Simple complexes on a flopped curve

TL;DR

This paper classifies certain complexes on flopped curves related to du Val singularities, revealing their structure, stability conditions, and the connectedness of the stability manifold, advancing understanding of derived categories in algebraic geometry.

Contribution

It provides a comprehensive classification of complexes with no negative self-extensions on flopped curves, linking them to algebraic and geometric structures and proving the connectedness of the stability manifold.

Findings

Classified complexes with no negative self-extensions on flopped curves.

Established the connectedness of the Bridgeland stability manifold.

Showed all basic tilting complexes are related by shifts and mutations.

Abstract

Studying crepant blow-ups of (compound) du Val singularities, we classify complexes of coherent sheaves which admit no negative self-extensions -- such a complex, up to flops and mutation equivalences, must either be (1) a module over a derived-equivalent algebra, or (2) a two-term extension of a coherent sheaf by skyscraper sheaves, or (3) a direct sum of shifts of skyscrapers. This translates into classifications of bricks, spherical objects, stability conditions, and algebraic t-structures in the local derived category; the lists populated by the homological minimal programme turn out exhaustive. We deduce that the Bridgeland stability manifold is connected, and that all basic tilting complexes on the variety (equivalently on the g-tame algebras derived-equivalent to it) are related by shifts and iterated mutation.