Derived Category of Fibrations

TL;DR

This paper provides a structure theorem for the derived category of a fibration of smooth complex projective varieties, assuming both fiber and base have full strongly exceptional collections of line bundles.

Contribution

It establishes a new framework for understanding derived categories of fibrations with specific exceptional collections on fiber and base.

Findings

Derived category structure theorem for fibrations

Conditions involving exceptional collections on fiber and base

Enhanced understanding of derived categories in algebraic geometry

Abstract

Let X ->Y be a Zariski locally trivial fibration of smooth complex projective varieties, with fiber F. We give a structure theorem for the derived category of X provided both F and Z have a full strongly exceptional collection of line bundles.