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

Abstract In 2005, Balmer defined the ringed space for a given tensor triangulated category, while in 2023, the second author introduced the ringed space 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 is an open ringed subspace of for a quasi‐projective variety . 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 for a smooth projective variety , which is constructed by gluing Fourier–Mukai partners of inside . As another application of our main theorem, we demonstrate that can be viewed as an open ringed subspace of . As a result, we show that all the Fourier–Mukai partners of an abelian variety can be reconstructed by topologically identifying the Fourier–Mukai locus within .