K-Theoretic Version of Fourier-Mukai Transforms Between Crepant Resolutions of Finite Quotient Singularities

• Autor: Serna Giraldo, Ivan Junnior
• Formato: tesis doctoral
• Fecha de Publicación: 2023
• Institución: Superintendencia Nacional de Educación Superior Universitaria
• Repositorio: Registro Nacional de Trabajos conducentes a Grados y Títulos - RENATI
• Lenguaje: inglés
• OAI Identifier: oai:repositorio.sunedu.gob.pe:20.500.14366/386
• Enlace del recurso: https://hdl.handle.net/20.500.14366/386
• Nivel de acceso: acceso abierto
• Materia: Categorías (Matemáticas); Funtores; Transformada de Fourier-Mukai; Resoluciones crepantes; Correspondencia de McKay; Teoría geométrica de los invariantes; Geometría; https://purl.org/pe-repo/ocde/ford#1.01.01

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.