Fourier-Mukai transforms and stability conditions on abelian threefolds
Piyaratne, Hathurusinghege Dulip Bandara
Construction of Bridgeland stability conditions on a given Calabi-Yau threefold is an important problem and this thesis realizes the rst known examples of such stability conditions. More precisely, we construct a dense family of stability conditions on the derived category of coherent sheaves on a principally polarized abelian threefold X with Picard rank one. In particular, we show that the conjectural construction proposed by Bayer, Macr and Toda gives rise to Bridgeland stability conditions on X. First we reduce the requirement of the Bogomolov-Gieseker type inequalities to a smaller class of tilt stable objects which are essentially minimal objects of the conjectural stability condition hearts for a given smooth projective threefold. Then we use the Fourier-Mukai theory to prove the strong Bogomolov-Gieseker type inequalities for these minimal objects of X. This is done by showing any Fourier-Mukai transform of X gives an equivalence of abelian categories which are double tilts of coherent sheaves.