FOURIER-MUKAI TRANSFORMS FOR SINGULAR PROJECTIVE VARIETIES

This Ph. D thesis deals with various aspects of the theory of derived categories of sheaves on projective varieties. The first result is a generalization of a representability theorem by Lunts and Orlov for fully faithful functors where the source derived category is the derived category of sheaves over the double point scheme. The second result is a description of the space of stability conditions for the same derived category. The final result is a descend criterion for equivariant sheaves to the quotient variety with respect to an action by a finite group.