A smooth projective counterexample to Bondal-Polishchuk's conjecture

Abstract

For a particular smooth projective (weak Fano) threefold $X$ we show that the braid group action on the set of full exceptional collections in $D^b(X)$ is not transitive. This provides a counterexample to a conjecture of Bondal and Polishchuk from 1993. The conjecture was first disproved by Chang, Haiden, and Schroll, who constructed a family of partially wrapped Fukaya categories for which the transitivity fails. However, no counterexample of the form $D^b(X)$ for $X$ a smooth projective variety was previously known. In addition, we show that the space of Bridgeland stability conditions $Stab(X)$ on $X$ has infinitely many connected components. This is the first known example of a smooth projective variety whose space of Bridgeland stability conditions is disconnected. Finally, we apply a similar method to establish that $Stab(Y)$ for a symmetric quintic threefold $Y$ is also disconnected.

The preprint has been significantly expanded to include some refinements concerning the orbits of the braid group action for the same counterexample as considered in the first version as well as new results on Bridgeland stability conditions