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

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 $\operatorname{Spec}_\vartriangle \operatorname{Perf} X$.