Dice Question Streamline Icon: https://streamlinehq.com

Classification of tt-structures on Perf(P^1) up to tt-geometry

Classify all tensor triangulated structures on the triangulated category Perf(P^1) up to tt-geometry by determining every tt-structure whose Balmer spectrum, viewed as a subspace of the Matsui spectrum, identifies the tt-geometry of Perf(P^1).

Information Square Streamline Icon: https://streamlinehq.com

Background

The paper completely classifies polarizations on Perf(P1) by describing the corresponding pt-spectra, but emphasizes that the analogous classification problem for tt-structures is not settled.

The authors point to existing partial results and examples, noting that understanding tt-structures up to tt-geometry remains an outstanding problem in this setting.

References

On the other hand, to author's knowledge, it is still not known if we have a complete classification of tt-structures on \perf \bb P1 up to tt-geometry.

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