Higher Triangulated Categories and Fourier-Mukai Transforms on Abelian Surfaces and Threefolds

Abstract

In this thesis, we develop a new concept of a higher triangulated structure that centers around a refined notion of distinguished objects and maps. Our approach builds on the earlier research of Balmer (1) and Neeman (44). We provide a new set of axioms and prove a crucial gluing theorem for higher triangles and we also prove that the homo topy category K(A) and D(A) satisfies the axioms of higher triangulated categories. Furthermore, we apply our techniques from higher triangulated categories to simplify several theorems. For instance, we introduce a new torsion theory for abelian surfaces X that arises on Mukai fibrations and cofiltrations of sheaves E in Coh(X). Specifi cally, we create a torsion theory via Fourier-Mukai transforms on X. We are motivated by Bridgeland’s (7) study of the case when X is an elliptic curve, which is already well-known. We set out to generalize this work to threefolds and show that there is a torsion theory with a heart Aα,Φ depending on the FM transform Φ, which we show is the same as the second tilt Aα,β, the heart of a Bridgeland stability condition on X. The motivation for introducing a new torsion theory for X is to show that we can link slope stability with weak index theorems.