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Realizing pt-spectra as tt-spectra on Perf(P^1)

Determine whether every polarization on Perf(P^1) yields a pt-spectrum that coincides with the Balmer spectrum of some tensor triangulated structure on Perf(P^1).

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Background

The authors observe that not all tt-spectra arise as fixed loci of autoequivalences, giving the Kronecker quiver tt-structure as a counterexample.

They then raise the converse problem: whether every pt-spectrum coming from a polarization of Perf(P1) can be realized as the tt-spectrum of a suitable tt-structure on Perf(P1).

References

We do not know if the converse is also false. That is, we do not know if given a polarization on \perf X, there exists a tt-structure on \perf \bb P1 whose tt-spectrum agrees with the pt-spectrum.

— Polarizations on a triangulated category (2502.15621 - Ito, 21 Feb 2025) in Section 2.2, Example “preoplarization in perf P^1”