A new proof of the Bondal-Orlov reconstruction using Matsui spectra

TL;DR

This paper introduces a new proof of the Bondal-Orlov reconstruction theorem using Matsui spectra, demonstrating the spectra's role in reconstructing derived categories of algebraic varieties and their Fourier-Mukai partners.

Contribution

It provides a novel proof of the Bondal-Orlov and Ballard reconstruction theorems via Matsui spectra and shows the Fourier-Mukai locus as an open subspace, enabling reconstruction of Fourier-Mukai partners.

Findings

$ ext{Spec}_{oxtimes_X^ ext{L}} ext{Perf} X$ is an open subspace of $ ext{Spec}_ riangle ext{Perf} X$

Fourier-Mukai partners can be reconstructed from the Fourier-Mukai locus within the spectra

The spectra framework unifies various reconstruction theorems in algebraic geometry

Abstract

In 2005, Balmer defined the ringed space $\operatorname{Spec}_\otimes \mathcal{T}$ for a given tensor triangulated category, while in 2023, the second author introduced the ringed space $\operatorname{Spec}_\vartriangle \mathcal{T}$ for a given triangulated category. In the algebro-geometric context, these spectra provided several reconstruction theorems using derived categories. In this paper, we prove that $\operatorname{Spec}_{\otimes_X^\mathbb{L}} \operatorname{Perf} X$ is an open ringed subspace of $\operatorname{Spec}_\vartriangle \operatorname{Perf} X$ for a quasi-projective variety $X$. As an application, we provide a new proof of the Bondal-Orlov and Ballard reconstruction theorems in terms of these spectra. Recently, the first author introduced the Fourier-Mukai locus $\operatorname{Spec}^\mathsf{FM} \operatorname{Perf} X$ for a smooth projective variety $X$, which is constructed by gluing Fourier-Mukai partners of $X$ inside $\operatorname{Spec}_\vartriangle \operatorname{Perf} X$. As another application of our main theorem, we demonstrate that $\operatorname{Spec}^\mathsf{FM} \operatorname{Perf} X$ can be viewed as an open ringed subspace of $\operatorname{Spec}_\vartriangle \operatorname{Perf} X$. As a result, we show that all the Fourier-Mukai partners of an abelian variety $X$ can be reconstructed by topologically identifying the Fourier-Mukai locus within $\operatorname{Spec}_\vartriangle \operatorname{Perf} X$.