Bounded derived categories of very simple manifolds

TL;DR

This paper explicitly constructs a non-representable cohomological functor of finite type for the bounded derived category of certain complex manifolds, extending previous foundational results in algebraic geometry.

Contribution

It generalizes a key result by Bondal and Van den Bergh to a broader class of complex manifolds with no proper closed subvarieties.

Findings

Explicit example of a non-representable cohomological functor

Extension of Bondal and Van den Bergh's result to higher-dimensional manifolds

Provides categorical characterization of these manifolds

Abstract

An unrepresentable cohomological functor of finite type of the bounded derived category of coherent sheaves of a compact complex manifold of dimension greater than one with no proper closed subvariety is given explicitly in categorical terms. This is a partial generalization of an impressive result due to Bondal and Van den Bergh.