UC Berkeley

The derived category Perf X of perfect complexes on a variety X is an algebraic invariant that reflects many geometric properties of X. Balmer proved that Perf X, together with the monoidal structure ⊗OX​L​ given by the derived tensor product of sheaves, recovers X. In the course of the proof, he constructed a locally ringed space Spec⊗​Perf X, called the tensor triangular spectrum, and showed that it is isomorphic to X.In this thesis, in order to study the geometry of varieties with equivalent perfect derived categories, called Fourier-Mukai partners, we consider the triangular spectrum SpecΔ​Perf X of Perf X, introduced by Matsui and defined using only the triangulated category structure of Perf X. Each Fourier-Mukai partner, realized as a tensor triangular spectrum, defines an open subspace of the common ambient triangular spectrum, and we prove that the interactions among these subspaces reflect the expected birational geometric behavior.Motivated by the observation that fixed loci of actions of autoequivalences on the triangular spectrum capture geometric information effectively, we introduce the framework of a polarized triangulated category, namely a pair (T,τ) consisting of a triangulated category T and an autoequivalence τ, called a polarization. To such a pair we associate a ringed space called the polarized triangular spectrum. As concrete applications, we generalize several reconstruction results of Bondal-Orlov, Ballard, and Favero, exhibit a universal way to compare birationally equivalent Fourier-Mukai partners with their common canonical model via Iitaka fibrations, and construct mirror partners in homological mirror symmetry of log Calabi-Yau surfaces as polarized triangular spectra. We further consider ∞-categorical enhancements of polarized triangulated categories, leading to precise comparisons with the relative triangular spectrum under the monodromy equivalence.This thesis is based on three papers of which the second is joint work with Hiroki Matsui. We reorganize the statements and proofs of results in using the simplified constructions provided in and provide several new results based on. We also draw several results from collaborations with Noah Olander and John Nolan, respectively.