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Polarizations on a triangulated category

Daigo Ito

TL;DR

This work develops the polarized triangulated category (pt-category) framework, defining the pt-spectrum $\mathrm{Spec}^\tau \mathcal T$ as fixed points of a polarization $\tau$ on a triangulated category. By leveraging the Serre functor, it shows how reconstruction theorems for varieties (originally due to Bondal–Orlov and Ballard) extend to broader settings, including nonprojective proper cases, via Serre-invariant loci and Matsui spectra. The authors connect pt-spectra to Balmer tt-spectra, introduce tt-ample and multi-polarizations, and illustrate how fixed-point subspaces recover strong geometric data (e.g., Iitaka fibrations, canonical models) and Fourier–Mukai partners. They further explore natural geometric sources of polarizations, apply the framework to noncommutative Proj and the birational landscape, and propose a homological mirror symmetry perspective in which algebraic mirrors arise as pt-spectra of Fukaya categories. Overall, the paper provides a unifying categorical approach to reconstruction, birational geometry, and mirror symmetry through pt-spectra, with concrete results for projective and certain nonprojective settings, and compelling links to noncommutative geometry and HMS.

Abstract

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.

Polarizations on a triangulated category

TL;DR

This work develops the polarized triangulated category (pt-category) framework, defining the pt-spectrum as fixed points of a polarization on a triangulated category. By leveraging the Serre functor, it shows how reconstruction theorems for varieties (originally due to Bondal–Orlov and Ballard) extend to broader settings, including nonprojective proper cases, via Serre-invariant loci and Matsui spectra. The authors connect pt-spectra to Balmer tt-spectra, introduce tt-ample and multi-polarizations, and illustrate how fixed-point subspaces recover strong geometric data (e.g., Iitaka fibrations, canonical models) and Fourier–Mukai partners. They further explore natural geometric sources of polarizations, apply the framework to noncommutative Proj and the birational landscape, and propose a homological mirror symmetry perspective in which algebraic mirrors arise as pt-spectra of Fukaya categories. Overall, the paper provides a unifying categorical approach to reconstruction, birational geometry, and mirror symmetry through pt-spectra, with concrete results for projective and certain nonprojective settings, and compelling links to noncommutative geometry and HMS.

Abstract

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 consisting of a triangulated category and an autoequivalence (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.

Paper Structure

This paper contains 11 sections, 25 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Geometry in Matsui spectrum
  3. Fixed points of autoequivalences and polarizations
  4. Generalization with tt-structures and $\otimes$-ample objects
  5. Generalization in the absence of tt-structures and polarizations
  6. Variants of polarizations
  7. Polarizations arising in nature
  8. Reconstruction of varieties
  9. Geometric realization of noncommutative projective schemes and relations to Iitaka fibrations
  10. Constructions of mirror partners as spectra of Fukaya categories
  11. Functoriality of spectra and topological lattices

Key Result

Theorem 1.1

Let $X$ be a Gorenstein proper variety of dimension $n$ over a field $k$ and suppose $\mathop{\mathrm{Spec}}\nolimits^\mathsf{Ser} \mathop{\mathrm{\mathsf{Perf}}}\nolimits X$ is a separated scheme over $k$ whose $n$-dimensional connected components (with reduced structure) are all isomorphic. Then,

Theorems & Definitions (79)

  • Theorem 1.1: Theorem \ref{['thm:Bondal--Orlov']}
  • Definition 1.2
  • Example 1.3: Example \ref{['example: tensor ample']}
  • Definition 1.4
  • Theorem 1.6
  • Lemma 2.4.1: Balmer_2002*https://arxiv.org/pdf/math/0111049.pdf
  • Theorem 2.6: Balmer_2005*https://arxiv.org/pdf/math/0409360.pdf, Matsui_2021*https://arxiv.org/pdf/2102.11317.pdf, Balmer_2002*https://www.math.ucla.edu/ balmer/Pubfile/Reconstr.pdf, matsukawa2025spectrum*https://arxiv.org/abs/2505.02724v1
  • proof
  • Theorem 2.7: Matsui_2021*https://arxiv.org/pdf/2102.11317.pdf, matsui2023triangular*https://arxiv.org/pdf/2301.03168.pdf, https://arxiv.org/pdf/2301.03168.pdf
  • Definition 2.8
  • ...and 69 more