Derived category of toric fibrations

TL;DR

This paper provides a structure theorem for the derived category of toric fiber bundles, assuming both base and fiber have full strongly exceptional collections of line bundles, advancing understanding of algebraic invariants in algebraic geometry.

Contribution

It establishes a structure theorem for the derived category of toric fibrations with specific exceptional collections on base and fiber.

Findings

Derived category structure theorem for toric fibrations

Conditions for existence of full strongly exceptional collections

Advancement in understanding algebraic invariants of varieties

Abstract

The derived category of bounded complexes of coherent sheaves is one of the most important algebraic invariants of a smooth projective variety. An important approach to understand derived categories is to construct full strongly exceptional sequences. The problem of characterizing smooth projective varieties which have a full strongly exceptional collection and investigate whether there is one consisting of line bundles is a classical and important question in Algebraic Geometry. Not all smooth projective varieties have a full strongly exceptional collection of coherent sheaves. In this paper we give a structure theorem for the derived category of a toric fiber bundle X over Z with fiber F provided that F and Z have both a full strongly exceptional collection of line bundles.