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

Abstract

In 2005, Balmer defined the ringed space ${Spec}_{\otimes }T$ for a given tensor triangulated category, while in 2023, the second author introduced the ringed space ${Spec}_{▵}T$ 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 ${Spec}_{{\otimes }_X^L}PerfX$ is an open ringed subspace of ${Spec}_{▵}PerfX$ 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 ${Spec}^{\mathrm{FM}}PerfX$ for a smooth projective variety $X$, which is constructed by gluing Fourier–Mukai partners of $X$ inside ${Spec}_{▵}PerfX$. As another application of our main theorem, we demonstrate that ${Spec}^{\mathrm{FM}}PerfX$ can be viewed as an open ringed subspace of ${Spec}_{▵}PerfX$. 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 ${Spec}_{▵}PerfX$.