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.
Parth Shimpi (Fri,) studied this question.