Bondal–Polishchuk transitivity conjecture for smooth projective varieties

Bondal and Polishchuk posed the conjecture in 1993: for a smooth projective variety $X$, the braid-group action on full exceptional collections in $D^b(X)$ should be transitive. The new construction answers this negatively.

Known results

• Rudakov: transitivity for $[1m\mathbb{P}^{2}$ and $[1m\mathbb{P}^{1}\times\mathbb{P}^{1}$.
• Kuleshov and Orlov: transitivity for all del Pezzo surfaces.
• Akira, Okawa, and Uehara: transitivity for the Hirzebruch surface $[1m\Sigma_2$.
• Counterexamples were known in partially wrapped Fukaya categories and in gentle-algebra settings, but not previously for $D^b(X)$ with $X$ smooth projective.

July 2026 counterexample

A preprint constructs $X=\operatorname{Bl}_{C}\mathbb{P}^{3}$, with $C$ a smooth rational sextic, and a six-term full exceptional collection $[1m\mathbb{E}$. Spherical twists produce $[1mT_{\mathcal{S}_1}(\mathbb{E})$, and an involution exchanging the relevant spherical objects proves these collections lie in distinct braid-group orbits. This is presented as the first counterexample for a smooth projective variety.

Current status (as of July 2026): The conjecture is resolved negatively in general by the arXiv counterexample; special cases remain valid, but no universal transitivity statement survives.