A generalized Bogomolov-Gieseker inequality for the three-dimensional projective space
Stability conditions on triangulated categories were introduced by \textit{T. Bridgeland} [Ann. Math. (2) 166, No. 2, 317--345 (2007; Zbl 1137.18008)], which was motivated by Douglas' work on \(\Pi\)-stability [\textit{M. R. Douglas}, 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/distributor. 395--408 (2002; Zbl 1008.81074)]. Their existence on the derived category \(D^{b}(X)\) of a projective variety with \(\dim_{\mathbb{C}}(X)\geq3\) is a very interesting and challenging problem. Motivated by Bridgeland's work on \(K3\) surfaces [\textit{T. Bridgeland}, Duke Math. J. 141, No. 2, 241--291 (2008; Zbl 1138.14022)], \textit{A. Bayer} et al., J. Algebr. Geom. 23, No. 1, 117--163 (2014; Zbl 1306.14005)] 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. In this paper, the author proved the inequality for \(\mathbb{P}^{3}\) which gives the first example when the generalized BG inequality is proved in full generality. As an application, the author gave a weaker version of Castelnuovo's theorem which bounds the genus of space curves by their degrees.
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