Bridgeland stability conditions on threefolds. II: An application to Fujita's conjecture (Q2922502)

Stability conditions on triangulated categories were introduced by \textit{T. Bridgeland} [Ann. Math. (2) 166, No. 2, 317--345 (2007; Zbl 1137.18008)], motivated by \textit{M. R. Douglas}' work on \(\Pi\)-stability [in: Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20--28, 2002. Vol. III: Invited lectures. Beijing: Higher Education Press; Singapore: World Scientific. 395--408 (2002; Zbl 1008.81074)]. Their existences on the derived category \(D^{b}(X)\) of a projective variety with \(\dim_{\mathbb{C}}(X)\geq3\) are very interesting and challenging problems.NEWLINENEWLINEMotivated by Bridgeland's work on \(K3\) surfaces [\textit{T. Bridgeland}, Duke Math. J. 141, No. 2, 241--291 (2008; Zbl 1138.14022)], in [\textit{A. Bayer} et al., J. Algebr. Geom. 23, No. 1, 117--163 (2014; Zbl 1306.14005)], the authors constructed a new t-structure \(\mathcal{A}_{\omega,B}\subseteq D^{b}(X)\) as a double tilt of the standard t-structure \(Coh(X)\subseteq D^{b}(X)\) for any smooth projective threefold \(X\). Along with the central charge \(Z_{\omega,B}\) constructed by string theorists before, they conjectured that \((Z_{\omega,B},\mathcal{A}_{\omega,B})\) is a stability condition on \(D^{b}(X)\) and proved that the conjecture is equivalent to a generalized Bogomolov-Gieseker type inequality for the third Chern character of certain stable complexes.NEWLINENEWLINEIn the paper under review, the authors show us how to use the inequality to prove Fujita's conjecture, namely \(K_{X}\otimes L^{m\geq6}\) is very ample for any ample line bundle \(L\rightarrow X\) which satisfies certain conditions involving BG type inequality mentioned above. The idea of the proof is close to Reider's original approach to the conjecture for algebraic surfaces [\textit{I. Reider}, Ann. Math. (2) 127, No. 2, 309--316 (1988; Zbl 0663.14010)].