Torsion pairs and 3-fold flops

TL;DR

This paper classifies intermediate t-structures on the local derived category of 3-fold flopping contractions, linking them to torsion classes, birational models, and algebraic structures, with applications to spherical modules and derived categories.

Contribution

It provides a complete classification of t-structures on the local derived category of 3-fold flops and resolutions, connecting geometric and algebraic perspectives.

Findings

Classified t-structures on 3-fold flopping contractions.

Described torsion classes for associated modification algebras.

Connected classifications to minimal model program and module categories.

Abstract

This paper classifies t-structures on the local derived category of a 3-fold flopping contraction, that are intermediate with respect to the heart of perverse coherent sheaves. Equivalently, this describes the complete lattice of torsion classes for the associated modification algebra. The intermediate hearts are (1) categories of coherent sheaves on birational models and tilts thereof in skyscrapers, (2) algebraic t-structures described in the homological minimal model programme, or (3) combinations of the above over appropriate open covers. An analogous classification is also proved for minimal (and partial) resolutions of Kleinian singularities, thus providing a description of all torsion pairs in the module categories of (contracted) affine preprojective algebras. The results have immediate applications to the classification of spherical modules and (semi)bricks, and are first steps towards describing all t-structures and spherical objects in derived categories of surfaces and 3-folds.