On the Iitaka volumes of log canonical surfaces and threefolds

Given positive integers \(d\geq\kappa\), and a subset \(\Gamma\subset [0,1]\), let \(\mathrm{Ivol}_{\mathrm{lc}}^{\Gamma}(d,\kappa)\) denote the set of Iitaka volumes of \(d\)-dimensional projective log canonical pairs \((X, B)\) such that the Iitaka--Kodaira dimension \(\kappa(K_X+B)=\kappa\) and the coefficients of \(B\) come from \(\Gamma\). In this paper, we show that, if \(\Gamma\) satisfies the descending chain condition, then so does \(\mathrm{Ivol}_\mathrm{lc}^\Gamma(d,\kappa)\) for \(d\leq 3\). In case \(d\leq 3\) and \(\kappa=1\), \(\Gamma\) and \(\mathrm{Ivol}_\mathrm{lc}^\Gamma(d,\kappa)\) are shown to share more topological properties, such as closedness in \(\mathbb{R}\) and local finiteness of accumulation complexity. In higher dimensions, we show that the set of Iitaka volumes for \(d\)-dimensional klt pairs with Iitaka dimension \(\geq d-2\) satisfies the DCC, partially confirming a conjecture of Zhan Li. We give a more detailed description of the sets of Iitaka volumes for the following classes of projective log canonical surfaces: (1) smooth properly elliptic surfaces, (2) projective log canonical surfaces with coefficients from \(\{0\}\) or \(\{0,1\}\). In particular, the minima as well as the minimal accumulation points are found in these cases.