A smooth projective counterexample to Bondal-Polishchuk's conjecture

Anya Nordskova

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arXiv · 2607.25391

A smooth projective counterexample to Bondal-Polishchuk's conjecture

Anya Nordskova

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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.

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Anya Nordskova. 2026-07-28. A smooth projective counterexample to Bondal-Polishchuk's conjecture. https://arxiv.org/abs/2607.25391

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