Disconnected Stab(X) and infinitely many braid group orbits

Abstract

We show that the space of Bridgeland stability conditions Stab(Y) on a very general quintic threefold Y has infinitely many connected components. Using similar methods, we also prove that for a particular smooth projective (weak Fano) threefold X the braid group action on the set of full exceptional collections in Dscr^b(X) has infinitely many orbits, in particular, it 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 Dscr^b(X) for X a smooth projective variety was previously known. In addition, we show that Stab(X) also has infinitely many connected components. These are the first known examples of a smooth projective varieties whose spaces of Bridgeland stability conditions are disconnected.

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BibTeX RIS

BibTeX

@misc{nordskova2026disconnected,
  title = {{Disconnected Stab(X) and infinitely many braid group orbits}},
  author = {Anya Nordskova},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.25391},
  eprint = {2607.25391},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.25391},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Disconnected Stab(X) and infinitely many braid group orbits
AU  - Anya Nordskova
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2607.25391
UR  - https://arxiv.org/abs/2607.25391
AB  - We show that the space of Bridgeland stability conditions Stab(Y) on a very general quintic threefold Y has infinitely many connected components. Using similar methods, we also prove that for a particular smooth projective (weak Fano) threefold X the braid group action on the set of full exceptional collections in Dscr^b(X) has infinitely many orbits, in particular, it 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 Dscr^b(X) for X a smooth projective variety was previously known. In addition, we show that Stab(X) also has infinitely many connected components. These are the first known examples of a smooth projective varieties whose spaces of Bridgeland stability conditions are disconnected.
ER  -

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