Polarizations on a triangulated category

In a recent collaboration, Hiroki Matsui and the author introduced a new proof of the reconstruction theorem of Bondal-Orlov and Ballard, using Matsui's construction of a ringed space associated to a triangulated category. This paper first shows that these ideas can be applied to reconstructions of more general varieties from their perfect derived categories. For further applications of these ideas, we introduce the framework of a polarized triangulated category, a pair $(\mathcal T,τ)$ consisting of a triangulated category $\mathcal T$ and an autoequivalence $τ$ (called a polarization), to which we can associate a ringed space called the pt-spectrum. As concrete applications, we observe that several reconstruction results of Favero naturally fit within this framework, leading to both generalizations and new proofs of these results. Furthermore, we explore broader implications of polarizations and pt-spectra in tensor triangular geometry, noncommutative projective geometry, birational geometry and homological mirror symmetry.