Bridgeland stability has attracted a lot of attention since Bridgeland presented his mathematical formulation of Douglas’s \(\Pi\)-stability in string theory and the theory of D-branes [T. Bridgeland, Ann. Math. (2) 166, No. 2, 317–345 (2007; Zbl 1137.18008)]. While the construction of Bridgeland stability conditions on surfaces was achieved early on, [T. Bridgeland, Duke Math. J. 141, No. 2, 241–291 (2008; Zbl 1138.14022)] and [D. Arcara and A. Bertram, J. Eur. Math. Soc. (JEMS) 15, No. 1, 1–38 (2013; Zbl 1259.14014)], the construction of Bridgeland stability conditions on threefolds has remained a largely open problem with important ramifications for mathematical physics, the original motivation for the subject in the first place. A conjectural construction was proposed in [A. Bayer et al., J. Algebr. Geom. 23, No. 1, 117–163 (2014; Zbl 1306.14005)], where the authors reduced the construction down to proving a weak Gieseker-Bogomolov inequality for so-called tilt-stable objects in an abelian category \(\mathcal B^{\omega,\beta}\) obtained from coherent sheaves \(\mathrm{Coh}(X)\) by tilting at a given slope with respect to classical slope stability. Those authors also proposed a stronger Gieseker-Bogomolov inequality. Until recently, however, stability conditions had only been constructed on three dimensional projective space [E. Macrì, Algebra Number Theory 8, No. 1, 173–190 (2014; Zbl 1308.14016)] and on smooth quadric threefolds [B. Schmidt, Bull. Lond. Math. Soc. 46, No. 5, 915–923 (2014; Zbl 1307.14024)], where in both cases the construction follows from proving the strong Gieseker-Bogomolov inequality for tilt-stable objects.
The paper under review, along with its prequel [A. Maciocia and D. Piyaratne, Algebr. Geom. 2, No. 3, 270–297 (2015; Zbl 1322.14040)], adds to this list by constructing Bridgeland stability conditions on a principally polarized abelian threefold \(X\) of Picard rank one. In the prequel the authors establish the weak Gieseker-Bogomolov inequality, while in the current paper they go on to establish the strong version of Bayer, Macrì, and Toda’s Gieseker-Bogomolov inequality for tilt-stable objects. Both papers use Fourier-Mukai transforms (FMT’s), i.e. autoequivalences of the derived category of sheaves \(\mathrm{D}^b(X)\), to establish equivalences between certain abelian subcategories of \(\mathrm{D}^b(X)\) which are obtained by further tilting \(\mathcal B^{\omega,\beta}\) with respect to tilt-stability. While the authors previously considered only the classical FMT using the Poincaré bundle as its kernel, by making use of more general non-trivial FMT’s, the authors are able to reduce the numerics of the strong form of the Gieseker-Bogomolov inequality to a similar but easily shown inequality for the corresponding objects in a different abelian category.