Obstructing subvarieties and non-standard autoequivalences

Determine whether every obstructing invariant subvariety of a toric variety gives rise to a non-standard autoequivalence of its perfect derived category that is associated with the subvariety.

Background

When the canonical bundle is tensor-generating, all derived autoequivalences are standard. The paper shows positive instances in which obstructing curves or divisors yield non-standard autoequivalences, including spherical twists and EZ-spherical twists. However, the authors state that the general implication from the existence of an obstructing invariant subvariety to a corresponding non-standard autoequivalence is unresolved and may fail in general.

References

If there is an obstructing (invariant) subvariety $Z$ of a toric variety $X$, then is there a non-standard autoequivalence of $\perf(X)$ "associated" to $Z$?

— Tensor-generating line bundles on toric varieties  (2610.22279 - Ito et al., 15 Oct 2026) in Question q:non-standard autoeq, Section 7.2 (Obstructing subvarieties and non-standard autoequivalences)

However, it would be interesting to see exactly when an obstructing subvariety to which the canonical bundle restricts as a nef or anti-nef line bundle gives rise to a twist-like autoequivalence. Moreover, it would also be interesting to see when obstructing subvarieties correspond to a suitable wall in VGIT whose associated window shift is a non-standard autoequivalence in the setting of . Another natural question is whether the data of the obstructing subvariety under the toric mirror symmetry provides a way to construct a non-standard autoequivalence on the other side. We leave these directions as future research.

— Tensor-generating line bundles on toric varieties  (2610.22279 - Ito et al., 15 Oct 2026) in Final paragraph of Section 7.2, immediately before the acknowledgments/end of paper