Proper Fourier-Mukai partners of abelian varieties and points outside the Fourier-Mukai loci in Matsui spectra

Abstract: We prove that any proper Fourier-Mukai partner of an abelian variety is again an abelian variety, by analyzing the Matsui spectrum of the derived category. This result was previously obtained by Huybrechts and Nieper-Wisskirchen in the case of smooth projective varieties. Our proof, however, extends the result to proper schemes using entirely different techniques. More generally, we show that any scheme of finite type that is derived equivalent to an open subscheme of an abelian variety is itself an open subscheme of an abelian variety. We also study the structure of the Matsui spectrum outside the Fourier-Mukai locus. For certain proper schemes, we show that the set of points lying outside the Fourier-Mukai locus in the Matsui spectrum has cardinality at least equal to that of the base field. This suggests the existence of additional geometric structures, such as moduli spaces, beyond the derived-equivalent part. As an application, we provide new counterexamples to conjectures of Ito, which predicted that the Serre invariant locus coincides with the Fourier-Mukai locus. While counterexamples involving K3 surfaces of Picard number one were previously given by Hirano-Ouchi, our examples arising from simple abelian varieties of dimension greater than two are the first of their kind.