Fourier-Mukai transforms and stability conditions on abelian threefolds
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Date
27/11/2014Author
Piyaratne, Hathurusinghege Dulip Bandara
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Abstract
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.