K-Theoretic Version of Fourier-Mukai Transforms Between Crepant Resolutions of Finite Quotient Singularities - Serna Giraldo, Ivan Junnior (2023)
We study crepant resolutions of singularities C³/G, where G is a finite abelian subgroup of SL(3,C). Using derived category methods, Bridgeland, King and Reid proved that the Hilbert scheme of G-clusters (G-Hilb)(C³) is a crepant resolution. Following Craw-Ishii, we study the moduli spaces Mθ of θ-stable G-constellations, in particular, (G-Hilb)(C³) is a moduli space of this type for a suitable parameters in the GIT-parameter space, while all crepant resolutions are of the form Mθ for some θ. The GIT-parameter space is divided into chambers, and for parameters in adjacent chambers, theMθ spaces are Fourier-Mukai partners. Following Craw-Ishii we study how the Fourier-Mukai transform between partners can induce a change in the tautological line bundles. As an application, we study the case of C³/Z₄. We outline the toric description of the singularity and its crepant resolution. Using Chern classes we determine the cohomological Fourier-Mukai transform between Fourier- Mukai partners, that are moduli spaces for adjacent chambers. In general, for the singularities C³/G, we also determine the cohomological Fourier- Mukai transform as a linear transformation between the cohomology rings.