from their derived categories in the sense of the Bondal--Orlov Reconstruction Theorem.